Multiresolution Hilbert Approach to Multidimensional Gauss-Markov Processes

Abstract

The study of the multidimensional stochastic processes involves complex computations in intricate functional spaces. In particular, the diffusion processes, which include the practically important Gauss-Markov processes, are ordinarily defined through the theory of stochastic integration. Here, inspired by the Lévy-Ciesielski construction of the Wiener process, we propose an alternative representation of multidimensional Gauss-Markov processes as expansions on well-chosen Schauder bases, with independent random coefficients of normal law with zero mean and unit variance. We thereby offer a natural multiresolution description of the Gauss-Markov processes as limits of finite-dimensional partial sums of the expansion, that are strongly almost-surely convergent. Moreover, such finite-dimensional random processes constitute an optimal approximation of the process, in the sense of minimizing the associated Dirichlet energy under interpolating constraints. This approach allows for a simpler treatment of problems in many applied and theoretical fields, and we provide a short overview of applications we are currently developing.

1. Introduction

Intuitively, multidimensional continuous stochastic processes are easily conceived as solutions to randomly perturbed differential equations of the form𝐗̇‌_𝑡=𝑓𝐗_𝑡,𝑡,𝝃_𝑡,(1.1) where the perturbative term 𝝃_𝑡 implicitly defines a probability space and 𝑓 satisfies some ad hoc regularity conditions. If the existence of such processes is well established for a wide range of equations through the standard Itô integration theory (see, e.g., [1]), studying their properties proves surprisingly challenging, even for the simplest multidimensional processes. Indeed, the high dimensionality of the ambient space and the nowhere differentiability of the sample paths conspire to heighten the intricacy of the sample paths spaces. In this regard, such spaces have been chiefly studied for multidimensional diffusion processes [2], and more recently, the development of rough paths theory has attracted renewed interest in the field (see [3–7]). However, aside from these remarkable theoretical works, little emphasis is put on the sample paths since most of the available results only make sense in distribution. This is particularly true in the Itô integration theory, where the sample path is completely neglected for the Itô map being defined up to null sets of paths.

To overcome the difficulty of working in complex multidimensional spaces, it would be advantageous to have a discrete construction of a continuous stochastic process as finite-dimensional distributions. Since we put emphasis on the description of the sample paths space, at stake is to write a process 𝐗 as an almost surely pathwise convergent series of random functions𝐗_𝑡=lim_𝑁→∞𝐗^𝑁_𝑡with𝐗^𝑁_𝑡=^𝑁_𝑛=0𝐟_𝑛(𝑡)⋅𝚵_𝑛,(1.2) where 𝐟_𝑛 is a deterministic function and Ξ_𝑛 is a given random variable.

The Lévy-Ciesielski construction of the 𝑑-dimensional Brownian motion 𝐖 (also referred to as Wiener process) provides us with an example of discrete representation for a continuous stochastic process. Noticing the simple form of the probability density of a Brownian bridge, it is based on completing sample paths by interpolation according to the conditional probabilities of the Wiener process [8]. More specifically, the coefficients Ξ_𝑛 are Gaussian independent and the elements 𝐟_𝑛, called the Schauder elements and denoted by 𝐬_𝑛, are obtained by time-dependent integration of the Haar basis elements: 𝐬_0,0(𝑡)=𝑡𝐈_𝑑 and 𝐬_𝑛,𝑘(𝑡)=𝑠_𝑛,𝑘(𝑡)𝐈_𝑑, with for all 𝑛>0𝑠_𝑛,𝑘(𝑡)=⎧⎪⎨⎪⎩2^(𝑛−1)/2𝑡−𝑙_𝑛,𝑘,𝑘2^−𝑛+1≤𝑡≤(2𝑘+1)2^−𝑛,2^(𝑛−1)/2𝑟_𝑛,𝑘−𝑡,(2𝑘+1)2^−𝑛≤𝑡≤(𝑘+1)2^−𝑛+1,0,otherwise.(1.3) This latter point is of relevance since, for being a Hilbert system, the introduction of the Haar basis greatly simplifies the demonstration of the existence of the Wiener process [9]. From another perspective, fundamental among discrete representations is the Karhunen-Loève decomposition giving a representation of stochastic processes by expanding it on a basis of orthogonal functions [10, 11]. The definition of the basis elements 𝑓_𝑛 depends only on the second-order statistics of the considered process and the coefficients 𝜉_𝑛 are pairwise uncorrelated random variables. Incidentally, such a decomposition is especially suited to study the Gaussian processes because the coefficients of the representation are Gaussian and independent. For these reasons, the Karhunen-Loève decomposition is of primary importance in exploratory data analysis, leading to methods referred to as “principal component analysis,” “Hotelling transform" [12] or “proper orthogonal decomposition” [13] according to the field of application. In particular, it was directly applied to the study of the stationary Gaussian Markov processes in the theory of random noise in radio receivers [14].

It is also important for our purpose to realize that the Schauder elements 𝐬_𝑛 have compact supports that exhibit a nested structure: this fact entails that the finite sums 𝐖_𝑁 are processes that interpolate the limit process 𝐖 on the endpoints of the supports, that is, on the dyadic points 𝑘2^−𝑁, 0≤𝑘≤2^𝑁. One of the specific goal of our construction is to maintain such a property in the construction of all multidimensional the Gauss-Markov processes 𝐗 (i.e., processes that are both Gaussian and satisfy the Markov property) of the form:𝐗_𝑡=𝐠(𝑡)^𝑡₀𝐟(𝑠)⋅𝑑𝐖_𝑠(1.4) (covering all 1-dimensional Gauss-Markov processes thanks to Doob’s representation of Gauss-Markov processes), being successively approximated by finite-dimensional processes 𝑋^𝑁 that interpolates 𝐗 at ever finer resolution. In that respect, it is only in that sense that we refer to our framework as a multiresolution approach as opposed to the wavelet multiresolution theory [15]. Other multiresolution approaches have been developed for certain Gaussian processes, most notably for the fractional Brownian motion [16].

In view of this, we propose a construction of the multidimensional Gaussian Markov processes using a multiresolution Schauder basis of functions. As for the Lévy-Ciesielski construction, and in contrast with Karhunen-Loève decomposition, our basis is not made of orthogonal functions but the elements are of nested compact support and the random coefficients Ξ_𝑛 are always independent and Gaussian (for convenience with law 𝒩(𝟎,𝐈_𝑑), i.e., with zero mean and unitary variance). We first develop a heuristic approach for the construction of stochastic processes reminiscent of the midpoint displacement technique [8, 9], before rigorously deriving the multiresolution basis that we will be using the paper. This set of functions is then studied as a multiresolution Schauder basis of functions: in particular, we derive explicitly from the multiresolution basis an Haar-like Hilbert basis, which is the underlying structure explaining the dual relationship between basis elements and coefficients. Based on these results, we study the construction application and its inverse, the coefficient applications, that relate coefficients on the Schauder basis to sample paths. We follow up by proving the almost-sure and strong convergence of the process having independent standard normal coefficients on the Schauder basis to a Gauss-Markov process. We also show that our decomposition is optimal in some sense that is strongly evocative of spline interpolation theory [17]: the construction yields successive interpolations of the process at the interval endpoints that minimize the Dirichlet energy induced by the differential operator associated with the Gauss-Markov process [18, 19]. We also provide a series of examples for which the proposed Schauder framework yields bases of functions that have simple closed form formulae: in addition to the simple one-dimensional Markov processes, we explicit our framework for two classes of multidimensional processes, the Gauss-Markov rotations and the iteratively integrated Wiener processes (see, e.g., [20–22]).

The ideas underlying this work can be directly traced back to the original work of Lévy. Here, we intend to develop a self-contained Schauder dual framework to further the description of multidimensional Gauss-Markov processes, and, in doing so, we extend some well-known results of interpolation theory in signal processing [23–25]. To our knowledge, such an approach is yet to be proposed. By restraining our attention to the Gauss-Markov processes, we obviously do not assume generality. However, we hope our construction proves of interest for a number of points, which we tentatively list in the following. First, the almost-sure pathwise convergence of our construction together with the interpolation property of the finite sums allows to reformulate results of the stochastic integration in term of the geometry of finite-dimensional sample paths. In this regard, we found it appropriate to illustrate how in our framework, the Girsanov theorem for the Gauss-Markov processes appears as a direct consequence of the finite-dimensional change of variable formula. Second, the characterization of our Schauder elements as the minimizer of a Dirichlet form paves the way to the construction of infinite-dimensional Gauss-Markov processes, that is, processes whose sample points themselves are infinite-dimensional [26, 27]. Third, our construction shows that approximating a Gaussian process by a sequence of interpolating processes relies entirely on the existence of a regular triangularization of the covariance operator, suggesting to further investigate this property for non-Markov Gaussian processes [28]. Finally, there is a number of practical applications where applying the Schauder basis framework clearly provides an advantage compared to standard stochastic calculus methods, among which first-hitting times of stochastic processes, pricing of multidimensional path-dependant options [29–32], regularization technique for support vector machine learning [33], and more theoretical work on uncovering the differential geometry structure of the space of the Gauss-Markov stochastic processes [34]. We conclude our exposition by developing in more detail some of these direct implications which will be the subjects of forthcoming papers.

2. Heuristic Approach to the Construction

In order to provide a discrete multiresolution description of the Gauss-Markov processes, we first establish basic results about the law of the Gauss-Markov bridges in the multidimensional setting. We then use them to infer the candidate expressions for our desired bases of functions, while imposing its elements to be compactly supported on nested sequence segments. Throughout this paper, we are working in a complete probability space (Ω,ℱ,𝐏).

2.1. Multidimensional Gauss-Markov Processes

After recalling the definition of the multidimensional Gauss-Markov processes in terms of stochastic integral, we use the well-known conditioning formula for the Gaussian vectors to characterize the law of the Gauss-Markov bridge processes.

2.1.1. Notations and Definitions

Let (𝐖_𝑡,ℱ_𝑡,𝑡∈[0,1]) be an 𝑚-dimensional Wiener process, consider the continuous functions 𝜶∶[0,1]→ℝ^𝑑×𝑑√Γ∶[0,1]→ℝ^𝑑×𝑚, and define the positive bounded continuous function Γ=√Γ⋅√Γ^𝑇∶[0,1]→ℝ^𝑑×𝑑. The 𝑑-dimensional Ornstein-Uhlenbeck process associated with these parameters is solution of the equation𝑑𝐗_𝑡=𝜶(𝑡)⋅𝐗_𝑡𝑑𝑡+√𝚪(𝑡)⋅𝑑𝐖_𝑡,(2.1) and with initial condition 𝐗_𝑡0 in 𝑡₀, it reads𝐗_𝑡=𝐅𝑡₀,𝑡⋅𝐗_𝑡0+𝐅𝑡₀,𝑡⋅^𝑡_𝑡0𝐅𝑠,𝑡₀⋅√𝚪(𝑠)⋅𝑑𝐖_𝑠,(2.2) where 𝐅(𝑡₀,𝑡) is the flow of the equation, namely, the solution in ℝ^𝑑×𝑑 of the linear equation:𝜕𝐅𝑡₀,𝑡𝜕𝑡=𝜶(𝑡)𝐅(𝑡),𝐅𝑡₀,𝑡₀=𝐈_𝑑.(2.3) Note that the flow 𝐅(𝑡₀,𝑡) enjoys the chain rule property:𝐅𝑡₀,𝑡=𝐅𝑡₁,𝑡⋅𝐅𝑡₀,𝑡₁.(2.4) For all 𝑡,𝑠 such that 𝑡₀<𝑠,𝑡, the vectors 𝐗_𝑡 and 𝐗_𝑠 admit the covariance𝐂_𝑡0(𝑠,𝑡)=𝐅𝑡₀,𝑡⎛⎜⎝^𝑡∧𝑠_𝑡0𝐅𝑤,𝑡₀𝚪(𝑤)𝐅𝑤,𝑡₀^𝑇𝑑𝑤⎞⎟⎠𝐅𝑡₀,𝑠^𝑇=𝐅𝑡₀,𝑡𝐡_𝑡0(𝑠,𝑡)𝐅𝑡₀,𝑠^𝑇,(2.5) where we further defined 𝐡_𝑢(𝑠,𝑡) the function𝐡_𝑢(𝑠,𝑡)=^𝑡_𝑠𝐅(𝑤,𝑢)⋅𝚪(𝑤)⋅𝐅(𝑤,𝑢)^𝑇𝑑𝑤,(2.6) which will be of particular interest in the sequel. Note that because of the chain rule property of the flow, we have 𝐡_𝑣(𝑠,𝑡)=𝐅(𝑣,𝑢)𝐡_𝑢(𝑠,𝑡)𝐅(𝑣,𝑢)^𝑇.(2.7) We suppose that the process 𝐗 is never degenerated, that is, for all 𝑡₀<𝑢<𝑣, all the components of the vector 𝐗_𝑣 taking into account 𝐗_𝑢 are nondeterministic random variables, which is equivalent to saying that the covariance matrix of 𝐗_𝑣 taking into account 𝐗_𝑢, denoted by 𝐂_𝑢(𝑣,𝑣) is symmetric positive definite for any 𝑢≠𝑣. Therefore, assuming the initial condition 𝐗₀=𝟎, the multidimensional centered process 𝐗 has a representation (similar to Doob’s representation for one-dimensional processes, see [35]) of form𝐗_𝑡=𝐠(𝑡)^𝑡₀𝐟(𝑠)⋅𝑑𝐖_𝑠,(2.8) with 𝐠(𝑡)=𝐅(0,𝑡) and 𝐟(𝑡)=𝐅(𝑡,0)⋅√Γ(𝑡).

Note that the processes considered in this paper are defined on the time interval [0,1]. However, because of the time-rescaling property of these processes, considering the processes on this time interval is equivalent to considering the process on any other bounded interval without loss of generality.

2.1.2. Conditional Law and Gauss-Markov Bridges

As stated in the introduction, we aim at defining a multiresolution description of Gauss-Markov processes. Such a description can be seen as a multiresolution interpolation of the process that is getting increasingly finer. This principle, in addition to the Markov property, prescribes to characterize the law of the corresponding Gauss-Markov bridge, that is, the Gauss-Markov process under consideration, conditioned on its initial and final values. The bridge process of the Gauss process is still a Gauss process and, for a Markov process, its law can be computed as follows.

Proposition 2.1. Let 𝑡_𝑥≤𝑡_𝑧 two times in the interval [0,1]. For any 𝑡∈[𝑡_𝑥,𝑡_𝑧], the random variable 𝐗_𝑡 conditioned on 𝐗_𝑡𝑥=𝐱 and 𝐗_𝑡𝑧=𝐳 is a Gaussian variable with covariance matrix 𝚺(𝑡) and mean vector 𝝁(𝑡) given by 𝚺𝑡;𝑡_𝑥,𝑡_𝑧=𝐡_𝑡𝑡_𝑥,𝑡𝐡_𝑡(𝑡_𝑥,𝑡_𝑧)⁻¹𝐡_𝑡𝑡,𝑡_𝑧,𝝁(𝑡)=𝝁^𝑙𝑡;𝑡_𝑥,𝑡_𝑧⋅𝐱+𝝁^𝑟𝑡;𝑡_𝑥,𝑡_𝑧⋅𝐳,(2.9) where the continuous matrix functions 𝝁^𝑙(⋅;𝑡_𝑥,𝑡_𝑧) and 𝝁^𝑟(⋅;𝑡_𝑥,𝑡_𝑧) of ℝ^𝑑×𝑑 are given by 𝝁^𝑙𝑡;𝑡_𝑥,𝑡_𝑧=𝐅𝑡_𝑥,𝑡𝐡_𝑡𝑥𝑡,𝑡_𝑧𝐡_𝑡𝑥𝑡_𝑥,𝑡_𝑧⁻¹,𝝁^𝑟𝑡;𝑡_𝑥,𝑡_𝑧=𝐅𝑡_𝑧,𝑡𝐡_𝑡𝑧𝑡_𝑥,𝑡𝐡_𝑡𝑧(𝑡_𝑥,𝑡_𝑧)⁻¹.(2.10)

Note that the functions 𝜇^𝑙 and 𝜇^𝑟 have the property that 𝝁^𝑙(𝑡_𝑥;𝑡_𝑥,𝑡_𝑧)=𝝁^𝑟(𝑡_𝑧;𝑡_𝑥,𝑡_𝑧)=𝐈_𝑑 and 𝝁^𝑙(𝑡_𝑧;𝑡_𝑥,𝑡_𝑧)=𝝁^𝑟(𝑡_𝑥;𝑡_𝑥,𝑡_𝑧)=𝟎 ensuring that the process is indeed equal to 𝐱 at time 𝑡_𝑥 and 𝐳 at time 𝑡_𝑧.

Proof. Let 𝑡_𝑥,𝑡_𝑧 be two times of the interval [0,1] such that 𝑡_𝑥<𝑡_𝑧, and let 𝑡∈[𝑡_𝑥,𝑡_𝑧]. We consider the Gaussian random variable 𝝃=(𝐗_𝑡,𝐗_𝑡𝑧) conditioned on the fact that 𝐗_𝑡𝑥=𝐱. Its mean can be easily computed from expression (2.2) and reads 𝐦_𝑡,𝐦_𝑡𝑧=𝐅𝑡_𝑥,𝑡𝐱,𝐅𝑡_𝑥,𝑡_𝑧𝐱=𝐠(𝑡)𝐠⁻¹𝑡_𝑥𝐱,𝐠𝑡_𝑧𝐠⁻¹𝑡_𝑥𝐱,(2.11) and its covariance matrix, from (2.5), reads ⎡⎢⎢⎣𝐂_𝑡,𝑡𝐂_{𝑡,𝑡𝑧}𝐂_{𝑡𝑧,𝑡}𝐂_{𝑡𝑧,𝑡𝑧}⎤⎥⎥⎦=⎡⎢⎢⎣𝐅𝑡_𝑥,𝑡𝐡_𝑡𝑥𝑡_𝑥,𝑡𝐅𝑡_𝑥,𝑡^𝑇𝐅𝑡_𝑥,𝑡𝐡_𝑡𝑥𝑡_𝑥,𝑡𝐅𝑡_𝑥,𝑡_𝑧^𝑇𝐅𝑡_𝑥,𝑡_𝑧𝐡_𝑡𝑥𝑡_𝑥,𝑡𝐅𝑡_𝑥,𝑡^𝑇𝐅𝑡_𝑥,𝑡_𝑧𝐡_𝑡𝑥𝑡_𝑥,𝑡_𝑧𝐅𝑡_𝑥,𝑡_𝑧^𝑇⎤⎥⎥⎦=⎡⎢⎢⎣𝐡_𝑡𝑡_𝑥,𝑡𝐡_𝑡𝑡_𝑥,𝑡𝐅𝑡,𝑡_𝑧^𝑇𝐅𝑡,𝑡_𝑧𝐡_𝑡𝑡_𝑥,𝑡𝐅𝑡,𝑡_𝑧𝐡_𝑡𝑡_𝑥,𝑡_𝑧𝐅𝑡,𝑡_𝑧^𝑇⎤⎥⎥⎦.(2.12) From there, we apply the conditioning formula for the Gaussian vectors (see, e.g., [36]) to infer the law of 𝐗_𝑡 conditioned on 𝐗_𝑡𝑥=𝐱 and 𝐗_𝑡𝑧=𝐳, that is the law 𝒩(𝝁(𝑡),𝚺(𝑡;𝑡_𝑥,𝑡_𝑧)) of 𝐁_𝑡 where 𝐁 denotes the bridge process obtained by pinning 𝐗 in 𝑡_𝑥 and 𝑡_𝑧. The covariance matrix is given by 𝚺𝑡;𝑡_𝑥,𝑡_𝑧=𝐂_𝑦,𝑦−𝐂_𝑦,𝑧𝐂⁻¹_𝑧,𝑧𝐂_𝑧,𝑦=𝐡_𝑡𝑡_𝑥,𝑡−𝐡_𝑡𝑡_𝑥,𝑡𝐡_𝑡𝑡_𝑥,𝑡_𝑧⁻¹𝐡_𝑡𝑡_𝑥,𝑡=𝐡_𝑡𝑡_𝑥,𝑡𝐡_𝑡𝑡_𝑥,𝑡_𝑧⁻¹𝐡_𝑡𝑡,𝑡_𝑧,(2.13) and the mean reads 𝝁(𝑡)=𝐦_𝑦+𝐂_𝑦,𝑧𝐂⁻¹_𝑧,𝑧𝐳−𝐦_𝑧=𝐅𝑡_𝑥,𝑡𝐈_𝑑−𝐡_𝑡𝑥𝑡_𝑥,𝑡𝐡_𝑡𝑥(𝑡_𝑥,𝑡_𝑧)⁻¹𝐱+𝐅𝑡_𝑧,𝑡𝐡_𝑡𝑧𝑡_𝑥,𝑡𝐡_𝑡𝑧(𝑡_𝑥,𝑡_𝑧)⁻¹𝐳,=𝐅(𝑡_𝑥,𝑡)𝐡_𝑡𝑥(𝑡,𝑡_𝑧)𝐡_𝑡𝑥(𝑡_𝑥,𝑡_𝑧)⁻¹_{𝝁𝑙(𝑡;𝑡𝑥,𝑡𝑧)}⋅𝐱+𝐅(𝑡_𝑧,𝑡)𝐡_𝑡𝑧(𝑡_𝑥,𝑡)𝐡_𝑡𝑧(𝑡_𝑥,𝑡_𝑧)⁻¹_{𝝁𝑟(𝑡;𝑡𝑥,𝑡𝑧)}⋅𝐳,(2.14) where we have used the fact that 𝐡_𝑡𝑥(𝑡_𝑥,𝑡_𝑧)=𝐡_𝑡𝑧(𝑡_𝑥,𝑡)+𝐡_𝑡𝑥(𝑡,𝑡_𝑧). The regularity of the thus-defined functions 𝝁_𝑥 and 𝝁_𝑧 directly stems from the regularity of the flow operator 𝐅. Moreover, since for any 0≤𝑡,𝑢≤1, we observe that 𝐅(𝑡,𝑡)=𝐈_𝑑 and ℎ_𝑢(𝑡,𝑡)=𝟎; we clearly have 𝝁_𝑥(𝑡_𝑥)=𝝁_𝑦(𝑡)=𝐈_𝑑 and 𝝁_𝑥(𝑡)=𝝁_𝑦(𝑡_𝑥)=0.

Remark 2.2. Note that these laws can also be computed using the expression of the density of the processes but involve more intricate calculations. An alternative approach also provides a representation of Gauss-Markov bridges with the use of integral and anticipative representation [37]. These approaches allow to compute the probability distribution of the Gauss-Markov bridge as a process (i.e., allows to compute the covariances), but since this will be of no use in the sequel, we do not provide the expressions.

2.2. The Multiresolution Description of Gauss-Markov Processes

Recognizing the Gauss property and the Markov property as the two crucial elements for a stochastic process to be expanded to Lévy-Cesielski, our approach first proposes to exhibit bases of deterministic functions that would play the role of the Schauder bases for the Wiener process. In this regard, we first expect such functions to be continuous and compactly supported on increasingly finer supports (i.e., subintervals of the definition interval [0,1]) in a similar nested binary tree structure. Then, as in the Lévy-Ciesielski construction, we envision that, at each resolution (i.e., on each support), the partially constructed process (up to the resolution of the support) has the same conditional expectation as the Gauss-Markov process when conditioned on the endpoints of the supports. The partial sums obtained with independent Gaussian coefficients of law 𝒩(0,1) will thus approximate the targeted Gauss-Markov process in a multiresolution fashion, in the sense that, at every resolution, considering these two processes on the interval endpoints yields finite-dimensional Gaussian vectors of the same law.

2.2.1. Nested Structure of the Sequence of Supports

Here, we define the nested sequence of segments that constitute the supports of the multiresolution basis. We construct such a sequence by recursively partitioning the interval [0,1].

More precisely, starting from 𝑆_1,0=[𝑙_1,0,𝑟_1,0] with 𝑙_1,0=0 and 𝑟_1,0=1, we iteratively apply the following operation. Suppose that, at the 𝑛th step, the interval [0,1] is decomposed into 2^𝑛−1 intervals 𝑆_𝑛,𝑘=[𝑙_𝑛,𝑘,𝑟_𝑛,𝑘], called supports, such that 𝑙_𝑛,𝑘+1=𝑟_𝑛,𝑘 for 0≤𝑘<2^𝑛−1. Each of these intervals is then subdivided into two child intervals, a left-child 𝑆_𝑛+1,2𝑘 and a right-child 𝑆_{𝑛+1,2𝑘+1}, and the subdivision point 𝑟_𝑛+1,2𝑘=𝑙_{𝑛+1,2𝑘+1} is denoted by 𝑚_𝑛,𝑘. Therefore, we have defined three sequences of real 𝑙_𝑛,𝑘, 𝑚_𝑛,𝑘, and 𝑟_𝑛,𝑘 for 𝑛>0 and 0≤𝑘<2^𝑛−1 satisfying 𝑙_0,0=0≤𝑙_𝑛,𝑘<𝑚_𝑛,𝑘<𝑟_𝑛,𝑘≤𝑟_0,0=1 and𝑙_𝑛+1,2𝑘=𝑙_𝑛,𝑘,𝑚_𝑛,𝑘=𝑟_𝑛+1,2𝑘=𝑙_{𝑛+1,2𝑘+1},𝑟_{𝑛+1,2𝑘+1}=𝑟_𝑛,𝑘(2.15) with the convention 𝑙_0,0=0 and 𝑟_0,0=1 and 𝑆_0,0=[0,1]. The resulting sequence of supports {𝑆_𝑛,𝑘;𝑛≤0,0≤𝑘<2^𝑛−1} clearly has a binary tree structure.

For the sake of compactness of notations, we define ℐ the set of indicesℐ=_𝑛<𝑁ℐ_𝑛withℐ_𝑁=(𝑛,𝑘)∈ℕ²∣0<𝑛≤𝑁,0≤𝑘<2^𝑛−1,(2.16) and for 𝑁>0, we define 𝐷_𝑁={𝑚_𝑛,𝑘,(𝑛,𝑘)∈ℐ_𝑁−1}∪{0,1}, the set of endpoints of the intervals 𝑆_𝑁,𝑘. We additionally require that there exists 𝜌∈(0,1) such that for all (𝑛,𝑘)∈ℐmax(𝑟_𝑛,𝑘−𝑚_𝑛,𝑘,𝑚_𝑛,𝑘−𝑙_𝑛,𝑘)<𝜌(𝑟_𝑛,𝑘−𝑙_𝑛,𝑘) which in particular implies thatlim_𝑛→∞sup_𝑘𝑟_𝑛,𝑘−𝑙_𝑛,𝑘=0(2.17) and ensures that the set of endpoints ∪_𝑁∈N𝐷_𝑁 is everywhere dense in [0,1]. The simplest case of such partitions is the dyadic partition of [0,1], where the endpoints for (𝑛,𝑘)∈ℐ read𝑙_𝑛,𝑘=𝑘2^−𝑛+1,𝑚_𝑛,𝑘=(2𝑘+1)2^−𝑛,𝑟_𝑛,𝑘=(𝑘+1)2^−𝑛+1,(2.18) in which case the endpoints are simply the dyadic points ∪_𝑁𝐷_𝑁={𝑘2^−𝑁|0≤𝑘≤2^𝑁}. Figure 1 represents the global architecture of the nested sequence of intervals.

Figure 1 

A sequence of nested intervals.

The nested structure of the supports, together with the constraint of continuity of the bases elements, implies that only a finite number of coefficients are needed to construct the exact value of the process at a given endpoint, thus providing us with an exact schema to simulate the sample values of the process on the endpoint up to an arbitrary resolution, as we will further explore.

2.2.2. Innovation Processes for Gauss-Markov Processes

For 𝐗_𝑡, a multidimensional Gauss-Markov process, we call the multiresolution description of a process the sequence of conditional expectations on the nested sets of endpoints 𝐷_𝑛. In detail, if we denote by ℱ_𝑁 the filtration generated by {𝐗_𝑡;𝑡∈𝐷_𝑁} given the values of the process at the endpoints 𝐷_𝑁 of the partition, we introduce the sequence of the Gaussian processes (𝐙^𝑁_𝑡)_𝑁≥1 defined by:𝐙^𝑁_𝑡=𝔼𝐗_𝑡∣ℱ_𝑁=𝔼_𝑁𝐗_𝑡.(2.19) These processes 𝐙^𝑁 can be naturally viewed as an interpolation of the process 𝐗 sampled at the increasingly finer times 𝐷_𝑁 since for all 𝑡∈𝐷_𝑁 we have 𝐙^𝑁_𝑡=𝐗^𝑁_𝑡. The innovation process (𝜹^𝑁_𝑡,ℱ_𝑡,𝑡∈[0,1]) is defined as the update transforming the process 𝐙^𝑁_𝑡 into 𝐙^𝑁+1_𝑡, that is, 𝜹^𝑁_𝑡=𝐙^𝑁+1_𝑡−𝐙^𝑁_𝑡.(2.20) It corresponds to the difference the additional knowledge of the process at the points 𝑚_𝑁,𝑘 make on the conditional expectation of the process. This process satisfies the following important properties that found our multiresolution construction.

Proposition 2.3. The innovation process 𝜹^𝑁_𝑡 is a centered Gaussian process independent of the processes 𝐙^𝑛_𝑡 for any 𝑛≤𝑁. For 𝑠∈𝑆_𝑁,𝑘 and 𝑡∈𝑆_𝑁,𝑝 with 𝑘,𝑝∈ℐ_𝑁, the covariance of the innovation process reads 𝔼_𝑁𝜹^𝑁_𝑡⋅𝜹^𝑁_𝑠^𝑇=𝝁_𝑁,𝑘(𝑡)⋅𝚺_𝑁,𝑘⋅𝝁_𝑁,𝑘(𝑡)^𝑇if𝑘=𝑝,𝟎if𝑘≠𝑝,(2.21) where 𝝁_𝑁,𝑘(𝑡)=𝝁^𝑟𝑡;𝑙_𝑁,𝑘,𝑚_𝑁,𝑘,𝑡∈𝑙_𝑁,𝑘,𝑚_𝑁,𝑘,𝝁^𝑙𝑡;𝑚_𝑁,𝑘,𝑟_𝑁,𝑘,𝑡∈𝑚_𝑁,𝑘,𝑟_𝑁,𝑘.(2.22) with 𝝁^𝑙, 𝝁^𝑟 and 𝚺_𝑁,𝑘=𝚺(𝑚_𝑁,𝑘;𝑙_𝑁,𝑘,𝑟_𝑁,𝑘) as defined in Proposition 2.1.

Proof. Because of the Markovian property of the process 𝐗, the law of the process 𝐙^𝑁 can be computed from the bridge formula derived in Proposition 2.1 and we have 𝐙^𝑁_𝑡=𝝁^𝑙𝑡;𝑙_𝑁,𝑘,𝑟_𝑁,𝑘⋅𝐗_{𝑙𝑁,𝑘}+𝝁^𝑟𝑡;𝑙_𝑁,𝑘,𝑟_𝑁,𝑘⋅𝐗_{𝑟𝑁,𝑘},𝐙^𝑁+1_𝑡=𝝁^𝑙𝑡;𝑙_𝑁,𝑘,𝑚_𝑁,𝑘⋅𝐗_{𝑙𝑁,𝑘}+𝝁^𝑟𝑡;𝑙_𝑁,𝑘,𝑚_𝑁,𝑘⋅𝐗_{𝑚𝑁,𝑘},for𝑡∈𝑙_𝑁,𝑘,𝑚_𝑁,𝑘,𝝁^𝑙𝑡;𝑚_𝑁,𝑘,𝑟_𝑁,𝑘⋅𝐗_{𝑚𝑁,𝑘}+𝝁^𝑟𝑡;𝑚_𝑁,𝑘,𝑟_𝑁,𝑘⋅𝐗_{𝑟𝑁,𝑘},for𝑡∈𝑙_𝑁,𝑘,𝑚_𝑁,𝑘.(2.23) Therefore, the innovation process can be written for 𝑡∈𝑆_𝑁,𝑘 as 𝜹^𝑁_𝑡=𝝁^𝑁_𝑁,𝑘(𝑡)⋅𝐗_{𝑚𝑁,𝑘}+𝝂^𝑁(𝑡)⋅𝐐^𝑁_𝑡,(2.24) where 𝐐^𝑁_𝑡 is a ℱ_𝑁 measurable process 𝝂^𝑁(𝑡) a deterministic matrix function and 𝝁_𝑁,𝑘(𝑡)=𝝁^𝑟𝑡;𝑙_𝑁,𝑘,𝑚_𝑁,𝑘,𝑡∈𝑙_𝑁,𝑘,𝑚_𝑁,𝑘,𝝁^𝑙𝑡;𝑚_𝑁,𝑘,𝑟_𝑁,𝑘,𝑡∈𝑚_𝑁,𝑘,𝑟_𝑁,𝑘.(2.25) The expressions of 𝝂 and 𝐐 are quite complex but are highly simplified when noting that 𝔼𝜹^𝑁_𝑡∣ℱ_𝑁=𝔼𝐙^𝑁+1_𝑡∣ℱ_𝑁−𝐙^𝑁_𝑡=𝔼𝔼𝐙_𝑡∣ℱ_𝑁+1∣ℱ_𝑁−𝐙^𝑁_𝑡=𝟎(2.26) directly implying that 𝝂(𝑡)⋅𝐐^𝑁_𝑡=𝝁^𝑁(𝑡)⋅𝐙^𝑁_{𝑚𝑁,𝑘} and yielding the remarkably compact expression 𝜹^𝑁_𝑡=𝝁_𝑁,𝑘(𝑡)⋅𝐗_{𝑚𝑁,𝑘}−𝐙^𝑁_{𝑚𝑁,𝑘}.(2.27) This process is a centered Gaussian process. Moreover, observing that it is ℱ_𝑁-measurable, it can be written as 𝜹^𝑁_𝑡=𝝁_𝑁,𝑘(𝑡)⋅𝐗_{𝑚𝑁,𝑘}∣ℱ_𝑁−𝐙^𝑁_{𝑚𝑁,𝑘},(2.28) and the process {𝐗_{𝑚𝑁,𝑘}∣ℱ_𝑁} appears as the Gauss-Markov bridge conditioned at times 𝑙_𝑁,𝑘 and 𝑟_𝑁,𝑘, and whose covariance is given by Proposition 2.1 and that has the expression 𝚺_𝑁,𝑘=𝚺𝑚_𝑁,𝑘;𝑙_𝑁,𝑘,𝑟_𝑁,𝑘=𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑚_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑟_𝑛,𝑘⁻¹𝐡_{𝑚𝑛,𝑘}𝑚_𝑛,𝑘,𝑟_𝑛,𝑘.(2.29) Let (𝑠,𝑡)∈[0,1]², and assume that 𝑠∈𝑆_𝑁,𝑘 and 𝑡∈𝑆_𝑁,𝑝. If 𝑘≠𝑝, then, because of the Markov property of the process 𝐗, the two bridges are independent and therefore the covariance 𝔼_𝑁[𝜹^𝑁_𝑡⋅(𝜹^𝑁_𝑠)^𝑇] is zero. If 𝑘=𝑝, we have 𝔼_𝑁𝜹^𝑁_𝑡⋅𝜹^𝑁_𝑠^𝑇=𝝁_𝑁,𝑘(𝑡)⋅𝚺_𝑁,𝑘⋅𝝁_𝑁,𝑘(𝑠)^𝑇.(2.30) Eventually, the independence property stems from the simple properties of the conditional expectation. Indeed, let 𝑛≤𝑁. We have 𝔼𝐙^𝑛_𝑡⋅𝜹^𝑁_𝑠^𝑇=𝔼𝐙^𝑛_𝑡⋅𝐙^𝑁+1_𝑠−𝐙^𝑁_𝑠^𝑇=𝔼𝔼𝐗_𝑡∣ℱ_𝑛⋅𝔼𝐗^𝑇_𝑠∣ℱ_𝑁+1−𝔼𝐗^𝑇_𝑠∣ℱ_𝑁=𝔼𝔼𝐗_𝑡∣ℱ_𝑛⋅𝔼𝐗^𝑇_𝑠∣ℱ_𝑁+1−𝔼𝔼𝐗_𝑡∣ℱ_𝑛⋅𝔼𝐗^𝑇_𝑠∣ℱ_𝑁=𝔼𝐙^𝑛_𝑡𝐙^𝑛_𝑠^𝑇−𝔼𝐙^𝑛_𝑡𝐙^𝑛_𝑠^𝑇=𝟎(2.31) and the fact that a zero covariance between two Gaussian processes implies the independence of these processes concludes the proof.

2.2.3. Derivation of the Candidate Multiresolution Bases of Functions

We deduce from the previous proposition the following fundamental theorem of this paper.

Theorem 2.4. For all 𝑁∈ℕ, there exists a collection of 𝝍_𝑁,𝑘∶[0,1]↦ℝ^𝑑×𝑑 that are zero outside the subinterval 𝑆_𝑁,𝑘 such that in distribution one has: 𝜹^𝑁_𝑡=_{𝑘∈ℐ𝑁}𝝍_𝑁,𝑘(𝑡)⋅𝚵_𝑁,𝑘,(2.32) where Ξ_𝑁,𝑘 are independent 𝑑-dimensional standard normal random variables (i.e., of law 𝒩(0,𝐈_𝑑)). This basis of functions is unique up to an orthogonal transformation.

Proof. The two processes 𝜹^𝑁_𝑡 and 𝐝=^𝑁_𝑡^def∑_{𝑘∈ℐ𝑁}𝝍_𝑁,𝑘(𝑡)⋅Ξ_𝑁,𝑘 are two Gaussian processes of mean zero. Therefore, we are searching for functions 𝝍_𝑁,𝑘 vanishing outside 𝑆_𝑁,𝑘 and ensuring that the two processes have the same probability distribution. A necessary and sufficient condition for the two processes to have the same probability distribution is to have the same covariance function (see, e.g., [36]). We therefore need to show the existence of a collection of functions 𝝍_𝑁,𝑘(𝑡) functions that vanish outside the subinterval 𝑆_𝑁,𝑘 and that ensure that the covariance of the process 𝐝^𝑁 is equal to the covariance of 𝜹^𝑁. Let (𝑠,𝑡)∈[0,1] such that 𝑠∈𝑆_𝑁,𝑘 and 𝑡∈𝑆_𝑁,𝑝. If 𝑘≠𝑝, the assumption fact that the functions 𝜓_𝑁,𝑘 vanish outside 𝑆_𝑁,𝑘 implies that 𝔼𝐝^𝑁_𝑡⋅𝐝^𝑁_𝑠^𝑇=𝟎.(2.33) If 𝑘=𝑝, the covariance reads 𝔼𝐝^𝑁_𝑡⋅𝐝^𝑁_𝑠^𝑇=𝔼𝝍_𝑁,𝑘(𝑡)⋅𝚵_𝑁,𝑘⋅𝚵^𝑇_𝑁,𝑘⋅𝝍_𝑁,𝑘(𝑠)^𝑇=𝝍_𝑁,𝑘(𝑡)⋅𝝍_𝑁,𝑘(𝑠)^𝑇,(2.34) which needs to be equal to the covariance of 𝜹^𝑁, namely, 𝝍_𝑁,𝑘(𝑡)⋅𝝍_𝑁,𝑘(𝑠)^𝑇=𝝁_𝑁,𝑘(𝑡)⋅𝚺_𝑁,𝑘⋅𝝁_𝑁,𝑘(𝑠)^𝑇.(2.35) Therefore, since 𝝁_𝑁,𝑘(𝑚_𝑁,𝑘)=𝐈_𝑑, we have 𝝍_𝑁,𝑘𝑚_𝑁,𝑘⋅𝝍_𝑁,𝑘𝑚_𝑁,𝑘^𝑇=𝚺_𝐍,𝐤.(2.36) We can hence now define 𝝍_𝑁,𝑘(𝑚_𝑁,𝑘) as a square root 𝝈_𝑁,𝑘 of the symmetric positive matrix 𝚺_𝐍,𝐤, by fixing 𝑠=𝑚_𝑁,𝑘 in (2.35) 𝝍_𝑁,𝑘(𝑡)⋅𝝈^𝑇_𝑁,𝑘=𝝁(𝑡)⋅𝝈_𝑁,𝑘⋅𝝈^𝑇_𝑁,𝑘.(2.37) Eventually, since by assumption we have that 𝚺_𝑁,𝑘 is invertible, so is 𝝈_𝑁,𝑘, and the functions 𝝍_𝑁,𝑘 can be written as 𝝍_𝑁,𝑘(𝑡)=𝝁_𝑁,𝑘(𝑡)⋅𝝈_𝑁,𝑘(2.38) with 𝝈_𝑁,𝑘 being a square root of 𝚺_𝑁,𝑘. Square roots of positive symmetric matrices are uniquely defined up to an orthogonal transformation. Therefore, all square roots of 𝚺_𝑁,𝑘 are related by orthogonal transformations 𝝈^_𝑁,𝑘=𝝈_𝑁,𝑘⋅𝐎_𝑁,𝑘, where 𝐎_𝑁,𝑘⋅𝐎^𝑇_𝑁,𝑘=𝐈_𝑑. This property immediately extends to the functions 𝝍_𝑁,𝑘 we are studying: two different functions 𝝍_𝑁,𝑘 and 𝝍^_𝑁,𝑘 satisfying the theorem differ from an orthogonal transformation 𝐎_𝑁,𝑘. We proved that, for 𝝍_𝑁,𝑘(𝑡)⋅Ξ_𝑁,𝑘 to have the same law as 𝜹^𝑁(𝑡) in the interval 𝑆_𝑁,𝑘, the functions 𝝍_𝑁,𝑘 with support in 𝑆_𝑁,𝑘 are necessarily of the form 𝝁_𝑁,𝑘(𝑡)⋅𝝈_𝑁,𝑘. It is straightforward to show the sufficient condition that provided such a set of functions, the processes 𝜹^𝑁_𝑡 and 𝐝^𝑁_𝑡 are equal in law, which ends the proof of the theorem.

Using the expressions obtained in Proposition 2.1, we can make completely explicit the form of the basis in terms of the functions 𝐟,𝐠, and 𝐡:𝝍_𝑛,𝑘(𝑡)=⎧⎪⎨⎪⎩𝐠(𝑡)𝐠⁻¹𝑚_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑡𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑚_𝑛,𝑘⁻¹𝝈_𝑛,𝑘,for𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,𝐠(𝑡)𝐠⁻¹𝑚_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑡,𝑟_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑚_𝑛,𝑘,𝑟_𝑛,𝑘⁻¹𝝈_𝑛,𝑘,for𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘,(2.39) and 𝝈_𝑛,𝑘 satisfies𝝈_𝑛,𝑘⋅𝝈^𝑇_𝑛,𝑘=𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑚_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑟_𝑛,𝑘⁻¹𝐡_{𝑚𝑛,𝑘}𝑚_𝑛,𝑘,𝑟_𝑛,𝑘.(2.40) Note that 𝝈_𝑛,𝑘 can be defined uniquely as the symmetric positive square root, or as the lower triangular matrix resulting from the Cholesky decomposition of 𝚺_𝑛,𝑘.

Let us now define the function 𝝍_0,0∶[0,1]↦ℝ^𝑑×𝑑 such that the process 𝝍_0,0(𝑡)⋅Ξ_0,0 has the same covariance as 𝐙⁰_𝑡, which is computed using exactly the same technique as that developed in the proof of Theorem 2.4 and that has the expression𝝍_0,0(𝑡)=𝐠(𝑡)𝐡₀𝑙_0,0,𝑡𝐡₀(𝑙_0,0,𝑟_0,0)⁻¹𝐠⁻¹𝑟_0,0𝝈_0,0,(2.41) for 𝝈_0,0, a square root of 𝐂_𝑟0,0 the covariance matrix of 𝐗_𝑟0,0 which from (2.5) reads𝐅(0,1)𝐡₀(1,1)𝐅(0,1)^𝑇=𝐠(1)𝐡₀(1,1)(𝐠(1))^𝑇.(2.42)

We are now in position to show the following corollary of Theorem 2.4.

Corollary 2.5. The Gauss-Markov process 𝐙^𝑁_𝑡 is equal in law to the process 𝐗^𝑁_𝑡=^𝑁−1_𝑛=0_{𝑘∈ℐ𝑛}𝝍_𝑛,𝑘(𝑡)⋅𝚵_𝑛,𝑘,(2.43) where Ξ_𝑛,𝑘 are independent standard normal random variables 𝒩(0,𝐈_𝑑).

Proof. We have 𝐙^𝑁_𝑡=𝐙^𝑁_𝑡−𝐙^𝑁−1_𝑡+𝐙^𝑁−1_𝑡−𝐙^𝑁−2_𝑡+⋯+𝐙²_𝑡−𝐙¹_𝑡+𝐙¹_𝑡=^𝑁−1_𝑛=1𝜹^𝑛_𝑡+𝐙¹_𝑡=^𝑁−1_𝑛=1_{𝑘∈ℐ𝑛}𝝍_𝑛,𝑘(𝑡)⋅𝚵_𝑛,𝑘+𝝍_0,0(𝑡)⋅𝚵_0,0=^𝑁−1_𝑛=0_{𝑘∈ℐ𝑛}𝝍_𝑛,𝑘(𝑡)⋅𝚵_𝑛,𝑘.(2.44)

We therefore identified a collection of functions {𝝍_𝑛,𝑘}_{(𝑛,𝑘)∈ℐ} that allows a simple construction of the Gauss-Markov process iteratively conditioned on increasingly finer partitions of the interval [0,1]. We will show that this sequence 𝐙^𝑁_𝑡 converges almost surely towards the Gauss-Markov process 𝐗_𝑡 used to construct the basis, proving that these finite-dimensional continuous functions 𝐙^𝑁_𝑡 form an asymptotically accurate description of the initial process. Beforehand, we rigorously study the Hilbertian properties of the collection of functions we just defined.

3. The Multiresolution Schauder Basis Framework

The above analysis motivates the introduction of a set of functions {𝝍_𝑛,𝑘}_{(𝑛,𝑘)∈ℐ} we now study in details. In particular, we enlighten the structure of the collection of functions 𝝍_𝑛,𝑘 as a Schauder basis in a certain space 𝒳 of continuous functions from [0,1] to ℝ^𝑑. The Schauder structure was defined in [38, 39], and its essential characterization is the unique decomposition property: namely that every element 𝑥 in 𝒳 can be written as a well-formed linear combination𝑥=_{(𝑛,𝑘)∈ℐ}𝝍_𝑛,𝑘⋅𝝃_𝑛,𝑘,(3.1) and that the coefficients satisfying the previous relation are unique.

3.1. System of Dual Bases

To complete this program, we need to introduce some quantities that will play a crucial role in expressing the family 𝝍_𝑛,𝑘 as a Schauder basis for some given space. In (2.39), two constant matrices ℝ^𝑑×𝑑 appear that will have a particular importance in the sequel for (𝑛,𝑘) in ℐ with 𝑛≠0:𝐋_𝑛,𝑘=𝐠^𝑇𝑚_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑚_𝑛,𝑘⁻¹𝝈_𝑛,𝑘=𝐡𝑙_𝑛,𝑘,𝑚_𝑛,𝑘⁻¹𝐠⁻¹𝑚_𝑛,𝑘𝝈_𝑛,𝑘,𝐑_𝑛,𝑘=𝐠^𝑇𝑚_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑚_𝑛,𝑘,𝑟_𝑛,𝑘⁻¹𝝈_𝑛,𝑘=𝐡𝑚_𝑛,𝑘,𝑟_𝑛,𝑘⁻¹𝐠⁻¹𝑚_𝑛,𝑘𝝈_𝑛,𝑘,(3.2) where 𝐡 stands for 𝐡₀. We further define the matrix𝐌_𝑛,𝑘=𝐠^𝑇𝑚_𝑛,𝑘𝝈⁻¹_𝑛,𝑘^𝑇(3.3) and we recall that 𝝈_𝑛,𝑘 is a square root of 𝚺_𝑛,𝑘, the covariance matrix of 𝐗_{𝑚𝑛,𝑘}, conditionally to 𝐗_{𝑙𝑛,𝑘} and 𝐗_{𝑟𝑛,𝑘}, given in (2.29). We stress that the matrices 𝐋_𝑛,𝑘,𝐑_𝑛,𝑘, 𝐌_𝑛,𝑘, and 𝚺_𝑛,k are all invertible and satisfy the important following properties.

Proposition 3.1. For all (𝑛,𝑘) in ℐ, 𝑛≠0, one has:(i)𝐌_𝑛,𝑘=𝐋_𝑛,𝑘+𝐑_𝑛,𝑘(ii)𝚺⁻¹_𝑛,𝑘=(𝐡_{𝑚𝑛,𝑘}(𝑙_𝑛,𝑘,𝑚_𝑛,𝑘))⁻¹+(𝐡_{𝑚𝑛,𝑘}(𝑚_𝑛,𝑘,𝑟_𝑛,𝑘))⁻¹.

To prove this proposition, we first establish the following simple lemma of linear algebra.

Lemma 3.2. Given two invertible matrices 𝐴 and 𝐵 in 𝐺𝐿_𝑛(ℝ) such that 𝐶=𝐴+𝐵 is also invertible, if one defines 𝐷=𝐴𝐶⁻¹𝐵, one has the following properties:(i)𝐷=𝐴𝐶⁻¹𝐵=𝐵𝐶⁻¹𝐴(ii)𝐷⁻¹=𝐴⁻¹+𝐵⁻¹.

Proof. (i)𝐷=𝐴𝐶⁻¹𝐵=(𝐶−𝐵)𝐶⁻¹𝐵=𝐵−𝐵𝐶⁻¹𝐵=𝐵(𝐼−𝐶⁻¹𝐵)=𝐵𝐶⁻¹(𝐶−𝐵)=𝐵𝐶⁻¹𝐴.(ii)(𝐴⁻¹+𝐵⁻¹)𝐷=𝐴⁻¹𝐷+𝐵⁻¹𝐷=𝐴⁻¹𝐴𝐶⁻¹𝐵+𝐵⁻¹𝐵𝐶⁻¹𝐴=𝐶⁻¹(𝐵+𝐴)=𝐶⁻¹𝐶=𝐼.

Proof of Proposition 3.1. (ii) Directly stems from Lemma 3.2, item (ii) by posing 𝐴=𝐡_{𝑚𝑛,𝑘}(𝑙_𝑛,𝑘,𝑚_𝑛,𝑘), 𝐵=𝐡_{𝑚𝑛,𝑘}(𝑚_𝑛,𝑘,𝑟_𝑛,𝑘), and 𝐶=𝐴+𝐵=𝐡_{𝑚𝑛,𝑘}(𝑙_𝑛,𝑘,𝑟_𝑛,𝑘). Indeed, the lemma implies that𝐷⁻¹=𝐴⁻¹𝐶𝐵⁻¹=𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑚_𝑛,𝑘⁻¹𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑟_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑚_𝑛,𝑘⁻¹=𝚺⁻¹_𝑛,𝑘.(3.4)(i)We have𝐋_𝑛,𝑘+𝐑_𝑛,𝑘=𝐠𝑚_𝑛,𝑘^𝑇𝐡𝑙_𝑛,𝑘,𝑚_𝑛,𝑘⁻¹+𝐡𝑚_𝑛,𝑘,𝑟_𝑛,𝑘⁻¹𝝈_𝑛,𝑘=𝐠𝑚_𝑛,𝑘^𝑇𝚺⁻¹_𝑛,𝑘𝝈^𝑛,𝑘=𝐠𝑚_𝑛,𝑘^𝑇𝝈⁻¹_𝑛,𝑘^𝑇,(3.5) which ends the demonstration of the proposition.

Let us define 𝐋_0,0=(𝐡(𝑙_0,0,𝑟_0,0))⁻¹𝐠⁻¹(𝑟_0,0)𝝈_0,0. With this notations we define the functions in a compact form as follows.

Definition 3.3. For every (𝑛,𝑘) in ℐ with 𝑛≠0, the continuous functions 𝝍_𝑛,𝑘 are defined on their support 𝑆_𝑛,𝑘 as 𝝍_𝑛,𝑘(𝑡)=𝐠(𝑡)𝐡𝑙_𝑛,𝑘,𝑡⋅𝐋_𝑛,𝑘,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,𝐠(𝑡)𝐡𝑡,𝑟_𝑛,𝑘⋅𝐑_𝑛,𝑘,𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘,(3.6) and the basis element 𝝍_0,0 is given on [0,1] by 𝝍_0,0(𝑡)=𝐠(𝑡)𝐡𝑙_0,0,𝑡⋅𝐋_0,0.(3.7)

The definition implies that 𝝍_𝑛,𝑘 are continuous functions in the space of piecewise derivable functions with piecewise continuous derivative which takes value zero at zero. We denote such a space by 𝐶¹₀([0,1],ℝ^𝑑×𝑑).

Before studying the property of the functions 𝝍_𝑛,𝑘, it is worth remembering that their definitions include the choice of a square root 𝝈_𝑛,𝑘 of 𝚺_𝑛,𝑘. Properly speaking, there is thus a class of bases 𝝍_𝑛,𝑘 and all the points we develop in the sequel are valid for this class. However, for the sake of simplicity, we consider from now on that the basis under scrutiny results from choosing the unique square root 𝝈_𝑛,𝑘 that is lower triangular with positive diagonal entries (the Cholesky decomposition).

3.1.1. Underlying System of Orthonormal Functions

We first introduce a family of functions 𝜙_𝑛,𝑘 and show that it constitutes an orthogonal basis on a certain Hilbert space. The choice of this basis can seem arbitrary at first sight, but the definition of these function will appear natural for its relationship with the functions 𝝍_𝑛,𝑘 and Φ_𝑛,𝑘 that is made explicit in the sequel, and the mathematical rigor of the argument lead us to choose this apparently artificial introduction.

Definition 3.4. For every (𝑛,𝑘) in ℐ with 𝑛≠0, we define a continuous function 𝜙_𝑛,𝑘∶[0,1]→ℝ^𝑚×𝑑 which is zero outside its support 𝑆_𝑛,𝑘 and has the expressions: 𝜙_𝑛,𝑘(𝑡)=𝐟(𝑡)^𝑇⋅𝐋_𝑛,𝑘if𝑙_𝑛,𝑘≤𝑡<𝑚_𝑛,𝑘,𝐟(𝑡)^𝑇⋅𝐑_𝑛,𝑘if𝑚_𝑛,𝑘≤𝑡<𝑟_𝑛,𝑘.(3.8) The basis element 𝜙_0,0 is defined on [0,1] by 𝜙_0,0(𝑡)=𝐟(𝑡)^𝑇⋅𝐋_0,0.(3.9)

Remark that the definitions make apparent the fact that these two families of functions are linked for all (𝑛,𝑘) in ℐ through the simple relation𝝍^_𝑛,𝑘=𝜶⋅𝝍_𝑛,𝑘+√𝚪⋅𝜙_𝑛,𝑘.(3.10) Moreover, this collection of functions 𝜙_𝑛,𝑘 constitutes an orthogonal basis of functions, in the following sense.

Proposition 3.5. Let 𝐿²_𝐟 be the closure of 𝐮∶[0,1]⟶ℝ^𝑚∣∃𝐯∈𝐿²[0,1],ℝ^𝑑,𝐮=𝐟^𝑇⋅𝐯,(3.11) equipped with the natural norm of 𝐿²([0,1],ℝ^𝑚). It is a Hilbert space, and moreover, for all 0≤𝑗<𝑑, the family of functions 𝑐_𝑗(𝜙_𝑛,𝑘) defined as the columns of 𝜙_𝑛,𝑘, namely 𝑐_𝑗𝜙_𝑛,𝑘=𝜙_𝑛,𝑘_𝑖,𝑗_{0≤𝑖<𝑚},(3.12) forms a complete orthonormal basis of 𝐿²_𝐟.

Proof. The space 𝐿²_𝐟 is clearly a Hilbert space as a closed subspace of the larger Hilbert space 𝐿²([0,1],ℝ^𝑚) is equipped with the standard scalar product: ∀𝐮,𝐯∈𝐿²[0,1],ℝ^𝑑,(𝐮,𝐯)=¹₀𝐮(𝑡)^𝑇⋅𝐯(𝑡)𝑑𝑡.(3.13) We now proceed to demonstrate that the columns of 𝜙_𝑛,𝑘 form an orthonormal family which generates a dense subspace of 𝐿²_𝐟. To this end, we define 𝑀([0,1],ℝ^𝑚×𝑑) as the space of functions 𝐀∶[0,1]⟶ℝ^𝑚×𝑑∣∀𝑗∶0≤𝑗<𝑑,𝑡⟼𝐀_𝑖,𝑗(𝑡)_{0≤𝑖<𝑚}∈𝐿²([0,1],ℝ^𝑚),(3.14) that is, the space of functions that take values in the set of 𝑚×𝑑-matrices whose columns are in 𝐿²([0,1],ℝ^𝑚). This definition allows us to define the bilinear function 𝒫∶𝑀([0,1],ℝ^𝑚×𝑑)×𝑀([0,1],ℝ^𝑚×𝑑)→ℝ^𝑑×𝑑 as 𝒫(𝐀,𝐁)=¹₀𝐀(𝑡)^𝑇⋅𝐁(𝑡)𝑑𝑡satisfying𝒫(𝐁,𝐀)=𝒫(𝐀,𝐁)^𝑇,(3.15) and we observe that the columns of 𝜙_𝑛,𝑘 form an orthonormal system if and only if ∀((𝑝,𝑞),(𝑛,𝑘))∈ℐ×ℐ,𝒫𝜙_𝑛,𝑘,𝜙_𝑝,𝑞=¹₀𝜙_𝑛,𝑘(𝑡)^𝑇⋅𝜙_𝑝,q(𝑡)𝑑𝑡=𝛿^𝑛,𝑘_𝑝,𝑞𝐈_𝑑,(3.16) where 𝛿^𝑛,𝑘_𝑝,𝑞 is the Kronecker delta function, whose value is 1 if 𝑛=𝑝 and 𝑘=𝑞, and 0 otherwise.
First of all, since the functions 𝜙_𝑛,𝑘 are zero outside the interval 𝑆_𝑛,𝑘, the matrix 𝒫(𝜙_𝑛,𝑘,𝜙_𝑝,𝑞) is nonzero only if 𝑆_𝑛,𝑘∩𝑆_𝑝,𝑞≠∅. In such cases, assuming that 𝑛≠𝑝 and, for example, that 𝑛<𝑝, we necessarily have 𝑆_𝑛,𝑘 strictly included in 𝑆_𝑝,𝑞: more precisely, 𝑆_𝑛,𝑘 is either included in the left-child support 𝑆_𝑝+1,2𝑞 or in the right-child support 𝑆_{𝑝+1,2𝑞+1} of 𝑆_𝑝,𝑞. In both cases, writing the matrix 𝒫(𝜙_𝑛,𝑘(𝑡),𝜙_𝑝,𝑞) shows that it is expressed as a matrix product whose factors include 𝒫(𝜙_𝑛,𝑘,𝐟^𝑇). We then show that 𝒫𝜙_𝑛,𝑘,𝐟^𝑇=¹₀𝜙_𝑛,𝑘(𝑡)^𝑇⋅𝐟(𝑡)^𝑇=𝐋^𝑇_𝑛,𝑘⋅^{𝑚𝑛,𝑘}_{𝑙𝑛,𝑘}𝐟(𝑢)⋅𝐟^𝑇(𝑢)𝑑𝑢−𝐑^𝑇_𝑛,𝑘⋅^{𝑟𝑛,𝑘}_{𝑚𝑛,𝑘}𝐟(𝑢)⋅𝐟^𝑇(𝑢)𝑑𝑢,=𝐋^𝑇_𝑛,𝑘⋅𝐡𝑙_𝑛,𝑘,𝑚_𝑛,𝑘−𝐑^𝑇_𝑛,𝑘⋅𝐡𝑚_𝑛,𝑘,𝑟_𝑛,𝑘=𝝈^𝑇_𝑛,𝑘𝐠⁻¹𝑚_𝑛,𝑘^𝑇−𝝈^𝑇_𝑛,𝑘𝐠⁻¹𝑚_𝑛,𝑘^𝑇,(3.17) which entails that 𝒫(𝜙_𝑛,𝑘,𝐟^𝑇)=0 if 𝑛<𝑝. If 𝑛>𝑝, we remark that 𝒫(𝜙_𝑛,𝑘,𝜙_𝑝,𝑞)=𝒫(𝜙_𝑝,𝑞,𝜙_𝑛,𝑘)^𝑇, and we conclude that 𝒫(𝜙_𝑛,𝑘,𝜙_𝑝,𝑞)=𝟎 from the preceding case. For 𝑛=𝑝, we directly compute for 𝑛>0 the only nonzero term 𝒫𝜙_𝑛,𝑘,𝜙_𝑛,𝑘=𝐋^𝑇_𝑛,𝑘⋅^{𝑚𝑛,𝑘}_{𝑙𝑛,𝑘}𝐟(𝑢)⋅𝐟^𝑇(𝑢)𝑑𝑢⋅𝐋_𝑛,𝑘+𝐑^𝑇_𝑛,𝑘⋅^{𝑟𝑛,𝑘}_{𝑚𝑛,𝑘}𝐟(𝑢)⋅𝐟^𝑇(𝑢)𝑑𝑢⋅𝐑_𝑛,𝑘,=𝝈^𝑇_𝑛,𝑘𝐠⁻¹𝑚_𝑛,𝑘^𝑇𝐡𝑙_𝑛,𝑘,𝑚_𝑛,𝑘⁻¹𝐠⁻¹𝑚_𝑛,𝑘𝝈_𝑛,𝑘+𝝈^𝑇_𝑛,𝑘𝐠⁻¹𝑚_𝑛,𝑘^𝑇𝐡𝑚_𝑛,𝑘,𝑟_𝑛,𝑘⁻¹𝐠⁻¹𝑚_𝑛,𝑘𝝈_𝑛,𝑘.(3.18) Using the passage relationship between the symmetric functions 𝐡 and 𝐡_{𝑚𝑛,𝑘} given in (2.7), we can then write 𝒫𝜙_𝑛,𝑘,𝜙_𝑛,𝑘=𝝈^𝑇_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑙_𝑛,𝑘,𝑚_𝑛,𝑘⁻¹𝝈_𝑛,𝑘+𝝈^𝑇_𝑛,𝑘𝐡_{𝑚𝑛,𝑘}𝑚_𝑛,𝑘,𝑟_𝑛,𝑘⁻¹𝝈_𝑛,𝑘.(3.19) Proposition 3.1 implies that 𝐡_{𝑚𝑛,𝑘}(𝑙_𝑛,𝑘,𝑚_𝑛,𝑘)⁻¹+𝐡_{𝑚𝑛,𝑘}(𝑚_𝑛,𝑘,𝑟_𝑛,𝑘)⁻¹=𝚺⁻¹_𝑛,𝑘=(𝝈⁻¹_𝑛,𝑘)^𝑇𝝈⁻¹_𝑛,𝑘 which directly implies that 𝒫(𝜙_𝑛,𝑘,𝜙^𝑇_𝑛,𝑘)=𝐈_𝑑. For 𝑛=0, a computation of the exact same flavor yields that 𝒫(𝜙_0,0,𝜙_0,0)=𝐈_𝑑. Hence, we have proved that the collection of columns of 𝜙_𝑛,𝑘 forms an orthonormal family of functions in 𝐿²_𝐟 (the definition of 𝜙_𝑛,𝑘 clearly states that its columns can be written in the form of elements of 𝐿²_𝐟).
The proof now amounts showing the density of the family of functions we consider. Before showing this density property, we introduce for all (𝑛,𝑘) in ℐ the functions 𝐏_𝑛,𝑘∶[0,1]→ℝ^𝑑×𝑑 with support on 𝑆_𝑛,𝑘 defined by 𝐏_𝑛,𝑘(𝑡)=𝐋_𝑛,𝑘if𝑙_𝑛,𝑘≤𝑡<𝑚_𝑛,𝑘,−𝐑_𝑛,𝑘if𝑚_𝑛,𝑘≤𝑡<𝑟_𝑛,𝑘,𝑛≠0,𝐏_0,0(𝑡)=𝐋_0,0.(3.20) Showing that the family of columns of 𝜙_𝑛,𝑘 is dense in 𝐿²_𝐟 is equivalent to show that the column vectors of the matrices 𝐏_𝑛,𝑘 seen as a function of 𝑡 are dense in 𝐿²([0,1],ℝ^𝑑). It is enough to show that the span of such functions contains the family of piecewise continuous ℝ^𝑑-valued functions that are to be constant on 𝑆_𝑛,𝑘, (𝑛,𝑘) in ℐ (the density of the endpoints of the partition ∪_𝑁∈ℕ𝐷_𝑁 entails that the latter family generates 𝐿²([0,1],ℝ^𝑑)).
In fact, we show that the span of functions 𝑉_𝑁=span𝑡⟼𝑐_𝑗𝐏_𝑛,𝑘(𝑡)∣0≤𝑗<𝑑,(𝑛,𝑘)∈ℐ_𝑁(3.21) is exactly equal to the space 𝐾_𝑁 of piecewise continuous functions from [0,1] to ℝ^𝑑 that are constant on the supports 𝑆_𝑁+1,𝑘, for any (𝑁+1,𝑘) in ℐ. The fact that 𝑉_𝑁 is included in 𝐾_𝑁 is clear from the fact that the matrix-valued functions 𝐏_𝑁,𝑘 are defined constant on the support 𝑆_𝑁+1,𝑘, for (𝑁,𝑘) in 𝐼.
We prove that 𝐾_𝑁 is included in 𝑉_𝑁 by induction on 𝑁≤0. The property is clearly true at rank 𝑁=0 since 𝐏_0,0 is then equal to the constant invertible matrix 𝐋_0,0. Assuming that the proposition true at rank 𝑁−1 for a given 𝑁>0, let us consider a piecewise continuous function 𝐜∶[0,1]→ℝ^𝑑 in 𝐾_𝑁−1. Remark that, for every (𝑁,𝑘) in ℐ, the function 𝐜 can only take two values on 𝑆_𝑁,𝑘 and can have discontinuity jump in 𝑚_𝑁,𝑘: let us denote these jumps as 𝐝_𝑁,𝑘=𝐜𝑚⁺_𝑁,𝑘−𝐜𝑚⁻_𝑁,𝑘.(3.22) Now, remark that for every (𝑁,𝑘) in ℐ, the matrix-valued functions 𝐏_𝑁,𝑘 take only two matrix values on 𝑆_𝑁,𝑘, namely, 𝐋_𝑁,𝑘 and −𝐑_𝑁,𝑘. From Proposition 3.1, we know that 𝐋_𝑁,𝑘+𝐑_𝑁,𝑘=𝐌_𝑁,𝑘 is invertible. This fact directly entails that there exist vectors 𝐚_𝑁,𝑘, for any (𝑁,𝑘) in ℐ, such that 𝐝_𝑁,𝑘=(𝐋_𝑁,𝑘+𝐑_𝑁,𝑘)(−𝐚_𝑁,𝑘). We then necessarily have that the function 𝐜′=𝐜+𝐏_𝑛,𝑘⋅𝐚_𝑛,𝑘 is piecewise constant on the supports 𝑆_𝑁,𝑘, (𝑁,𝑘) in ℐ. By recurrence hypothesis, 𝐜′ belongs to 𝑉_𝑁−1, so that 𝐜 belongs to 𝑉_𝑁, and we have proved that 𝐾_𝑁⊂𝑉_𝑁. Therefore, the space generated by the column vectors 𝑃_𝑛,𝑘 is dense in 𝐿²[0,1], which completes the proof that the functions 𝑡↦[(𝜙_𝑛,𝑘(𝑡))_𝑖,𝑗]_{0≤𝑖<𝑚} form a complete orthonormal family of 𝐿²[0,1].

The fact that the column functions of 𝜙_𝑛,𝑘 form a complete orthonormal system of 𝐿²_𝐟 directly entails the following decomposition of the identity on 𝐿²_𝐟.

Corollary 3.6. If 𝛿 is the real delta Dirac function, one has _(𝑛,_𝑘)∈ℐ𝜙_𝑛,𝑘(𝑡)⋅𝜙^𝑇_𝑛,𝑘(𝑠)=𝛿(𝑡−𝑠)𝐼𝑑_𝐿²_𝐟.(3.23)

Proof. Indeed it easy to verify that, for all 𝐯 in 𝐿²_𝐟, we have for all 𝑁>0_𝑈_{(𝑛,𝑘)∈ℐ𝑁}𝜙_𝑛,𝑘(𝑡)⋅𝜙^𝑇_𝑛,𝑘(𝑠)𝐯(𝑠)𝑑𝑠=_{(𝑛,𝑘)∈ℐ𝑁}𝜙_𝑛,𝑘(𝑡)⋅𝒫𝜙_𝑛,𝑘,𝐯=_{(𝑛,𝑘)∈ℐ𝑁}^𝑑−1_𝑝=0𝑐_𝑝𝜙_𝑛,𝑘𝑐_𝑝𝜙_𝑛,𝑘,𝐯,(3.24) where (𝑐_𝑝(𝜙_𝑛,𝑘),𝐯) denotes the inner product in 𝐿²_𝐟 between 𝐯 and the 𝑝-column of 𝝍_𝑛,𝑘. Therefore, by the Parseval identity, we have in the 𝐿²_𝐟 sense _𝑈_{(𝑛,𝑘)∈ℐ}𝜙_𝑛,𝑘(𝑡)⋅𝜙^𝑇_𝑛,𝑘(𝑠)𝐯(𝑠)𝑑𝑠=𝐯(𝑡).(3.25)

From now on, abusing language, we will say that the family of ℝ^𝑚×𝑑-valued functions 𝜙_𝑛,𝑘 is an orthonormal family of functions to refer to the fact that the columns of such matrices form orthonormal set of 𝐿²_𝐟. We now make explicit the relationship between this orthonormal basis and our functions (𝝍_𝑛,𝑘) derived in our analysis of the multidimensional Gauss-Markov processes.

3.1.2. Generalized Dual Operators

The Integral Operator 𝓚
The basis 𝜙_𝑛,𝑘 is of great interest in this paper for its relationship to the functions 𝝍_𝑛,𝑘 that naturally arise in the decomposition of the Gauss-Markov processes. Indeed, the collection 𝝍_𝑛,𝑘 can be generated from the orthonormal basis 𝜙_𝑛,𝑘 through the action of the integral operator 𝒦 defined on 𝐿²([0,1],ℝ^𝑚) into 𝐿²([0,1],ℝ^𝑑) by 𝐮↦𝒦[𝐮]=𝑡⟼𝐠(𝑡)⋅_𝑈𝟙_[0,𝑡](𝑠)𝐟(𝑠)𝐮(𝑠)𝑑𝑠,(3.26) where 𝑈⊃[0,1] is an open set and, for any set 𝐸⊂𝑈,𝟙_𝐸(⋅) denotes the indicator function of 𝐸. Indeed, realizing that 𝒦 acts on 𝑀([0,1],ℝ^𝑚×𝑑) into 𝑀([0,1],ℝ^𝑑×𝑑) through ∀𝐀∈𝑀[0,1],ℝ^m×𝑑,𝒦[𝐀]=𝒦𝑐₀(𝐀),…,𝒦𝑐_𝑑−1(𝐀),(3.27) where 𝑐_𝑗(𝐀) denotes the 𝑗th ℝ^𝑚-valued column function of 𝐀, we easily see that for all (𝑛,𝑘) in ℐ, 0≤𝑡≤1, 𝝍_𝑛,𝑘(𝑡)=𝐠(𝑡)⋅^𝑡₀𝐟(𝑠)⋅𝜙_𝑛,𝑘(𝑠)𝑑𝑠=𝒦𝜙_𝑛,𝑘(𝑡).(3.28) It is worth noticing that the introduction of the operator 𝒦 can be considered natural since it characterizes the centered Gauss-Markov process 𝐗 through loosely writing 𝐗=𝒦[𝐝𝐖].
In order to exhibit a dual family of functions to the basis 𝝍_𝑛,𝑘, we further investigate the property of the integral operator 𝒦. In particular, we study the existence of an inverse operator 𝒟, whose action on the orthonormal basis 𝜙_𝑛,𝑘 will conveniently provide us with a dual basis to 𝝍_𝑛,𝑘. Such an operator does not always exist; nevertheless, under special assumptions, it can be straightforwardly expressed as a generalized differential operator.

The Differential Operator 𝓓
Here, we make the assumptions that 𝑚=𝑑, that, for all 𝑡, 𝐟(𝑡) is invertible in ℝ^𝑑×𝑑, and that 𝐟 and 𝐟⁻¹ have continuous derivatives, which especially implies that 𝐿²_𝐟=𝐿²(ℝ^𝑑). In this setting, we define the space 𝐷₀(𝑈,ℝ^𝑑) of functions in 𝐶^∞₀(𝑈,ℝ^𝑑) that are zero at zero and denote by 𝐷^₀(𝑈,ℝ^𝑑) its dual in the space of distributions (or generalized functions). Under the assumptions just made, the operator 𝒦∶𝐷₀(𝑈,ℝ^𝑑)↦𝐷₀(𝑈,ℝ^𝑑) admits the differential operator 𝒟∶𝐷₀(𝑈,ℝ^𝑑)↦𝐷₀(𝑈,ℝ^𝑑) defined by 𝐮∈𝐷₀𝑈,ℝ^𝑑⟼𝒟[𝐮]=𝑡⟼𝐟⁻¹(𝑡)𝑑𝑑𝑡𝐠⁻¹(𝑡)𝐮(𝑡)(3.29) as its inverse, that is, when restricted to 𝐷₀(𝑈,ℝ^𝑑), we have 𝒟∘𝒦=𝒦∘𝒟=𝐼𝑑 on 𝐷₀(𝑈,ℝ^𝑑). The dual operators of 𝒦 and 𝒟 are expressed, for any 𝐮 in 𝐷₀(𝑈,ℝ^𝑑), as 𝒟^∗[𝐮]=𝑡⟼−𝐠⁻¹(𝑡)^𝑇𝑑𝑑𝑡𝐟⁻¹(𝑡)^𝑇𝐮(𝑡),𝒦^∗[𝐮]=𝑡⟼−𝐟(𝑡)^𝑇_𝑈𝟙_[0,𝑡](𝑠)𝐠^𝑇(𝑠)𝐮(𝑠)𝑑𝑠.(3.30) They satisfy (from the properties of 𝒦 and 𝒟) 𝒟^∗∘𝒦^∗=𝒦^∗∘𝒟^∗=𝐼𝑑 on 𝐷₀(𝑈,ℝ^𝑑). By dual pairing, we extend the definition of the operators 𝒦, 𝒟 as well as their dual operators, to the space of generalized function 𝐷^₀(𝑈,ℝ^𝑑). In details, for any distribution 𝑇 in 𝐷^₀(𝑈,ℝ^𝑑) and test function 𝐮 in 𝐷₀(𝑈,ℝ^𝑑), define 𝒦 and 𝒦^∗ by (𝒟[𝑇],𝐮)=𝑇,𝒟^∗[𝐮],(𝒦[𝑇],𝐮)=𝑇,𝒦^∗[𝐮],(3.31) and reciprocally for the dual operators 𝒟^∗ and 𝒦^∗.

Candidate Dual Basis
We are now in a position to use the orthonormality of 𝜙_𝑛,𝑘 to infer a dual family of the basis 𝝍_𝑛,𝑘. For any function 𝐮 in 𝐿²(𝑈,ℝ^𝑑), the generalized function 𝒦[𝐮] belongs to 𝐶₀(𝑈,ℝ^𝑑), the space of continuous functions that are zero at zero. We equip this space with the uniform norm and denote its topological dual 𝑅₀(𝑈,ℝ^𝑑), the set of 𝑑-dimensional Radon measures with 𝑅₀(𝑈,ℝ^𝑑)⊂𝐷^₀(𝑈,ℝ^𝑑). Consequently, operating in the Gelfand triple 𝐶₀𝑈,ℝ^𝑑⊂𝐿²𝑈,ℝ^𝑑⊂𝑅₀𝑈,ℝ^𝑑,(3.32) we can write, for any function 𝐮, 𝐯 in 𝐿²(𝑈,ℝ^𝑑)⊂𝑅₀(𝑈,ℝ^𝑑), (𝐮,𝐯)=((𝒟∘𝒦)[𝐮],𝐯)=𝒦[𝐮],𝒟^∗[𝐯].(3.33) The first equality stems from the fact that, when 𝒦 and 𝒟 are seen as generalized functions, they are still inverse of each other, so that in particular 𝒟∘𝒦=𝐼𝑑 on 𝐿²(𝑈,ℝ^𝑑). The dual pairing associated with the Gelfand triple (3.32) entails the second equality where 𝒟^∗ is the generalized operator defined on 𝐷^₀(𝑈,ℝ^𝑑) and where 𝒟^∗[𝐯] is in 𝑅₀(𝑈,ℝ^𝑑).
As a consequence, defining the functions 𝜹_𝑛,𝑘 in 𝑅₀(𝑈,ℝ^𝑑×𝑑), the 𝑑×𝑑-dimensional space of Radon measures, by 𝜹_𝑛,𝑘=𝒟^∗𝜙_𝑛,𝑘=𝒟^∗𝑐_𝑖𝜙_𝑛,𝑘,…,𝒟^∗𝑐_𝑗𝜙_𝑛,𝑘(3.34) provides us with a family of 𝑑×𝑑-generalized functions which are dual to the family 𝝍_𝑛,𝑘 in the sense that, for all ((𝑛,𝑘),(𝑝,𝑞)) in ℐ×ℐ, we have 𝒫𝜹_𝑛,𝑘,𝝍_𝑝,𝑞=𝛿^𝑛,𝑘_𝑝,𝑞𝐈_𝑑,(3.35) where the definition of 𝒫 has been extended through dual pairing: given any 𝐀 in 𝑅₀(𝑈,ℝ^𝑚×𝑑) and any 𝐁 in 𝐶₀(𝑈,ℝ^𝑚×𝑑), we have 𝒫(𝐀,𝐁)=𝑐_𝑖(𝐀),𝑐_𝑗(𝐁)_{0≤𝑖,𝑗<𝑑}(3.36) with (𝑐_𝑖(𝐀),𝑐_𝑗(𝐁)) denoting the dual pairing between the 𝑖th column of 𝐀 taking value in 𝑅₀(𝑈,ℝ^𝑑) and the 𝑗th column of 𝐁 taking value in 𝐶₀(𝑈,ℝ^𝑑). Under the favorable hypothesis of this section, the 𝑑×𝑑-generalized functions 𝜹_𝑛,𝑘 can actually be easily computed since considering the definition of 𝜙_𝑛,𝑘 shows that the functions (𝐟⁻¹)^𝑇⋅𝜙_𝑛,𝑘 have support 𝑆_𝑛,𝑘 and are constant on 𝑆_𝑛+1,2𝑘 and S_{𝑛+1,2𝑘+1} in ℝ^𝑑×𝑑. Only the discontinuous jumps in 𝑙_𝑛,𝑘, 𝑚_𝑛,𝑘, and 𝑟_𝑛,𝑘 intervene, leading to expressing for (𝑛,𝑘) in ℐ, 𝑛≠0𝜹_𝑛,𝑘(𝑡)=𝐠(𝑡)⁻¹^𝑇⋅𝐌_𝑛,𝑘𝛿𝑡−𝑚_𝑛,𝑘−𝐋_𝑛,𝑘𝛿𝑡−𝑙_𝑛,𝑘+𝐑_𝑛,𝑘𝛿𝑡−𝑟_𝑛,𝑘(3.37) and 𝜹_0,0(𝑡)=(𝐠(𝑡)⁻¹)^𝑇⋅𝐋_0,0, where 𝛿(⋅) denotes the standard delta Dirac function (centered in 0). These functions can be extended to the general setting of the paper since its expressions do not involve the assumptions made on the invertibility and smoothness of 𝐟(𝑡). We now show that these functions, when defined in the general setting, still provide a dual basis of the functions 𝝍_𝑛,𝑘.

3.1.3. Dual Basis of Generalized Functions

The expression of the basis 𝜹_𝑛,𝑘 that has been found under favorable assumptions makes no explicit reference to these assumptions. It suggests defining functions 𝜹_𝑛,𝑘 formally as linear combination of Dirac functions acting by duality on 𝐶₀(𝑈,ℝ^𝑑×𝑑).

Definition 3.7. For (𝑛,𝑘) in ℐ, the family of generalized functions 𝜹_𝑛,𝑘 in 𝑅₀(𝑈,ℝ^𝑑×𝑑) is given by (𝑛≠0) 𝜹_𝑛,𝑘(𝑡)=𝐠(𝑡)⁻¹^𝑇⋅𝐌_𝑛,𝑘𝛿𝑡−𝑚_𝑛,𝑘−𝐋_𝑛,𝑘𝛿𝑡−𝑙_𝑛,𝑘+𝐑_𝑛,𝑘𝛿𝑡−𝑟_𝑛,𝑘,(3.38) and 𝜹_0,0(𝑡)=(𝐠(𝑡)⁻¹)^𝑇⋅𝐋_0,0, where 𝛿 is the standard Dirac distribution.

Notice that the basis 𝜹_𝑛,𝑘 is defined for the open set 𝑈. For the sake of consistency, we extend the definition of the families 𝝍_𝑛,𝑘 and 𝜙_𝑛,𝑘 on 𝑈 by setting them to zero on 𝑈⧵[0,1], except for 𝝍_0,0, which is continued for 𝑡>1 by a continuous function 𝐜 that is compactly supported in [1,𝑎) for a given 𝑎 in 𝑈, 𝑎>1 and satisfies 𝐜(1)=𝝍_0,0(1).

We can now formulate the following.

Proposition 3.8. Given the dual pairing in 𝐶₀(𝑈)⊂𝐿²(𝑈)⊂𝑅(𝑈) where 𝑈 is a bounded open set of ℝ containing [0,1], the family of continuous functions 𝝍_𝑛,𝑘 in 𝐶₀(𝑈) admits, for dual family in 𝑅(𝑈), the set of distributions 𝜹_𝑛,𝑘.

Proof. We have to demonstrate that, for all ((𝑛,𝑘),(𝑝,𝑞)) in ℐ×ℐ, 𝒫𝜹_𝑝,𝑞,𝝍_𝑛,𝑘=𝛿^𝑛,𝑘_𝑝,𝑞𝐈_𝑑.(3.39) Suppose first that 𝑛,𝑝>0. If 𝑝<𝑛, 𝒫(𝜹_𝑛,𝑘,𝝍_𝑝,𝑞) can only be nonzero if the support 𝑆_𝑝,𝑞 is strictly included in 𝑆_𝑛,𝑘. We then have 𝒫𝜹_𝑝,𝑞,𝝍_𝑛,𝑘=𝐌^𝑇_𝑝,𝑞𝐠⁻¹𝑚_𝑝,𝑞𝝍_𝑛,𝑘𝑚_𝑝,𝑞−𝐋^𝑇_𝑝,𝑞𝐠⁻¹𝑙_𝑝,𝑞𝝍_𝑛,𝑘𝑙_𝑝,𝑞+𝐑^𝑇_𝑝,𝑞𝐠⁻¹𝑟_𝑝,𝑞𝝍_𝑛,𝑘𝑟_𝑝,𝑞.(3.40) Assume that 𝑆_𝑝,𝑞 is to the left of 𝑚_𝑛,𝑘, that is, 𝑆_𝑝,𝑞 is a left child of 𝑆_𝑛,𝑘 in the nested binary tree of supports and write 𝒫𝜹_𝑝,𝑞,𝝍_𝑛,𝑘=𝐌^𝑇_𝑝,𝑞𝐡𝑙_𝑛,𝑘,𝑚_𝑝,𝑞−𝐋^𝑇_𝑝,𝑞𝐡𝑙_𝑛,𝑘,𝑙_𝑝,𝑞−𝐑^𝑇_𝑝,𝑞𝐡𝑙_𝑛,𝑘,𝑟_𝑝,𝑞𝐋_𝑛,𝑘.(3.41) Using the fact that 𝐌_𝑝,𝑞=𝐋_𝑝,𝑞+𝐑_𝑝,𝑞 and that the function ℎ(𝑥,𝑦), as any integral between 𝑥 and 𝑦, satisfies the chain rule 𝐡(𝑥,𝑦)=𝐡(𝑥,𝑧)+𝐡(𝑧,𝑦) for all (𝑥,𝑦,𝑧), we obtain 𝒫𝜹_𝑝,𝑞,𝝍_𝑛,𝑘=−𝐋^𝑇_𝑝,𝑞𝐡𝑙_𝑛,𝑘,𝑚_𝑝,𝑞−𝐡𝑙_𝑛,𝑘,𝑙_𝑝,𝑞+𝐑^𝑇_𝑝,𝑞ℎ𝑙_𝑛,𝑘,𝑚_𝑝,𝑞−ℎ𝑙_𝑛,𝑘,𝑟_𝑝,𝑞𝐋_𝑛,𝑘=−𝐋^𝑇_𝑝,𝑞𝐡𝑙_𝑝,𝑞,𝑚_𝑝,𝑞+𝐑^𝑇_𝑝,𝑞ℎ𝑟_𝑝,𝑞,𝑚_𝑝,𝑞𝐋_𝑛,𝑘=−𝝈^𝑇_𝑝,𝑞𝐠⁻¹𝑚_𝑝,𝑞^𝑇𝐡𝑙_𝑝,𝑞,𝑚_𝑝,𝑞⁻¹^𝑇⋅𝐡𝑙_𝑝,𝑞,𝑚_𝑝,𝑞+𝝈^𝑇_𝑝,𝑞𝐠⁻¹𝑚_𝑝,𝑞^𝑇𝐡𝑚_𝑝,𝑞,𝑟_𝑝,𝑞⁻¹^𝑇⋅𝐡𝑚_𝑝,𝑞,𝑟_𝑝,𝑞⋅𝐋_𝑛,𝑘=𝟎.(3.42) The same result is true if 𝑆_𝑝,𝑞 is a right child of 𝑆_𝑛,𝑘 in the nested binary tree of supports. If 𝑝=𝑛, necessarily the only nonzero term is for 𝑞=𝑝, that is, 𝒫𝜹_𝑝,𝑞,𝝍_𝑛,𝑘=𝐌^𝑇_𝑛,𝑘𝐠⁻¹𝑚_𝑛,𝑘𝝍𝑚_𝑛,𝑘=𝐌^𝑇_𝑛,𝑘𝐡𝑙_𝑛,𝑘,𝑚_𝑛,𝑘𝐋_𝑛,𝑘=𝝈⁻¹_𝑝,𝑞𝐠𝑚_𝑛,𝑘𝐡l_𝑛,𝑘,𝑚_𝑛,𝑘𝐡𝑙_𝑛,𝑘,𝑚_𝑛,𝑘⁻¹𝐠⁻¹𝑚_𝑛,𝑘𝝈_𝑝,𝑞=𝐈_𝑑.(3.43) If 𝑝>𝑛, 𝒫(𝜹_𝑛,𝑘,𝝍_𝑝,𝑞) can only be nonzero if the support 𝑆_𝑛,𝑘 is included in 𝑆_𝑝,𝑞, but then 𝝍_𝑛,𝑘 is zero in 𝑙_𝑝,𝑞, 𝑚_𝑝,𝑞, 𝑟_𝑝,𝑞 so that 𝒫(𝜹_𝑛,𝑘,𝝍_𝑝,𝑞)=𝟎.
Otherwise, if 𝑛=0 and 𝑝>0, we directly have 𝒫𝜹_𝑝,𝑞,𝝍_0,0=𝐌^𝑇_𝑝,𝑞𝐠⁻¹𝑚_𝑝,𝑞𝝍_0,0𝑚_𝑝,𝑞−𝐋^𝑇_𝑝,𝑞𝐠⁻¹𝑙_𝑝,𝑞𝝍_0,0𝑙_𝑝,𝑞+𝐑^𝑇_𝑝,𝑞𝐠⁻¹𝑟_𝑝,𝑞𝝍_0,0𝑟_𝑝,𝑞,=−𝐋^𝑇_𝑝,𝑞𝐡𝑙_𝑝,𝑞,𝑚_𝑝,𝑞+𝐑^𝑇_𝑝,𝑞𝐡𝑚_𝑝,𝑞,𝑟_𝑝,𝑞𝐋_0,0,=−𝝈^𝑇_𝑝,𝑞𝐠⁻¹𝑚_𝑝,𝑞^𝑇+𝝈^𝑇_𝑝,𝑞𝐠⁻¹𝑚_𝑝,𝑞^𝑇𝐋_𝑛,𝑘,=𝟎.(3.44) Finally, if 𝑝=0, given the simple form of 𝜹_0,0 with a single Dirac function centered in 𝑟_0,0, we clearly have 𝒫(𝜹_0,0,𝝍_𝑛,𝑘)=0, and if 𝑛>0𝒫𝜹_0,0,𝝍_0,0,=𝐋^𝑇_0,0𝐡𝑙_0,0,𝑟_0,0𝐋_0,0,=𝝈^𝑇_0,0𝐠⁻¹𝑟_0,0^𝑇𝐋_0,0,=𝝈^𝑇_0,0𝐠⁻¹𝑟_0,0^𝑇𝐡𝑙_0,0,𝑟_0,0𝐋_0,0⁻¹𝐠⁻¹𝑟_0,0𝝈_0,0,=𝝈^𝑇_0,0𝐂⁻¹_𝑟0,0𝝈_0,0,(3.45) and using the fact that (by definition of 𝝈_0,0) we have 𝝈_0,0⋅𝝈^𝑇_0,0=𝐂_𝑟0,0, this last expression is equal to 𝒫𝜹_0,0,𝝍_0,0=𝝈^𝑇_0,0𝝈^𝑇_0,0⁻¹⋅𝝈_0,0⁻¹𝝈_0,0=𝐼_𝑑,(3.46) which completes the proof.

This proposition directly implies the main result of the section.

Theorem 3.9. The collection of functions (𝝍_𝑛,𝑘;(𝑛,𝑘)∈𝐼) constitutes a Schauder basis of functions on 𝐶₀(𝑈,ℝ^𝑑), that is, any element of 𝐶₀(𝑈,ℝ^𝑑) can be written in a unique way as a sum of coefficients 𝑎_𝑛,𝑘 multiplied by 𝝍_𝑛,𝑘.

This theorem provides us with a complementary view of stochastic processes: in addition to the standard sample paths view, this structure allows to see the Gauss-Markov processes as coefficients on the computed basis. This duality is developed in the sequel.

3.2. The Sample Paths Space

3.2.1. The Construction Application

The Schauder basis of functions with compact supports constructed allows to define functions by considering the coefficients on this basis, which constitute sequences of real numbers in the space_𝜉Ω=𝝃=𝝃_𝑛,𝑘_𝐼;∀(𝑛,𝑘)∈ℐ,𝝃_𝑛,𝑘∈ℝ^𝑑=ℝ^𝑑^ℐ.(3.47) We equip _𝜉Ω with the uniform norm ‖𝝃‖_∞=sup_{(𝑛,𝑘)∈ℐ}|𝝃_𝑛,𝑘|, where we write |𝝃_𝑛,𝑘|=sup_{0≤𝑖<𝑑}|(𝝃_𝑛,𝑘)_𝑖|. We denote by ℬ(_𝜉Ω) the Borelian sets of the topology induced by the uniform norm and we recall that 𝐶(_𝜉Ω), the cylinder sets of _𝜉Ω, form a generative family of Borelian sets. Remark that not any sequence of coefficients provides a continuous function, and one needs to assume a certain decrease in the coefficients to get convergence. A sufficient condition to obtain convergent sequences is to consider coefficients in the space_𝜉Ω^=𝝃∈_𝜉Ω∣∃𝛿∈(0,1),∃𝑁∈ℕ,∀(𝑛,𝑘)∈ℐ⧵ℐ_𝑁,||𝝃_𝑛,𝑘||<2^𝑛𝛿/2.(3.48) This set is clearly a Borelian set of _𝜉Ω since it can be written as a countable intersection and union of cylinder, namely, by denoting by 𝒥 the set of finite subset of ℕ and 𝛿_𝑝=1−1/𝑝, 𝑝>1,_𝜉Ω^=_𝑝>1_𝐽∈𝒥_{𝑛∈ℕ⧵𝐽}𝝃∈_𝜉Ω∣max_{0≤𝑘<2𝑛−1}||𝝃_𝑛,𝑘||<2^{𝑛𝛿𝑝/2}.(3.49) It is also easy to verify that it forms a vectorial subspace of _𝜉Ω.

After these definitions, we are in position to introduce the following useful function.

Definition 3.10. One denotes by Ψ^𝑁 the partial construction application: 𝚿^𝑁=⎧⎪⎨⎪⎩^𝜉Ω,⟶𝐶₀[0,1],ℝ^𝑑,𝝃,⟼_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘(𝑡)⋅𝝃_𝑛,𝑘,(3.50) where the 𝐶₀([0,1],ℝ^𝑑) is the 𝑑-dimensional Wiener space, which is complete under the uniform norm ‖𝐱‖_∞=sup_0≤𝑡≤1|𝐱(𝑡)|.

This sequence of partial construction applications is shown to converge to the construction application in the following.

Proposition 3.11. For every 𝝃 in _𝜉Ω^, Ψ^𝑁(𝝃) converges uniformly toward a continuous function in 𝐶₀([0,1],ℝ^𝑑). One will denote this function Ψ(𝝃), defined as 𝚿∶⎧⎪⎨⎪⎩^𝜉Ω^⟶𝐶₀[0,1],ℝ^𝑑,𝝃⟼_{(𝑛,𝑘)∈ℐ}𝜓_𝑛,𝑘(𝑡)⋅𝝃_𝑛,𝑘,(3.51) and this application will be referred to as the construction application.

This proposition is proved in Appendix D. The image of this function constitutes a subset of the Wiener space continuous functions 𝐶₀([0,1],ℝ^𝑑). Let us now define the vectorial subspace _𝑥Ω^=Ψ(_𝜉Ω^) of 𝐶₀([0,1],ℝ^𝑑) so that Ψ appears as a bijection.

It is important to realize that, in the multidimensional case, the space _𝑥Ω^ depends on Γ and 𝜶 in a nontrivial way. For instance, assuming that 𝜶=0, the space _𝑥Ω^ depends obviously crucially on the rank of Γ. To fix the idea, for a given constant √Γ(𝐭)=[0,0⋯1]^𝑇 in ℝ^𝑑×1, we expect the space _𝑥Ω^ to only include sample paths of 𝐶₀([0,1],ℝ^𝑑) for which the 𝑛−1 first components are constant. Obviously, a process with such sample paths is degenerated in the sense that its covariance matrix is not invertible.

Yet, if we additionally relax the hypothesis that 𝜶≠0, the space _𝑥Ω^ can be dramatically altered: if we take𝜶(𝑡)=⎡⎢⎢⎢⎢⎢⎢⎣01⋱⋱⋱10⎤⎥⎥⎥⎥⎥⎥⎦,(3.52) the space _𝑥Ω^ will represent the sample space of the 𝑑−1-integrated Wiener process, a nondegenerate 𝑑-dimensional process we fully develop in the example section.

However, the situation is much simpler in the one-dimensional case: because the uniform convergence of the sample paths is preserved as long as 𝛼 is continuous and Γ is nonzero through (D.8), the definition _𝑥Ω^ does not depend on 𝛼 or Γ. Moreover, in this case, the space _𝑥Ω^ is large enough to contain reasonably regular functions as proved in Appendix D, Proposition 3.

In the case of the 𝑑−1-integrated Wiener process, the space _𝑥Ω^ clearly contains the functions {𝐟=(𝑓_𝑑−1,…,𝑓₀)∣𝑓₀∈𝐻,𝑓^_𝑖=𝑓_𝑖−1,0<𝑖<𝑑}.

This remark does not hold that the space _𝑥Ω^ does not depend on 𝛼 as long as 𝛼 is continuous because the uniform convergence of the sample paths is preserved through the change of basis of expansion 𝜓_𝑛,𝑘 through (D.8).

We equip the space _𝑥Ω^ with the topology induced by the uniform norm on 𝐶₀([0,1],ℝ^𝑑). As usual, we denote ℬ(_𝑥Ω^) the corresponding Borelian sets. We prove in Appendix D the following.

Proposition 3.12. The function Ψ∶(_𝜉Ω^,ℬ(_𝜉Ω^))→(_𝑥Ω^,ℬ(_𝑥Ω^)) is a bounded continuous bijection.

We therefore conclude that we dispose of a continuous bijection mapping the coefficients onto the sample paths, Ψ. We now turn to study its inverse, the coefficient application, mapping sample paths on coefficients over the Schauder basis.

3.2.2. The Coefficient Application

In this section, we introduce and study the properties of the following function.

Definition 3.13. One calls coefficient application and denotes by Ξ the function defined by 𝚵∶𝐶₀[0,1],ℝ^𝑑⟶_𝜉Ω=ℝ^𝑑^ℐ,𝐱⟼𝚫(𝐱)={𝚫(𝐱)}_{(𝑛,𝑘)∈ℐ}with{𝚫(𝐱)}_𝑛,𝑘=𝒫𝜹_𝑛,𝑘,𝐱.(3.53)

Should a function 𝑥 admit a uniformly convergent decomposition in terms of the basis of elements 𝝍_𝑛,𝑘, the function Δ gives its coefficients in such a representation. More precisely, we have the following.

Theorem 3.14. The function Δ∶(_𝑥Ω^,ℬ(_𝑥Ω^))→(_𝜉Ω^,ℬ(_𝜉Ω^)) is a measurable linear bijection whose inverse is Ψ=Δ⁻¹.

The proof of this theorem is provided in Appendix D.

4. Representation of Gauss-Markov Processes

4.1. Inductive Construction of Gauss-Markov Processes

Up to this point, we have rigorously defined the dual spaces of sample paths _𝑥Ω^ and coefficients _𝜉Ω^. Through the use of the Schauder basis 𝝍_𝑛,𝑘 and its dual family of generalized functions 𝜹_𝑛,𝑘, we have defined the inverse measurable bijections Ψ and Δ transforming one space into the other. In doing so, we have unraveled the fundamental role played by the underlying orthonormal basis 𝜙_𝑛,𝑘. We now turn to use this framework to formulate a pathwise construction of the Gauss-Markov processes in the exact same flavor as the Lévy-Ciesielski construction of the Wiener process.

4.1.1. Finite-Dimensional Approximations

Considering the infinite-dimensional subspace _𝑥Ω^ of 𝐶₀([0,1],ℝ^𝑑), let us introduce the equivalence relation ∼_𝑁 as𝐱∼_𝑁𝐲⟺∀𝑡∈𝐷_𝑁,𝐱(𝑡)=𝐲(𝑡).(4.1) We can use the functions Ψ to carry through the structure of ∼_𝑁 on the infinite-dimensional space of coefficients _𝜉Ω^:𝝃∼_𝑁𝜼⟺𝚿(𝜉)∼_𝑁𝚿(𝜼)⟺∀(𝑛,𝑘)∈ℐ_𝑁,𝝃_𝑛,𝑘=𝜼_𝑛,𝑘,(4.2) which clearly entails that 𝐱∼_𝑁𝐲 if and only if Δ(𝐱)∼_𝑁Δ(𝐲). We denote the sets of equivalence classes of _𝑥Ω^/∼_𝑁=_𝑥Ω_𝑁 and _𝜉Ω^/∼_𝑁=_𝜉Ω_𝑁, which are both clearly isomorphic _𝑥Ω_𝑁=(ℝ^𝑑)^ℐ=_𝜉Ω_𝑁. For every 𝑁>0, we define the finite-dimensional operators Ψ_𝑁=_𝑥𝐢_𝑁∘Ψ∘_𝜉𝐩_𝑁 and Δ_𝑁=_𝜉𝐢_𝑁∘Δ∘_𝑥𝐩_𝑁, with the help of the canonical projections _𝜉𝐩_𝑁∶_𝜉Ω^→_𝜉Ω_𝑁, _𝑥𝐩_𝑁∶_𝑥Ω^→_𝑥Ω_𝑁 and the inclusion map _𝜉𝐢_𝑁∶_𝜉Ω_𝑁→_𝜉Ω^, _𝑥𝐢_𝑁∶_𝑥Ω_𝑁→_𝑥Ω^.

The results of the preceding sections straightforwardly extend on the equivalence classes, and in particular we see that the functions Ψ_𝑁∶_𝜉Ω_𝑁→_𝑥Ω_𝑁 and Δ_𝑁∶_𝑥Ω_𝑁→_𝜉Ω_𝑁 are linear finite-dimensional bijections satisfying Ψ_𝑁=Δ_𝑁⁻¹. We write 𝐞={𝐞_𝑝,𝑞}_{(𝑝,𝑞)∈ℐ} (resp., 𝐟={𝐟_𝑝,𝑞}_{(𝑝,𝑞)∈ℐ}), the canonical basis of _𝜉Ω_𝑁 (resp., _𝑥Ω_𝑁) when listed in the recursive dyadic order. In these bases, the matrices Ψ_𝑁 and Δ_𝑁 are lower block triangular. Indeed, denoting Ψ_𝑁 in the natural bases 𝐞={𝐞_𝑝,𝑞}_{(𝑝,𝑞)∈ℐ} and 𝐟={𝐟_𝑝,𝑞}_{(𝑝,𝑞)∈ℐ} by𝚿_𝑁=𝝍_𝑛,𝑘𝑚_𝑖,𝑗=𝚿^𝑖,𝑗_𝑛,𝑘,(4.3) where Ψ^𝑖,𝑗_𝑛,𝑘 is a 𝑑×𝑑 matrix, the structure of the nested support 𝑆_𝑛,𝑘 entails the block-triangular structure (where only possibly nonzero coefficients are written):_𝛼Ψ_𝑁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝝍^0,0_0,0𝝍^1,0_0,0𝝍^1,0_1,0𝝍^2,0_0,0𝝍^2,0_1,0𝝍^2,0_2,0𝝍^2,1_0,0𝝍^2,1_1,0𝝍^2,1_2,1𝝍^3,0_0,0𝝍^3,0_1,0𝝍^3,0_2,0𝝍^3,0_3,0𝝍^3,1_0,0𝝍^3,1_1,0𝝍^3,1_2,0𝝍^3,1_3,1𝝍^3,2_0,0𝝍^3,2_1,0𝝍^3,2_2,1𝝍^3,2_3,2𝝍^3,3_0,0𝝍^3,3_1,0𝝍^3,3_2,1𝝍^3,3_3,3⋮⋱⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(4.4)

Similarly, for the matrix representation of Δ_𝑁 in the natural bases 𝑒_𝑛,𝑘 and 𝑓_𝑖,𝑗𝚫_𝑁=𝚫^𝑛,𝑘_𝑖,𝑗(4.5) proves to have the following triangular form:𝚫_𝑁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐠⁻¹𝑡_0,0^𝑇𝐌_0,0−𝐠⁻¹𝑡_0,0^𝑇𝐑_1,0𝐠⁻¹𝑡_1,0^𝑇𝐌_1,0−𝐠⁻¹𝑡_0,0^𝑇𝐑_2,0𝐠⁻¹𝑡_1,0^𝑇𝐌_2,0−𝐠⁻¹𝑡_0,0^𝑇𝐑_2,1−𝐠⁻¹𝑡_1,0^𝑇𝐋_2,1𝐠⁻¹𝑡_2,1^𝑇𝐌_2,1⋮⋱⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(4.6) The duality property, Proposition 3.8, simply reads for all 0≤𝑛<𝑁 and 0≤𝑘<2^𝑛−1, 0≤𝑝<𝑁 and 0≤𝑝<2^𝑞−1𝒫𝜹_𝑝,𝑞,𝝍_𝑛,𝑘=_{(𝑛,𝑘)∈ℐ𝑁}𝚫^𝑝,𝑞_𝑖,𝑗⋅𝚿^𝑖,𝑗_𝑛,𝑘=𝛿^𝑝,𝑞_𝑛,𝑘𝐈_𝑑,(4.7) that is, Δ_𝑁⋅Ψ_𝑁=𝐼𝑑_𝜉Ω𝑁. But because we are now in a finite-dimensional setting, we also have Ψ_𝑁⋅Δ_𝑁=𝐼𝑑_𝑥Ω𝑁:𝛿^𝑖,𝑗_𝑘,𝑙𝐈_𝑑=_{(𝑝,𝑞)∈ℐ𝑁}𝚿^𝑖,𝑗_𝑝,𝑞⋅𝚫^𝑝,𝑞_𝑘,𝑙.(4.8) Realizing that 𝛿^𝑖,𝑗_𝑘,𝑙𝐈_𝑑 represents the class of functions 𝐱 in _𝑥Ω^ whose values are zero on every dyadic point of 𝐷_𝑁 except for 𝐱(𝑙2^𝑘)=𝐈_𝑑, {Δ^𝑝,𝑞_𝑘,𝑙}_{(𝑝,𝑞)∈ℐ𝑁} clearly appear as the coefficients of the decomposition of such functions in the bases 𝝍_𝑝,𝑞 for (𝑝,𝑞) in ℐ_𝑁.

Denoting Ξ={Ξ_𝑛,𝑘}_{(𝑛,𝑘)∈𝐼}, a set of independent Gaussian variables of law 𝒩(𝟎,𝐈_𝑑) on (Ω,ℱ,𝐏), and for all 𝑁>0, we form the finite dimensional Gauss-Markov vector [𝐗^𝑁_𝑖,𝑗]_{(𝑖,𝑗)∈ℐ𝑁} as𝐗^𝑁_𝑖,𝑗=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘𝑚_𝑖,𝑗⋅𝚵_𝑛,𝑘,(4.9) which, from Corollary 2.5, has the same law as [𝐗_𝑡]_{𝑡∈𝐷𝑁}, the finite-dimensional random vector obtained from sampling 𝐗 on 𝐷_𝑁 (modulo a permutation on the indices). We then prove the following lemma that sheds light on the meaning of the construction.

Lemma 4.1. The Cholesky decomposition of the finite-dimensional covariance block matrix 𝚺_𝑁 is given by 𝚺_𝑁=Ψ_𝑁⋅Ψ_𝑁^𝑇.

Proof. For every 0≤𝑡,𝑠≤1, we compute the covariance of the finite-dimensional process 𝐗^𝑁 as 𝐂_𝑁(𝑡,𝑠)=𝔼𝐗^𝑁_𝑡⋅𝐗^𝑁_𝑠^𝑇=^𝑁_𝑛=0_{0≤𝑘<2𝑛−1}𝝍_𝑛,𝑘(𝑡)⋅𝝍_𝑛,𝑘(𝑠)^𝑇.(4.10) From there, we write the finite-dimensional covariance block matrix 𝚺_𝑁 in the recursively ordered basis 𝐟_𝑖,𝑗 for 0≤𝑖≤𝑁, 0≤𝑗<2^𝑖−1, as 𝚺_𝑁^𝑖,𝑗_𝑘,𝑙=𝐂_𝑁𝑚_𝑖,𝑗,𝑚_𝑘,𝑙=^𝑁_𝑛=0_{0≤𝑘<2𝑛−1}𝚿^𝑖,𝑗_𝑛,𝑘⋅𝝍^𝑘,𝑙_𝑛,𝑘.(4.11) We already established that the matrix Ψ_𝑁 was triangular with positive diagonal coefficient, which entails that the preceding equality provides us with the Cholesky decomposition of 𝚺.

In the finite-dimensional case, the inverse covariance or potential matrix is a well-defined quantity and we straightforwardly have the following corollary.

Corollary 4.2. The Cholesky decomposition of the finite-dimensional inverse covariance matrix 𝚺⁻¹_𝑁 is given by 𝚺⁻¹_𝑁=Δ_𝑁^𝑇⋅Δ_𝑁.

Proof. The result stems for the equalities 𝚺⁻¹_𝑁=(Ψ_𝑁⋅Ψ_𝑁^𝑇)⁻¹=(Ψ⁻¹_𝑁)^𝑇⋅Ψ⁻¹_𝑁=Δ_𝑁^𝑇⋅Δ_𝑁.

4.1.2. The Lévy-Ciesielski Expansion

We now show that, asymptotically, the bases 𝝍_𝑛,𝑘 allow us to faithfully build the Gauss-Markov process from which we have derived its expression. In this perspective we consider Ξ={Ξ_𝑛,𝑘}_{(𝑛,𝑘)∈ℐ}, a set of independent Gaussian variables of law 𝒩(𝟎,𝐈_𝑑) on (Ω,ℱ,𝐏), and, for all 𝑁>0, we form the finite-dimensional continuous Gaussian process 𝐙^𝑁, defined for 0≤𝑡≤1 by𝐗^𝑁_𝑡=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘(𝑡)⋅𝚵_𝑛,𝑘,(4.12) which, from the result of Theorem 2.4, has the same law 𝐙^𝑁_𝑡=𝔼[𝑋_𝑡∣ℱ_𝑁]. We prove the following lemma.

Lemma 4.3. The sequence of processes 𝐗^𝑁 almost surely converges towards a continuous Gaussian process denoted by 𝐗^∞.

Proof. For all fixed 𝑁>0 and for any 𝜔 in Ω, we know that 𝑡↦𝐗^𝑁_𝑡(𝜔) is continuous. Moreover, we have established, that, for every 𝝃 in _𝜉Ω^, 𝐗^𝑁(𝝃) converges uniformly in 𝑡 toward a continuous limit denoted by 𝐗^𝑁(𝝃). Therefore, in order to prove that lim_𝑁→∞𝐗^𝑁 defines almost surely a process 𝐗 with continuous paths, it is sufficient to show that 𝐏_𝜉(_𝜉Ω^)=1, where 𝐏_𝜉=𝐏_Ξ−1 is the Ξ-induced measure on _𝜉Ω, which stems from a classical Borel-Cantelli argument. For 𝜉 a random variable of normal law 𝒩(0,1), and 𝑎>0, we have 𝐏||𝜉||>𝑎=2𝜋^∞_𝑎𝑒^−𝑢2/2𝑑𝑢≤2𝜋^∞_𝑎𝑢𝑎𝑒^−𝑢2/2𝑑𝑢=2𝜋𝑒^−𝑎2/2𝑎.(4.13) Then, for any 𝛿>0𝐏_𝜉max_{0≤𝑘<2𝑛−1}||𝝃_𝑛,𝑘||_∞>2^𝑛𝛿/2≤𝑑2^𝑛𝐏||𝜉||>𝑑2^𝑛𝛿/2=2𝜋2^{(1−𝛿/2)𝑛}exp−2^𝑛𝛿−1.(4.14) Since the series ^∞_𝑛=02𝜋2^{(1−𝛿/2)𝑛}exp−2^𝑛𝛿−1(4.15) is convergent, the Borel-Cantelli argument implies that 𝐏_𝜉(_𝜉Ω^)=1. Eventually, the continuous almost-sure limit process 𝐗^∞_𝑡 is Gaussian as a countable sum of Gaussian processes.

Now that these preliminary remarks have been made, we can evaluate, for any 𝑡 and 𝑠 in [0,1], the covariance of 𝐗 as the limit of the covariance of 𝐗^𝑁.

Lemma 4.4. For any 0≤𝑡,𝑠≤1, the covariance of 𝐗^∞={𝐗^∞_𝑡=Ψ_𝑡∘Ξ;0≤𝑡≤1} is 𝐂(𝑡,𝑠)=𝔼𝐗^∞_𝑡⋅𝐗^∞_𝑠^𝑇=𝐠(𝑡)𝐡(𝑡∧𝑠)𝐠(𝑠)^𝑇.(4.16)

Proof. As Ξ_𝑛,𝑘 are independent Gaussian random variables of normal law 𝒩(𝟎,𝐈_𝑑), we see that the covariance of 𝐗^𝑁 is given by 𝐂^𝑁(𝑡,𝑠)=𝔼𝐗^𝑁_𝑡⋅𝐗^𝑁_𝑠^𝑇=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘(𝑡)⋅𝝍_𝑛,𝑘(𝑠)^𝑇.(4.17) To compute the limit of the right-hand side, we need to remember that the element of the bases 𝝍_𝑛,𝑘 and the functions 𝜙_𝑛,𝑘 are linked by the following relation: 𝝍_𝑛,𝑘(𝑡)=𝒦𝜙_𝑛,𝑘=𝐠(𝑡)_𝑈𝟙_[0,𝑡](𝑠)𝐟(𝑠)𝜙_𝑛,𝑘(𝑠)𝑑𝑠,(4.18) from which we deduce 𝐂^𝑁(𝑡,𝑠)=𝐠(𝑡)⎛⎜⎝_(𝑛,𝑘_)∈ℐ𝑁_𝑈𝟙_[0,𝑡](𝑢)𝐟(𝑢)𝜙_𝑛,𝑘(𝑢)𝑑𝑢_𝑈𝟙_[0,𝑠](𝑣)𝐟(𝑣)𝜙_𝑛,𝑘(𝑣)𝑑𝑣^𝑇⎞⎟⎠𝐠(𝑠)^𝑇.(4.19) Defining the auxiliary ℝ^𝑑×𝑑-valued function 𝜿_𝑛,𝑘(𝑡)=_𝑈𝟙_[0,𝑡](𝑢)𝐟(𝑢)𝜙_𝑛,𝑘(𝑢)𝑑𝑢,(4.20) we observe that the (𝑖,𝑗)-coefficient function reads 𝜿_𝑛,𝑘_𝑖,𝑗(𝑡)=_𝑈𝟙_[0,𝑡](𝑢)𝑙_𝑖(𝐟(𝑢))^𝑇⋅𝑐_𝑗𝜙_𝑛,𝑘(𝑢)𝑑𝑢=_𝑈𝟙_[0,𝑡](𝑢)𝑐_𝑖𝐟^𝑇(𝑢)^𝑇⋅𝑐_𝑗𝜙_𝑛,𝑘(𝑢)𝑑𝑢,(4.21) where 𝟙_[0,𝑡] is the real function that is one if 0≤𝑢≤𝑡 and zero otherwise. As we can write 𝟙_[0,𝑡](𝑢)𝑐_𝑖𝐟^𝑇(𝑢)=𝐟^𝑇(𝑢)⋅⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣0⋮𝟙_[0,𝑡](𝑢)⋮0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⟵𝑖,(4.22) we see that the function 𝐟_𝑖,𝑡=𝟙_[0,𝑡]𝑐_𝑖(𝐟^𝑇) belongs to 𝐿²_𝐟, so that we can write (𝜿_𝑛,𝑘)_𝑖,𝑗(𝑡) as a scalar product in the Hilbert space 𝐿²_𝐟: 𝜿_𝑛,𝑘_𝑖,𝑗(𝑡)=_𝑈𝐟^𝑇_𝑖,𝑡(𝑢)⋅𝑐_𝑗𝜙_𝑛,𝑘(𝑢)𝑑𝑢=𝐟_𝑖,𝑡,𝑐_𝑗𝜙_𝑛,𝑘.(4.23) We then specify the (𝑖,𝑗)-coefficient of 𝐠⁻¹(𝑡)𝐂^𝑁(𝑡,𝑠)(𝐠⁻¹(𝑠))^𝑇 writing _(𝑛,𝑘_)∈ℐ𝑁𝜿(𝑡)⋅𝜿(𝑠)^𝑇_𝑖,𝑗=_{(𝑛,𝑘)∈ℐ𝑁}^𝑑−1_𝑝=0𝐟_𝑖,𝑡,𝑐_𝑗𝜙_𝑛,𝑘𝐟_𝑗,𝑠,𝑐_𝑗𝜙_𝑛,𝑘,(4.24) and, remembering that the family of functions 𝑐_𝑗(𝜙_𝑛,𝑘) forms a complete orthonormal system of 𝐿²_𝐟, we can use the Parseval identity, which reads _(𝑛,_𝑘)∈ℐ𝜿(𝑡)⋅𝜿(𝑠)^𝑇_𝑖,𝑗=𝐟_𝑖,𝑡,𝐟_𝑗,𝑠=_𝑈𝟙_[0,𝑡](𝑢)𝑐_𝑖𝐟^𝑇(𝑢)^𝑇⋅𝟙_[0,𝑠](𝑢)𝑐_𝑗𝐟^𝑇(𝑢)𝑑𝑢=^𝑡∧𝑠₀𝐟⋅𝐟^𝑇_𝑖,𝑗(𝑢)𝑑𝑢.(4.25) Thanks to this relation, we can conclude the evaluation of the covariance since lim_𝑁→∞𝐂^𝑁(𝑡,𝑠)=𝐠(𝑡)^𝑡∧𝑠₀𝐟⋅𝐟^𝑇(𝑢)𝑑𝑢𝐠(𝑠)^𝑇=𝐠(𝑡)𝐡(𝑡∧𝑠)𝐠(𝑠)^𝑇.(4.26)

We stress the fact that the relation𝐂(𝑡,𝑠)=_{(𝑛,𝑘)∈ℐ}𝝍_𝑛,𝑘(𝑡)⋅𝝍_𝑛,𝑘(𝑠)^𝑇=𝚿(𝑡)∘𝚿^𝑇(𝑠)(4.27) provides us with a continuous version of the Cholesky decomposition of the covariance kernel 𝐂. Indeed, if we chose 𝝈_𝑛,𝑘 as the Cholesky square root of 𝚺_𝑛,𝑘, we remark that the operators Φ are triangular in the following sense: consider the chain of nested vectorial spaces {𝐹_𝑛,𝑘}_{(𝑛,𝑘)∈ℐ}𝐹_0,0⊂𝐹_1,0⊂𝐹_2,0⊂𝐹_2,1⋯⊂𝐹_𝑛,0⊂⋯⊂𝐹_{𝑛,2𝑛−1}⋯⊂_𝜉Ω^(4.28) with 𝐹_𝑛,𝑘=span{𝐟_𝑖,𝑗∣0≤𝑖≤𝑛,0≤𝑗≤𝑘}; then, for every (𝑛,𝑘) in ℐ, the operator Ψ transforms the chain {𝐹_𝑛,𝑘}_{(𝑛,𝑘)∈ℐ} into the chainℰ_0,0⊂ℰ_1,0⊂ℰ_2,0⊂ℰ_2,1⋯⊂ℰ_𝑛,0⊂⋯⊂ℰ_{𝑛,2𝑛−1}⋯⊂_𝑥Ω^(4.29) with ℰ_𝑛,𝑘=span{Ψ_𝑖,𝑗∣0≤𝑖≤𝑛,0≤𝑗≤𝑘}.

The fact that this covariance is equal to the covariance of the process 𝐗, solution of (2.1) implies that we have the following fundamental result.

Theorem 4.5. The process 𝐗^∞ is equal in law to the initial Gauss-Markov process 𝐗 used to construct the basis of functions.

Remark 4.6. Our multiresolution representation of the Gauss-Markov processes appears to be the direct consequence of the fact that, because of the Markov property, the Cholesky decomposition of the finite-dimensional covariance admits a simple inductive continuous limit. More generally, triangularization of the kernel operators has been studied in depth [28, 40–42], and it would be interesting to investigate if these results make possible a similar multiresolution approach for non-Markov Gaussian processes. In this regard, we naturally expect to lose the compactness of the supports of a putative basis.

Remark 4.7. We eventually underline the fact that large deviations related to this convergence can be derived through the use of the Baldi and Caramellino good rate function related to the Gaussian pinned processes [43, 44].

4.2. Optimality Criterion of the Decomposition

In the following, we draw from the theory of interpolating splines to further characterize the nature of our proposed basis for the construction of the Gauss-Markov processes. Essentially adapting the results from the previous works [45, 46], we first show that the finite-dimensional sample paths of our construction induce a nested sequence ℰ_𝑁 of the reproducing Hilbert kernel space (RKHS). In turn, the finite-dimensional process 𝐗^𝑁 naturally appears as the orthogonal projection of the infinite-dimensional process 𝐗 onto ℰ_𝑁. We then show that such an RKHS structure allows us to define a unicity criterion for the finite-dimensional sample path as the only functions of ℰ that minimize a functional, called Dirichlet energy, under constraint of interpolation on 𝐷_𝑁 (equivalent to conditioning on the times 𝐷_𝑁). In this respect, we point out that the close relation between the Markov processes and the Dirichlet forms is the subject of a vast literature, largely beyond the scope of the present paper (see, e.g., [18]).

4.2.1. Sample Paths Space as a Reproducing Hilbert Kernel Space

In order to define the finite-dimensional sample paths as a nested sequence of RKHSs, let us first define the infinite-dimensional operator𝚽∶⎧⎪⎨⎪⎩𝑙²_𝜉Ω⟼𝐿²_𝐟,𝝃⟼𝚽[𝝃]=⎧⎨⎩𝑡⟼_{(𝑛,𝑘)∈ℐ}𝜙_𝑛,𝑘(𝑡)⋅𝝃_𝑛,𝑘⎫⎬⎭.(4.30) Since we know that the column functions of 𝜙_𝑛,𝑘 form a complete orthonormal system of 𝐿²_𝐟, the operator Φ is an isometry and its inverse satisfies Φ⁻¹=Φ^𝑇, which reads for all 𝐯 in 𝐿²_𝐟𝚽⁻¹[𝐯]_𝑛,𝑘=_𝑈𝜙^𝑇_𝑛,𝑘(𝑡)⋅𝐯(𝑡)𝑑𝑡=𝒫𝜙_𝑛,𝑘,𝐯.(4.31) Equipped with this infinite-dimensional isometry, we then consider the linear operator ℒ=Φ∘Δ suitably defined on the setℰ=𝐮∈𝐶₀𝑈,ℝ^𝑑∣ℒ[𝐮]∈𝐿²_𝐟=𝐮∈𝐶₀𝑈,ℝ^𝑑∣𝚫[𝐮]∈𝑙²_𝜉Ω(4.32) with ‖𝝃‖₂²=∑_{𝑛,𝑘∈ℐ}|𝝃_𝑛,𝑘|₂², the 𝑙² norm of _𝜉Ω. The set ℰ form an infinite-dimensional vectorial space that is naturally equipped with the inner product∀(𝐮,𝐯)∈ℰ²,⟨𝐮,𝐯⟩=_𝑈ℒ[𝐮](𝑡)^𝑇⋅ℒ[𝐯](𝑡)𝑑𝑡=(ℒ[𝐮],ℒ[𝐯]).(4.33) Moreover since 𝐮(0)=𝐯(0)=𝟎, such an inner product is definite positive and, consequently, ℰ forms an Hilbert space.

Remark 4.8. Two straightforward remarks are worth making. First, the space ℰ is strictly included in the infinite-dimensional sample paths space _𝑥Ω^. Second, notice that, in the favorable case 𝑚=𝑑, if 𝐟 is everywhere invertible with continuously differentiable inverse, we have ℒ=𝒟=𝒦⁻¹. More relevantly, the operator ℒ can actually consider a first-order differential operator from ℰ to 𝐿²_𝐟 as a general left inverse of the integral operator 𝒦. Indeed, realizing that on 𝐿²_𝐟,𝒦 can be expressed as 𝒦=Ψ∘Φ⁻¹, we clearly have ℒ∘𝒦=𝚽∘𝚫∘𝚿∘𝚽⁻¹=𝐼𝑑_𝐿²_𝐟.(4.34)

We know motivate the introduction of the Hilbert space ℰ by the following claim.

Proposition 4.9. The Hilbert space (ℰ,⟨,⟩) is a reproducing kernel Hilbert space (RKHS) with ℝ^𝑑×𝑑-valued reproducing kernel 𝐂, the covariance function of the process 𝐗.

Proof. Consider the problem of finding all elements 𝐮 of ℰ solution of the equation ℒ[𝐮]=𝐯 for 𝐯 in 𝐿²_𝐟. The operator 𝒦 provides us with a continuous ℝ^𝑑×𝑚-valued kernel function 𝐤: ∀(𝑡,𝑠)∈𝑈²,𝐤(𝑡,𝑠)=𝟙_[0,𝑡](𝑠)𝐠(𝑡)⋅𝐟(𝑠),(4.35) which is clearly the Green function for our differential equation. This entails that the following equalitiy holds for every 𝐮 in ℰ: 𝐮(𝑡)=_𝑈𝐤(𝑡,𝑠)ℒ[𝐮](𝑠)𝑑𝑠.(4.36) Moreover, we can decompose the kernel 𝐤 in the 𝐿²_𝐟 sense as 𝐤(𝑡,𝑠)=_{(𝑛,𝑘)∈ℐ}𝝍_𝑛,𝑘(𝑡)⋅𝜙^𝑇_𝑛,𝑘(𝑠)(4.37) since we have 𝐤(𝑡,𝑠)=𝒦𝛿_𝑠𝐼𝑑_𝐿²_𝐟(𝑡)=𝒦⎡⎢⎣_(𝑛,_𝑘)∈ℐ𝜙_𝑛,𝑘⋅𝜙^𝑇_𝑛,𝑘(𝑠)⎤⎥⎦(𝑡)=_{(𝑛,𝑘)∈ℐ}𝒦𝜙_𝑛,𝑘(𝑡)⋅𝜙^𝑇_𝑛,𝑘(𝑠),(4.38) with 𝛿_𝑠=𝛿(⋅−𝑠). Then, we clearly have 𝐂(𝑡,𝑠)=_𝑈𝐤(𝑡,𝑢)⋅𝐤(𝑠,𝑢)^𝑇𝑑𝑢=_{(𝑛,𝑘)∈ℐ}𝝍_𝑛,𝑘(𝑡)⋅𝝍^𝑇_𝑛,𝑘(𝑠),(4.39) where we recognize the covariance function of 𝐗, which implies 𝐤(𝑡,𝑠)=_{(𝑛,𝑘)∈ℐ}𝝍_𝑛,𝑘(𝑡)⋅ℒ𝜙_𝑛,𝑘^𝑇(𝑠)=ℒ[𝐂(𝑡,⋅)].(4.40) Eventually, for all 𝐮 in 𝐿²_𝐟, we have 𝐮(𝑡)=_𝑈ℒ[𝐂(𝑡,⋅)](𝑠)⋅ℒ[𝐮](𝑠)𝑑𝑠=𝒫⟨𝐂(𝑡,⋅),𝐮⟩,(4.41) where we have introduced the 𝒫-operator associated with the inner product ⟨,⟩: for all ℝ^𝑑×𝑑-valued functions 𝐀 and 𝐁 defined on 𝑈 such that the columns 𝑐_𝑖(𝐀) and 𝑐_𝑖(𝐁), 0≤𝑖<𝑑, are in ℰ, we define the matrix 𝒫⟨𝐀,𝐁⟩ in ℝ^𝑑×𝑑 by ∀0≤𝑖,𝑗<𝑑,𝒫⟨𝐀,𝐁⟩_𝑖,𝑗=𝑐_𝑖(𝐀),𝑐_𝑗(𝐁).(4.42) By the Moore-Aronszajn theorem [47], we deduce that there is a unique reproducing kernel Hilbert space associated with a given covariance kernel. Thus, ℰ is the reproducing subspace of 𝐶₀(𝑈,ℝ^𝑑) corresponding to the kernel 𝐂, with respect to the inner product ⟨,⟩.

Remark 4.10. From a more abstract point of view, it is well known that the covariance operator of a Gaussian measure defines an associated Hilbert structure [48, 49].

In the sequel, we will use the space ℰ as the ambient Hilbert space to define the finite-dimensional sample paths spaces as a nested sequence of RKHS. More precisely, let us write for ℰ_𝑁 the finite-dimensional subspace of ℰℰ_𝑁=𝐮∈𝐶₀𝑈,ℝ^𝑑∣ℒ[𝐮]∈𝐿²_𝐟,𝑁,(4.43) with the space 𝐿²_𝐟,𝑁 being defined as𝐿²_𝐟,𝑁=span𝑐_𝑖(𝜙_𝑛,𝑘)_{𝑛,𝑘∈ℐ𝑁,0≤𝑖<𝑑}.(4.44)

We refer to such spaces as finite-dimensional approximation spaces since we remark thatℰ_𝑁=span𝑐_𝑖(𝝍_𝑛,𝑘)_{𝑛,𝑘∈ℐ𝑁,0≤𝑖<𝑑}=𝚿_𝑁_𝜉Ω_𝑁,(4.45) which means that the space ℰ_𝑁 is made of the sample space of the finite-dimensional process 𝐗_𝑁. The previous definition makes obvious the nested structure ℰ₀⊂ℰ₁⊂⋯⊂ℰ, and it is easy to characterize each space ℰ_𝑁 as a reproducing Hilbert kernel space.

Proposition 4.11. The Hilbert spaces (ℰ_𝑁,⟨,⟩) are reproducing kernel Hilbert spaces (RKHSs) with ℝ^𝑑×𝑑-valued reproducing kernel 𝐂_𝑁, the covariance function of the process 𝐗_𝑁.

Proof. The proof this proposition follows the exact same argument as that in the case of ℰ, but with the introduction of finite-dimensional kernels 𝐤_𝑁∀(𝑡,𝑠)∈𝑈²,𝐤_𝑁(𝑡,𝑠)=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘(𝑡)⋅𝜙^𝑇_𝑛,𝑘(𝑠),(4.46) and the corresponding covariance function ∀(𝑡,𝑠)∈[0,1]²,𝐂_𝑁(𝑡,𝑠)=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘(𝑡)⋅𝝍^𝑇_𝑛,𝑘(𝑠).(4.47)

4.2.2. Finite-Dimensional Processes as Orthogonal Projections

The framework set in the previous section offers a new interpretation of our construction. Indeed, for all 𝑁>0, the columns of {𝝍_𝑛,𝑘}_{(𝑛,𝑘)∈ℐ𝑁} form an orthonormal basis of the space ℰ_𝑁:𝒫𝝍_𝑛,𝑘,𝝍_𝑝,𝑞=𝒫ℒ𝝍_𝑛,𝑘,ℒ𝝍_𝑝,𝑞=𝒫𝜙_𝑛,𝑘,𝜙_𝑝,𝑞=𝛿^𝑛,𝑘_𝑝,𝑞.(4.48) This leads to defining the finite-dimensional approximation 𝐱_𝑁 of an sample path 𝐱 of ℰ as the orthogonal projection of 𝐱 on ℰ_𝑁 with respect to the inner product ⟨,⟩. At this point, it is worth remembering that the space ℰ is strictly contained in _𝑥Ω^ and does not coincide with _𝑥Ω^: actually one can easily show that 𝐏(ℰ)=𝟎. We devote the rest of this section to defining the finite-dimensional processes 𝐙^𝑁=𝔼_𝑁[𝑋] resulting from the conditioning on 𝐷_𝑁, as pathwise orthogonal projection of the original process 𝐗 on the sample space ℰ_𝑁.

Proposition 4.12. For any 𝑁>0, the conditioned processes 𝔼_𝑁[𝐗] can be written as the orthogonal projection of 𝐗 on ℰ_𝑁 with respect to ⟨,⟩: 𝔼_𝑁[𝐗]=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘⋅𝒫𝝍_𝑛,𝑘,𝐗.(4.49)

The only hurdle to prove Proposition 4.12 is purely technical in the sense that the process 𝐗 exists in a larger space than ℰ: we need to find a way to extend the definition of ⟨,⟩ so that the expression bears a meaning. Before answering this point quite straightforwardly, we need to establish the following lemma.

Lemma 4.13. Writing the Gauss-Markov process 𝐗_𝑡=∫¹₀𝐤(𝑡,𝑠)𝑑𝐖_𝑠, for all 𝑁>0, the conditioned process 𝐙^𝑁=𝔼_𝑁[𝐗] is expressed as the stochastic integral 𝐙^𝑁=¹₀𝐤_𝑁(𝑡,𝑠)𝑑𝐖_𝑠with𝐤_𝑁(𝑡,𝑠)=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘(𝑡)⋅𝜙^𝑇_𝑛,𝑘(𝑠).(4.50)

Proof. In the previous section, we have noticed that the kernel 𝐤_𝑁 converges toward the kernel 𝐤 (the Green function) in the 𝐿²_𝐟 sense 𝐤(𝑡,𝑠)=_{(𝑛,𝑘)∈ℐ}𝝍_𝑛,𝑘(𝑡)⋅𝜙^𝑇_𝑛,𝑘(𝑠)=lim_𝑁→∞_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘(𝑡)⋅𝜙^𝑇_𝑛,𝑘(𝑠)𝐤_𝑁=lim_𝑁→∞𝐤_𝑁(𝑡,𝑠).(4.51) This implies that the process 𝐗 as the stochastic integral can also be written as 𝐗_𝑡=𝐠(𝑡)_𝑈𝟙_[0,𝑡](𝑠)(𝑠)𝑑𝐖_𝑠=¹₀𝐤(𝑡,𝑠)𝑑𝐖_𝑠=lim_𝑁→∞¹₀𝐤_𝑁(𝑡,s)𝑑𝐖_𝑠.(4.52) Specifying the decomposition of 𝐤_𝑁, we can then naturally express 𝐗 as the convergent sum 𝐗_𝑡=_{(𝑛,𝑘)∈ℐ}𝝍_𝑛,𝑘⋅𝚵_𝑛,𝑘with𝚵_𝑛,𝑘=¹₀𝜙^𝑇_𝑛,𝑘(𝑠)𝑑𝐖_𝑠,(4.53) where the orthonormality property of the 𝜙_𝑛,𝑘 with respect to (,) makes the vectors Ξ_𝑛,𝑘 appear as independent 𝑑-dimensional Gaussian variables of law 𝒩(0,𝐈_𝑑). It is then easy to see that by definition of the elements 𝝍_𝑛,𝑘, for almost every 𝜔 in Ω, we then have ∀𝑁>0,0≤𝑡≤1,𝐙^𝑁(𝜔)=𝔼_𝑁[𝐗](𝜔)=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘⋅𝚵_𝑛,𝑘(𝜔),(4.54) and we finally recognize in the previous expression that for all 0≤𝑡≤1𝐙^𝑁_𝑡=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘⋅𝚵_𝑛,𝑘=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘(𝑡)⋅_𝑈𝜙^𝑇_𝑛,𝑘(𝑠)𝑑𝐖_s=¹₀𝐤_𝑁(𝑡,𝑠)𝑑𝐖_𝑠.(4.55)

We can now proceed to justify the main result of Proposition 4.12.

Proof. The finite-dimensional processes 𝐙^𝑁 defined through Lemma 4.13 have sample paths 𝑡↦𝐙^𝑁_𝑡(𝜔) belonging to ℰ_𝑁. Moreover, for almost every 𝜔 in Ω and for all 𝑛,𝑘 in ℐ_𝑁, 𝒫𝝍_𝑛,𝑘,𝐙^𝑁(𝜔)=𝒫𝝍_𝑛,𝑘,¹₀𝐤_𝑁(𝑡,𝑠)𝑑𝐖_𝑠(𝜔)=𝒫𝝍_𝑛,𝑘,_{(𝑝,𝑞)∈𝐼𝑁}𝝍_𝑝,𝑞(𝜔)⋅¹₀𝜙^𝑇_𝑝,𝑞(𝑠)𝑑𝐖_𝑠(𝜔)=¹₀𝜙^𝑇_𝑛,𝑘(𝑠)𝑑𝐖_𝑠(𝜔),(4.56) because of the orthonormality property of 𝝍_𝑛,𝑘 with respect to ⟨,⟩. As the previous equalities hold for every 𝑁>0, the applications 𝐱↦𝒫⟨𝝍_𝑛,𝑘,𝐱⟩ can naturally be extended on _𝑥Ω^ by continuity. Therefore, it makes sense to write, for all (𝑛,𝑘) in ℐ_𝑁, 𝒫⟨𝝍_𝑛,𝑘,𝐙^𝑁⟩=lim_𝑁→∞𝒫⟨𝝍_𝑛,𝑘,𝐙^𝑁⟩=^def𝒫⟨𝝍_𝑛,𝑘,𝐗⟩ even if 𝐗 is defined into a larger sample space than ℰ. In other words, we have 𝒫⟨𝝍_𝑛,𝑘,𝐗⟩=¹₀𝜙^𝑇_𝑛,𝑘(𝑠)𝑑𝐖_𝑠=𝚵_𝑛,𝑘,(4.57) and we can thus express the conditioned process 𝐙^𝑁=𝔼_𝑁[𝐗] as the orthogonal projection of 𝐗 onto the finite sample path ℰ_𝑁 by writing 𝐙^𝑁=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘⋅𝒫𝝍_𝑛,𝑘,𝐗.(4.58)

4.2.3. Optimality Criterion of the Sample Paths

Proposition 4.12 elucidates the structure of the conditioned processes 𝐙_𝑁 as pathwise orthogonal projections of 𝐗 on the finite-dimensional RKHS ℰ_𝑁. It allows us to cast the finite sample paths in a geometric setting and incidentally, to give a characterization of them as the minimizer of some functionals. In doing so, we shed a new light on well-known results of the interpolation theory [50–52] and extend them to the multidimensional case.

The central point of this section reads as follows.

Proposition 4.14. Given a function 𝐱 in ℰ, the function 𝐱_𝑁=(Ψ∘Δ_𝑁)[𝐱] belongs to ℰ_𝑁 and is defined by the following optimal criterion: 𝐱_𝑁 is the only function in ℰ interpolating 𝐱 on 𝐷_𝑁 such that the functional ⟨𝐲,𝐲⟩=‖‖ℒ𝐲(𝑡)‖‖₂²=¹₀||ℒ𝐲(𝑡)||₂²𝑑𝑡(4.59) takes its unique minimal value over ℰ in 𝐱_𝑁.

Proof. The space ℰ_𝑁 has been defined as ℰ_𝑁=Ψ_𝑁[_𝜉Ω_𝑁]=Ψ∘Δ_𝑁[ℰ] so that, for all 𝐱 in ℰ, 𝐱_𝑁 clearly belongs to ℰ_𝑁. Moreover, 𝐱_𝑁 interpolates 𝐱 on 𝐷_𝑁: indeed, we know that the finite-dimensional operators Δ_𝑁 and Ψ⁻¹_𝑁 are inverse of each other Δ_𝑁=Ψ⁻¹_𝑁, which entails that for all 𝑡 in 𝐷_𝑁𝐱_𝑁(𝑡)=𝚿∘𝚫_𝑁[𝐱](𝑡)=𝚿_𝑁∘𝚫_𝑁[𝐱](𝑡)=𝐱(𝑡),(4.60) where we use the fact that, for any 𝝃 in _𝜉Ω^ and for all 𝑡 in 𝐷_𝑁, Ψ_𝑁[𝝃](𝑡)=Ψ[𝝃](𝑡) (recall that 𝝍_𝑛,𝑘(𝑡)=𝟎 if 𝑛>𝑁 and 𝑡 belongs to 𝐷_𝑁).
Let us now show that 𝐱_𝑁 is determined in ℰ by the announced optimal criterion. Suppose 𝐲 belongs to ℰ and interpolates 𝐱 on 𝐷_𝑁, and remark that we can write ⟨𝐲,𝐲⟩=‖ℒ[𝐲]‖₂²=‖(𝚽∘𝚫)[𝐲](𝑡)‖₂²=‖𝚫[𝐲]‖₂²,(4.61) since Φ is an isometry. Then, consider Δ[𝐲] in 𝑙²(_𝜉Ω) and remark that, since for all (𝑛,𝑘) in ℐ_𝑁, 𝜹_𝑛,𝑘 are Dirac measures supported by 𝐷_𝑁, we have ∀(𝑛,𝑘)∈ℐ_𝑁,𝚫_𝑛,𝑘𝐲=𝒫𝜹_𝑛,𝑘,𝐲=𝒫𝜹_𝑛,𝑘,𝐱=𝚫_𝑛,𝑘[𝐱]=𝚫_𝑛,𝑘𝐱_𝑁.(4.62) This entails ‖‖𝚫𝐲‖‖₂²𝑑𝑡=_(𝑛,_𝑘)∈ℐ||𝚫_𝑛,𝑘𝐲||₂²≥_{(𝑛,𝑘)∈ℐ𝑁}||𝚫_𝑛,𝑘𝐲||₂²=‖‖𝚫[𝐱_𝑁]‖‖₂²𝑑𝑡.(4.63) Since, by definition of 𝐱_𝑁, 𝜹_𝑛,𝑘[𝐱_𝑁]=𝟎 if 𝑛>𝑁. Moreover, the minimum ⟨𝐱_𝑁,𝐱_𝑁⟩ is only attained for 𝐲 such that 𝜹_𝑛,𝑘[𝐲]=𝟎 if 𝑛>𝑁 and 𝜹_𝑛,𝑘[𝐲]=𝜹_𝑛,𝑘[𝐱] if 𝑛≤𝑁, which defines univocally 𝐱_𝑁. This establishes that, for all 𝐲 in ℰ such that for all 𝑡 in 𝐷_𝑁, 𝐲(𝑡)=𝐱(𝑡), we have ⟨𝐱_𝑁,𝐱_𝑁⟩≤⟨𝐲,𝐲⟩ and the equality case holds if and only if 𝐲=𝐱_𝑁.

Remark 4.15. When ℒ represents a regular differential operator of order 𝑑, ∑^𝑑_𝑖=1𝑎_𝑖(𝑡)𝐷^𝑖, where 𝐷=𝑑/𝑑𝑡, that is, for 𝑑𝐗_𝑡=𝜶(𝑡)⋅𝐗_𝑡+√𝚪(𝑡)⋅𝑑𝑊_𝑡,(4.64) with 𝜶(𝑡)=⎡⎢⎢⎢⎢⎢⎢⎣01⋱⋱⋱1𝑎_𝑑𝑎_𝑑−1…𝑎₁⎤⎥⎥⎥⎥⎥⎥⎦,√𝚪(𝐭)=⎡⎢⎢⎢⎢⎢⎢⎣00⋮1⎤⎥⎥⎥⎥⎥⎥⎦.(4.65) The finite-dimensional sample paths coincide exactly with the spline interpolation of order 2𝑑+1, which are well known to satisfy the previous criterion [46]. This example will be further explored in the example section.

The Dirichlet energy simply appears as the squared norm induced on ℰ by the inner product ⟨,⟩, which in turn can be characterized as a Dirichlet quadratic form on ℰ. Actually, such a Dirichlet form can be used to define the Gauss-Markov process, extending the Gauss-Markov property to processes indexed on the multidimensional spaces parameter [19]. In particular, for an 𝑛-dimensional parameter space, we can condition such Gauss-Markov processes on a smooth 𝑛−1-dimensional boundary. Within the boundary, the sample paths of the resulting conditioned process (the solution to the prediction problem in [19]) are the solutions to the corresponding Dirichlet problems for the elliptic operator associated with the Dirichlet form.

The characterization of the basis as the minimizer of such a Dirichlet energy (4.59) gives rise to an alternative method to compute the basis as the solution of a Dirichlet boundary value problem for an elliptic differential operator.

Proposition 4.16. Let us assume that 𝜶 and √Γ are continuously differentiable and that √Γ is invertible. Then, the functions 𝝁_𝑛,𝑘 are defined as 𝝁_𝑛,𝑘(𝑡)=⎧⎪⎨⎪⎩𝝁^𝑙(𝑡),𝑡∈𝑙_𝑛,𝑘,𝑚_𝑛,𝑘,𝝁^𝑟(𝑡),𝑡∈𝑚_𝑛,𝑘,𝑟_𝑛,𝑘,𝟎,else,(4.66) where 𝝁^𝑙 and 𝝁^𝑟 are the unique solutions of the second-order 𝑑-dimensional linear differential equation 𝐮^+𝚪⁻¹𝜶^𝑇𝚪−𝚪^−𝜶𝐮^−𝚪⁻¹𝜶^𝑇𝚪−𝚪^𝜶+𝜶^𝐮=𝟎(4.67) with the following boundary value conditions: 𝝁^𝑙𝑙_𝑛,𝑘=𝟎,𝝁^𝑙𝑚_𝑛,𝑘=𝐈_𝑑,𝝁^𝑟𝑚_𝑛,𝑘=𝐈_𝑑,𝝁^𝑟𝑟_𝑛,𝑘=𝟎.(4.68)

Proof. By Proposition 4.14, we know that 𝝁_𝑛,𝑘(𝑡) minimizes the convex functional ¹₀||ℒ[𝐮](𝑠)||₂²𝑑𝑠(4.69) over ℰ, being equal to zero outside the interval [𝑙_𝑛,𝑘,𝑟_𝑛,𝑘] and equal to one at the point 𝑡=𝑚_𝑛,𝑘. Because of the hypotheses on 𝜶 and √Γ, we have ℒ=𝒟 and we can additionally restrain our search to functions that are twice continuously differentiable. Incidentally, we only need to minimize separately the contributions on the interval [𝑙_𝑛,𝑘,𝑚_𝑛,𝑘] and [𝑚_𝑛,𝑘,𝑟_𝑛,𝑘]. On both intervals, this problem is a classical Euler-Lagrange problem (see, e.g., [53]) and is solved using basic principles of calculus of variations. We easily identify the Lagrangian of our problem as 𝐿𝑡,𝐮,𝐮^=|||𝐮^−𝜶(𝑡)𝐮(𝑡)√𝚪(𝐭)⁻¹|||₂²=𝐮^(𝑡)−𝜶(𝑡)𝐮(𝑡)^𝑇(𝚪(𝑡))⁻¹𝐮^(𝑡)−𝜶(𝑡)𝐮(𝑡).(4.70) From there, after some simple matrix calculations, the Euler-Lagrange equations 𝜕𝐿(𝑡,𝐮,𝐮′)𝜕𝑢_𝑖−𝑑𝑑𝑡⎛⎜⎝𝜕𝐿𝑡,𝐮,𝐮^𝜕𝑢′_𝑖⎞⎟⎠=0,𝑖=1,…,𝑑,(4.71) can be expressed under the form: 𝐮^+𝚪⁻¹𝜶^𝑇𝚪−𝚪^−𝜶𝐮^−𝚪⁻¹𝜶^𝑇𝚪−Γ^𝜶+𝜶′𝐮=𝟎,(4.72) which ends the proof.

Remark 4.17. It is a simple matter of calculus to check that the expression of 𝝁 given in Proposition 2.1 satisfies (4.67). Notice also that, in the case Γ=𝐈_𝑑, the differential equation becomes 𝐮^+𝜶^𝑇−𝜶𝐮^−𝜶^𝑇𝜶+𝜶^𝐮=𝟎,(4.73) which is further simplified for constant 𝜶.

Under the hypotheses of Proposition 4.16, we can thus define 𝝁_𝑛,𝑘 as the unique solution to the second-order linear differential equation (4.67) with the appropriate boundary values conditions. From this definition, it is then easy to derive the bases 𝝍_𝑛,𝑘 by completing the following program.(1)Compute the 𝑡↦𝝁_𝑛,𝑘(𝑡) by solving the linear ordinary differential problem.(2)Apply the differential operator 𝒟 to get the functions 𝒟[𝝁_𝑛,𝑘].(3)Orthonormalize the column functions 𝑡↦𝑐_𝑗(𝒟[𝝁_𝑛,𝑘(𝑡)]) by the Gram-Schmidt process.(4)Apply the integral operator 𝒦 to get the desired functions 𝝍_𝑛,𝑘 (or equivalently multiply the original function 𝑡↦𝝁_𝑛,𝑘(𝑡) by the corresponding Gram-Schmidt triangular matrix).

Notice finally that each of these points is easily implemented numerically.

5. Examples: Derivation of the Bases for Some Classical Processes

5.1. One-Dimensional Case

In the one-dimensional case, the construction of the Gauss-Markov process is considerably simplified since we do not have to consider the potential degeneracy of matrix-valued functions. Indeed, in this situation, the centered Gauss-Markov process 𝑋 is solution of the one-dimensional stochastic equation𝑑𝑋_𝑡=𝛼(𝑡)𝑋_𝑡𝑑𝑡+√Γ(𝑡)𝑑𝑊_𝑡,(5.1) with 𝛼 homogeneously Hölder continuous and Γ positive continuous function. We then have the Doob representation𝑋_𝑡=𝑔(𝑡)^𝑡₀𝑓(𝑠)𝑑𝑊_𝑠,with𝑔(𝑡)=𝑒^{∫𝑡0𝛼(𝑣)𝑑𝑣},𝑓(𝑡)=√Γ(𝑡)𝑒^{−∫𝑡0𝛼(𝑣)𝑑𝑣}.(5.2) Writing the function ℎ asℎ(𝑡)=^𝑡₀𝑓²(𝑠)𝑑𝑠,(5.3) the covariance of the process reads for any 0≤𝑡,𝑠≤0𝐶(𝑡,𝑠)=𝑔(𝑡)𝑔(𝑠)ℎ(𝑡∧𝑠).(5.4) The variance of the Gauss-Markov bridge 𝐵_𝑡 pinned in 𝑡_𝑥 and 𝑡_𝑧 yields𝜎_{𝑡𝑥,𝑡𝑧}(𝑡)²=𝑔(𝑡)²ℎ(𝑡)−ℎ𝑡_𝑥ℎ𝑡_𝑧−ℎ(𝑡)ℎ𝑡_𝑧−ℎ𝑡_𝑥.(5.5) These simple relations entail that the functions 𝜓_𝑛,𝑘 are defined on their supports 𝑆_𝑛,𝑘 by 𝜓_𝑛,𝑘(𝑡)²=𝔼[(𝛿^𝑛(𝑡))²] with𝔼(𝛿^𝑛(𝑡))²=𝜎_{𝑙𝑛,𝑘,𝑟𝑛,𝑘}(𝑡)²−𝟙_{𝑆𝑛+1,2𝑘}(𝑡)𝜎_{𝑙𝑛,𝑘,𝑚𝑛,𝑘}(𝑡)²+𝟙_{𝑆𝑛+1,2𝑘+1}(𝑡)𝜎_{𝑙𝑛,𝑘,𝑚𝑛,𝑘}(𝑡)².(5.6)

This reads on 𝑆_𝑛+1,2𝑘𝜓_𝑛,𝑘(𝑡)²=𝑔(𝑡)²⎡⎢⎣ℎ(𝑡)−ℎ𝑙_𝑛,𝑘ℎ𝑟_𝑛,𝑘−ℎ(𝑡)ℎ𝑟_𝑛,𝑘−ℎ𝑙_𝑛,𝑘−ℎ(𝑡)−ℎ𝑙_𝑛,𝑘ℎ𝑚_𝑛,𝑘−ℎ(𝑡)ℎ𝑚_𝑛,𝑘−ℎ𝑙_𝑛,𝑘⎤⎥⎦(5.7) and on 𝑆_{𝑛+1,2𝑘+1} as𝜓_𝑛,𝑘(𝑡)²=𝑔(𝑡)²⎡⎢⎣ℎ(𝑡)−ℎ𝑙_𝑛,𝑘ℎ𝑟_𝑛,𝑘−ℎ(𝑡)ℎ𝑟_𝑛,𝑘−ℎ𝑙_𝑛,𝑘−ℎ(𝑡)−ℎ𝑚_𝑛,𝑘ℎ𝑟_𝑛,𝑘−ℎ(𝑡)ℎ𝑟_𝑛,𝑘−ℎ𝑚_𝑛,𝑘⎤⎥⎦(5.8) and therefore we have𝜓_𝑛,𝑘(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝜎_𝑛,𝑘𝑔(𝑡)ℎ(𝑡)−ℎ𝑙_𝑛,𝑘𝑔𝑚_𝑛,𝑘ℎ𝑚_𝑛,𝑘−ℎ𝑙_𝑛,𝑘,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,𝜎_𝑛,𝑘𝑔(𝑡)ℎ𝑟_𝑛,𝑘−ℎ(𝑡)𝑔𝑚_𝑛,𝑘ℎ𝑟_𝑛,𝑘−ℎ𝑚_𝑛,𝑘,𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘,(5.9) with𝜎_𝑛,𝑘=⎷ℎ𝑟_𝑛,𝑘−ℎ𝑚_𝑛,𝑘ℎ𝑚_𝑛,𝑘−ℎ𝑙_𝑛,𝑘ℎ𝑟_𝑛,𝑘−ℎ𝑙_𝑛,𝑘.(5.10) As for the first element, it simply results from the conditional expectation of the one-dimensional bridge pinned in 𝑙_0,0=0 and 𝑟_0,0=1:𝜓_0,0(𝑡)=𝑔(𝑡)ℎ(𝑡)−ℎ𝑙_0,0ℎ𝑟_0,0−ℎ𝑙_0,0.(5.11) In this class of processes, two paradigmatic process are the Wiener process and the Ornstein-Uhlenbeck processes with constant coefficients. In the case of the Wiener process, ℎ(𝑡)=𝑡 and 𝑔(𝑡)=1, which yields the classical triangular-shaped Schauder functions used by Lévy [8]. As for the Ornstein-Uhlenbeck process with constant coefficients 𝛼 and √Γ, we have 𝑔(𝑡)=exp(𝛼𝑡), 𝑓(𝑡)=√Γexp(−𝛼𝑡) and ℎ(𝑡)=(Γ/2𝛼)(1−𝑒^−2𝛼𝑡), yielding for the construction basis the expressions𝜓_𝑛,𝑘(𝑡)=⎧⎪⎪⎪⎨⎪⎪⎪⎩Γ𝛼sinh𝛼𝑡−𝑙_𝑛,𝑘sinh𝛼𝑚_𝑛,𝑘−𝑙_𝑛,𝑘,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,Γ𝛼sinh𝛼𝑟_𝑛,𝑘−𝑡sinh𝛼𝑚_𝑛,𝑘−𝑙_𝑛,𝑘,𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘,𝜓_0,0(𝑡)=Γ𝛼𝑒^−𝛼/2sinh(𝛼𝑡)√sinh(𝛼),(5.12) which were already evidenced in [54].

5.2. Multidimensional Case

In the multidimensional case, the explicit expressions for the basis functions 𝝍_𝑛,𝑘 make fundamental use of the flow 𝐅 of the underlying linear equation (2.3) for a given function 𝜶. For commutative forms of 𝜶 (i.e., such that 𝜶(𝑡)⋅𝜶(𝑠)=𝜶(𝑠)⋅𝜶(𝑡) for all 𝑡,𝑠), the flow can be formally expressed as an exponential operator. It is, however, a notoriously difficult problem to find a tractable expression for general 𝛼. As a consequence, it is only possible to provide closed-from formulae for our basis functions in very specific cases.

5.2.1. Multidimensional Gauss-Markov Rotations

We consider in this section that 𝜶 is antisymmetric and constant and √Γ∈ℝ^𝑑×𝑚 such that Γ=𝜎²𝐈_𝑑. For 𝜶 antisymmetric, since 𝜶^𝑇(𝑡)=−𝜶(𝑡), we have𝐅(𝑠,𝑡)^𝑇=𝐅(𝑠,𝑡)⁻¹,(5.13) that is, the flow is unitary. This property implies that𝐡_𝑢(𝑠,𝑡)=𝜎²^𝑡_𝑠𝐅(𝑤,𝑢)𝐅(𝑤,𝑢)^𝑇𝑑𝑤=𝜎²(𝑡−𝑠)𝐈_𝑑,(5.14) which yields by definition of 𝝈_𝑛,𝑘𝝈_𝑛,𝑘⋅𝝈^𝑇_𝑛,𝑘=𝜎²𝑚_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘−𝑚_𝑛,𝑘𝑟_𝑛,𝑘−𝑙_𝑛,𝑘𝐈_𝑑.(5.15) The square root 𝝈_𝑛,𝑘 is then uniquely defined (by choosing both Cholesky and symmetrical square roots) by𝝈_𝑛,𝑘=𝜎⎷𝑚_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘−𝑚_𝑛,𝑘𝑟_𝑛,𝑘−𝑙_𝑛,𝑘𝐈_𝑑,(5.16) and 𝝍_𝑛,𝑘(𝑡) reads𝝍_𝑛,𝑘(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝜎⎷𝑟_𝑛,𝑘−𝑚_𝑛,𝑘𝑚_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘−𝑙_𝑛,𝑘𝑡−𝑙_𝑛,𝑘𝐅𝑚_𝑛,𝑘,𝑡,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,𝜎⎷𝑚_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘−𝑚_𝑛,𝑘𝑟_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘−𝑡𝐅𝑚_𝑛,𝑘,𝑡,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘.(5.17)

Recognizing the (𝑛,𝑘) element of the Schauder basis for the construction of the one-dimensional Wiener process𝑠_𝑛,𝑘(𝑡)=⎧⎪⎪⎨⎪⎪⎩⎷𝑟_𝑛,𝑘−𝑚_𝑛,𝑘𝑟_𝑛,𝑘−𝑙_𝑛,𝑘𝑚_𝑛,𝑘−𝑙_𝑛,𝑘𝑡−𝑙_𝑛,𝑘,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,⎷𝑚_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘−𝑙_n,𝑘𝑟_𝑛,𝑘−𝑚_𝑛,𝑘𝑟_𝑛,𝑘−𝑡,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,(5.18) we obtain the following formula:𝝍_𝑛,𝑘(𝑡)=𝜎𝑠_𝑛,𝑘(𝑡)𝐅𝑡−𝑚_𝑛,𝑘.(5.19)

This form shows that the Schauder basis for multidimensional rotations results from the multiplication of the triangular-shaped elementary function used for the Lévy-Ciesielski construction of the Wiener process with the flow of the equation, that is, the elementary rotation.

The simplest example of this kind is the stochastic sine and cosine process corresponding to𝜶=⎛⎜⎜⎝01−10⎞⎟⎟⎠,√𝚪=𝜎²𝐈₂.(5.20) In that case, 𝝍_𝑛,𝑘 has the expression𝝍_𝑛,𝑘(𝑡)=𝑠_𝑛,𝑘(𝑡)⎛⎜⎜⎝cos𝑡−𝑚_𝑛,𝑘−sin𝑡−𝑚_𝑛,𝑘sin𝑡−𝑚_𝑛,𝑘cos𝑡−𝑚_𝑛,𝑘⎞⎟⎟⎠.(5.21) Interestingly, the different basis functions have the structure of the solutions of the nonstochastic oscillator equation. One of the equations perturbs the trajectory in the radial component of the deterministic solution and the other one in the tangential direction. We represent such a construction in Figure 2 with the additional conditioning that 𝐗₁=𝐗₀, that is, imposing that the trajectory forms a loop between time 0 and 1.

Figure 2 

Construction of the stochastic sine and cos Ornstein-Uhelenbeck processes for the parameters given in (5.20): multiresolution construction of the sample path.

(a)

(b)

Figure 3 

(a) Basis for the construction of the Integrated Wiener process (𝑑=2). Plain red curve: 𝜓1,1, dashed red: 𝜓2,1, plain blue: 𝜓1,2 and dashed blue: 𝜓2,2. (b) 10-steps construction of the process. Moreover, we observe that each basis accounts separately for different aspects of sample path: 𝜓2,1 fixes the value of the integrated process at the middle point 𝑚𝑛,𝑘 and 𝜓2,2 the value of the derivative of the process at the endpoints {𝑙𝑛,𝑘,𝑟𝑛,𝑘} in relationship with the value of the Wiener process at the middle point, whose contributions are split between functions 𝜓1,1 and 𝜓1,2 (see Figure 3).

5.2.2. The Successive Primitives of the Wiener Process

In applications, it often occurs that people use smooth stochastic processes to model the integration of noisy signals. This is for instance the case of a particular subject of a Brownian forcing or of the synaptic integration of noisy inputs [55]. Such smooth processes involves in general integrated martingales, and the simplest example of such processes are the successive primitives of a standard Wiener process.

Let 𝑑>2, and denote by 𝑋^𝑑_𝑡 the 𝑑−1th order primitive of the Wiener process. This process can be defined via the lower-order primitives 𝑋^𝑘_𝑡 for 𝑘<𝑑 via the relations𝑑𝑋^𝑘+1_𝑡=𝑋^𝑘_𝑡𝑑𝑡,𝑘<𝑑,𝑑𝑋¹_𝑡=𝑑𝑊_𝑡,(5.22) where 𝑊_𝑡 is a standard real Wiener process. These equations can be written in our formalism as𝑑𝐗_𝑡=𝜶(𝑡)⋅𝐗_𝑡+√𝚪(𝑡)⋅𝑑𝑊_𝑡,(5.23) with𝜶(𝑡)=⎡⎢⎢⎢⎢⎢⎢⎣01⋱⋱⋱10⎤⎥⎥⎥⎥⎥⎥⎦,√𝚪(𝐭)=⎡⎢⎢⎢⎢⎢⎢⎣00⋮1⎤⎥⎥⎥⎥⎥⎥⎦.(5.24) In particular, though none of the integrated processes 𝑋^𝑘 for 𝐾>1 is Markov by itself, the 𝑑-uplet 𝐗=(𝑋^𝑑,…,𝑋¹) is a Gauss-Markov process.

Furthermore, because of the simplicity and the sparsity of the matrices involved, we can identify in a compact form all the variables used in the computation of the construction basis for these processes. In particular, the flow 𝐅 of the equation is the exponential of the matrix 𝜶, and since 𝛼 is nilpotent, it is easy to show that 𝐅 has the expression,𝐅(𝑠,𝑡)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1(𝑡−𝑠)(𝑡−𝑠)²2⋯(𝑡−𝑠)^𝑑−1(𝑑−1)!⋱⋱⋱⋮⋱⋱(𝑡−𝑠)²2⋱(𝑡−𝑠)1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(5.25) and the only nonzero entry of the 𝑑×𝑑 matrix Γ is one at position (𝑑−1,𝑑−1). Using this expression and the highly simple expression of Γ, we can compute the general element of the matrix 𝐡_𝑢(𝑡,𝑠), which reads𝐡_𝑢(𝑠,𝑡)_𝑖,𝑗=(−1)^𝑖+𝑗(𝑡−𝑢)^{2𝑑−1−(𝑖+𝑗)}−(𝑠−𝑢)^{2𝑑−1−(𝑖+𝑗)}(2𝑑−1−(𝑖+𝑗))(𝑑−1−𝑖)!(𝑑−1−𝑗)!.(5.26) Eventually, we observe that the functions 𝝍_𝑛,𝑘, yielding the multiresolution description of the integrated Wiener processes, are directly deduced from the matrix-valued function𝐜_𝑛,𝑘(𝑡)_𝑖,𝑗=𝝍_𝑛,𝑘⋅𝐋⁻¹_𝑛,𝑘=𝐠(𝑡)𝐡𝑙_𝑛,𝑘,𝑡,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,𝝍_𝑛,𝑘⋅𝐑⁻¹_𝑛,𝑘=𝐠(𝑡)𝐡𝑡,𝑟_𝑛,𝑘,𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘,(5.27) whose components are further expressed as𝐜_𝑛,𝑘(𝑡)_𝑖,𝑗=^𝑑−1_𝑝=𝑖(−1)^𝑝+𝑗𝑡^𝑖−𝑝(𝑖−𝑝)!𝑡^{2𝑑−1−(𝑝+𝑗)}−𝑙^{2𝑑−1−(𝑝+𝑗)}_𝑛,𝑘(2𝑑−1−(𝑝+𝑗))(𝑑−1−𝑝)!(𝑑−1−𝑗)!,(5.28) for 𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘 and as𝐜_𝑛,𝑘(𝑡)_𝑖,𝑗=^𝑑−1_𝑝=𝑖(−1)^𝑝+𝑗𝑡^𝑖−𝑝(𝑖−𝑝)!𝑚^{2𝑑−1−(𝑝+𝑗)}_𝑛,𝑘−𝑡^{2𝑑−1−(𝑝+𝑗)}(2𝑑−1−(𝑝+𝑗))(𝑑−1−𝑝)!(𝑑−1−𝑗)!,(5.29) for 𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘. The final computation of the 𝝍_𝑛,𝑘 involves the computation of 𝐋_𝑛,𝑘 and 𝐑_𝑛,𝑘, which in the general case can become very complex. However, this expression is highly simplified when assuming that 𝑚_𝑛,𝑘 is the middle of the interval [𝑙_𝑛,𝑘,𝑟_𝑛,𝑘]. Indeed, in that case, we observe that, for any (𝑖,𝑗) such that 𝑖+𝑗 is odd, (𝐡_𝑚(𝑙,𝑟))_𝑖,𝑗=𝟎, which induces the same property on the covariance matrix 𝚺_𝑛,𝑘 and on the polynomials (𝐜_𝑛,𝑘(𝑡))_𝑖,𝑗. This property gives therefore a preference to the dyadic partition that provides simple expressions for the basis elements in any dimensions, and allows simple computations of the basis.

Remark 5.1. Observe that, for all 0≤𝑖<𝑑−1, we have 𝐜_𝑛,𝑘(𝑡)^_𝑖,𝑗=𝐜_𝑛,𝑘(𝑡)_𝑖+1,𝑗±^𝑑−1_𝑝=𝑖(−1)^𝑝+𝑗𝑡^𝑖−𝑝(𝑖−𝑝)!𝑡^{2𝑑−1−(𝑝+𝑗)}−𝑙^{2𝑑−2−(𝑝+𝑗)}_𝑛,𝑘(𝑑−1−𝑝)!(𝑑−1−𝑗)!,=𝐜_𝑛,𝑘(𝑡)_𝑖+1,𝑗±𝑡^{𝑑−𝑗−1}(𝑑−𝑗−1)!^{𝑑−1−𝑖}_𝑞=0(−𝑡)^𝑞𝑡^{(𝑑−1−𝑖)−𝑝}𝑝!((𝑑−1−𝑖)−𝑝)!_{0}.(5.30) As 𝐋_𝑛,𝑘 and 𝐑_𝑛,𝑘 are constant, we immediately deduce the important relation that, for all 0≤𝑖≤𝑑−1,(𝝍_𝑛,𝑘(𝑡))^(𝑖)_0,𝑗=(𝝍_𝑛,𝑘(𝑡))_𝑖,𝑗. This indicates that each finite-dimensional sample paths of our construction has components that satisfies the nondeterministic equation associated with the iteratively integrated Wiener process. Actually, this fact is better stated remembering that the Schauder basis 𝝍_𝑛,𝑘 and the corresponding orthonormal basis 𝜙_𝑛,𝑘∶[0,1]→ℝ^1×𝑑 are linked through (3.10), which reads ⎡⎢⎢⎢⎢⎢⎢⎣𝝍_𝑛,𝑘^_0,0…𝝍_𝑛,𝑘^_0,𝑑−1⋮⋮𝝍_𝑛,𝑘^_𝑑−2,0…𝝍_𝑛,𝑘^_{𝑑−2,𝑑−1}𝝍_𝑛,𝑘^_𝑑−1,0…𝝍_𝑛,𝑘^_{𝑑−1,𝑑−1}⎤⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣𝝍_𝑛,𝑘_1,0…𝝍_𝑛,𝑘_1,𝑑−1⋮⋮𝝍_𝑛,𝑘_𝑑−1,0…𝝍_𝑛,𝑘_{𝑑−1,𝑑−1}0…0⎤⎥⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎢⎣0…0⋮⋮0…0𝝍_𝑛,𝑘_0,0…𝝍_𝑛,𝑘_0,𝑑−1⎤⎥⎥⎥⎥⎥⎥⎦.(5.31) Additionally, we realize that the orthonormal basis is entirely determined by the one-dimensional families (𝜙_𝑛,𝑘)_0,𝑗, which are mutually orthogonal functions satisfying (𝜙_𝑛,𝑘)_0,𝑗=(𝝍_𝑛,𝑘)^(𝑑)_0,𝑗.

We study in more detail the case of the integrated and doubly-integrated Wiener process (𝑑=2 and 𝑑=3), for which closed-form expressions are provided in Appendices A and B. As expected, the first row of the basis functions for the integrated Wiener process turns out to be the well-known cubic Hermite splines [56]. These functions have been widely used in numerical analysis and actually constitute the basis of the lowest degree in a wider family of bases known as the natural basis of polynomial splines of interpolation [25]. Such bases are used to interpolate data points with constraint of smoothness of different degrees (e.g., the cubic Hermite splines ensure that the resulting interpolation is in 𝐶¹[0,1]). The next family of splines of interpolation (corresponding to the 𝐶² constraint) is naturally retrieved by considering the construction of the doubly-integrated Wiener process: we obtain a family of three 3-dimensional functions that constitutes the columns of a 3×3 matrix that we denote by 𝝍. The top row is made of polynomials of degree five, which have again simple expressions when 𝑚_𝑛,𝑘 is the middle of the interval [𝑙_𝑛,𝑘,𝑟_𝑛,𝑘].

6. Stochastic Calculus from the Hilbert Point of View

Thus far, all calculations, propositions, and theorems are valid for any finite-dimensional the Gauss-Markov process and all the results are valid pathwise, that is, for each sample path. The analysis provides a Hilbert description of the processes as a series of standard Gaussian random variables multiplied by certain specific functions, that form a Schauder basis in the suitable spaces. This new description of Gauss-Markov processes provides a new way for treating problems arising in the study of stochastic processes. As examples of this, we derive the Itô formula and the Girsanov theorem from the Hilbertian viewpoint. Note that these results are equalities in law, that is, dealing with the distribution of stochastic processes, which is a weaker notion compared to the pathwise analysis. In this section, we restrict our analysis to the one-dimensional case for technical simplicity.

The closed-form expressions of the basis of functions 𝜓_𝑛,𝑘 in the one-dimensional case are given in Section 5.1. The differential and integral operators associated, introduced in Section 3.1.2 are highly simplified in the one-dimensional case. Let 𝑈 be a bounded open set of [0,1], we denote by 𝐶(𝑈) the space of continuous real functions on 𝑈 and we recall that the topological dual of 𝐶(𝑈) is 𝑅(𝑈), the space of Radon measures on 𝑈. We also introduce 𝐷₀(𝑈), the space of test functions in 𝐶^∞(𝑈) that are zero in zero, and whose dual space 𝐷^₀(𝑈) satisfies 𝐷^₀(𝑈)⊂𝑅(𝑈).Let 𝑈 be a bounded open neighborhood of [0,1], and denote by 𝐶(𝑈) is the space of continuous real functions on 𝑈, 𝑅(𝑈) its topological dual, the space of Radon measures, 𝐷₀(𝑈) the space of test function in 𝐶^∞(𝑈) which are zero at zero and it dual 𝐷^₀(𝑈)⊂𝑅(𝑈). We consider the Gelfand triple𝐷₀(𝑈)⊂𝐶(𝑈)⊂𝐿²(𝑈)⊂𝐷^₀(𝑈)⊂𝑅(𝑈).(6.1)

The integral operator 𝒦 is defined (and extended by dual pairing) by𝒦[⋅](𝑡)=_𝑈𝟙_[0,𝑡](𝑠)𝑔_𝛼(𝑡)𝑓_𝛼(𝑠)⋅𝑑𝑠,(6.2) and the inverse differential operator 𝒟 reads𝒟[⋅](𝑡)=1𝑔_𝛼(𝑡)𝑑𝑑𝑡⋅𝑓_𝛼(𝑡).(6.3)

Now that we dispose of all the explicit forms of the basis functions and related operators, we are in position to complete our program and start by proving the very important Itô formula and its finite-dimensional counterpart before turning to the Girsanov theorem.

6.1. Itô’s Formula

A very useful theorem in the stochastic processes theory is the Itô formula. We show here that this formula is consistent with the Hilbert framework introduced. Most of the proofs can be found in Appendix E. The proof of Itô formula is based on demonstrating the integration by parts property.

Proposition 6.1 (Integration by parts). Let (𝑋_𝑡) and (𝑌_𝑡) be two one-dimensional Gauss-Markov processes starting from zero. Then one has the following equality in law: 𝑋_𝑡𝑌_𝑡=^𝑡₀𝑋_𝑠∘𝑑𝑌_𝑠+^𝑡₀𝑌_𝑠∘𝑑𝑋_𝑠,(6.4) where, for two stochastic processes 𝐴_𝑡 and 𝐵_𝑡, ∫^𝑡₀𝐴_𝑠∘𝑑𝐵_𝑠 denotes the Stratonovich integral. In terms of the Itô integral, this formula is written as 𝑋_𝑡𝑌_𝑡=^𝑡₀𝑋_𝑠𝑑𝑌_𝑠+^𝑡₀𝑌_𝑠𝑑𝑋_𝑠+⟨𝑋,𝑌⟩_𝑡,(6.5) where the brackets denote the mean quadratic variation.

The proof of this proposition is quite technical and is provided in Appendix E. It is based a thorough analysis of the finite-dimensional processes 𝑋^𝑁_𝑡 and 𝑌^𝑁_𝑡. For this integration by parts formula and using a density argument, one can recover the more general Itô formula.

Theorem 6.2 (Itô). Let (𝑋_𝑡)_𝑡 be a Gauss-Markov process and 𝐹∈𝐶²(ℝ). The process 𝑓(𝑋_𝑡) is a Markov process and satisfies the following relation in law: 𝑓𝑋_𝑡=𝑓𝑋₀+^𝑡₀𝑓^𝑋_𝑠𝑑𝑋_𝑠+12^𝑡₀𝑓^𝑋_𝑠𝑑⟨𝑋⟩_𝑠.(6.6)

This theorem is proved in Appendix E.

The Itô formula implies in particular that the multiresolution description developed in the paper is valid for every smooth functional of a Gauss-Markov process. In particular, it allows a simple description of exponential functionals of Gaussian Markovian processes, which are of particular interest in mathematics and have many applications, in particular in economics (see, e.g., [57]).

Therefore, we observe that in the view of the paper, Itô formula stems from the nonorthogonal projections of basis element. For multidimensional processes, the proof of the Itô formula is deduced from the one-dimensional proof and would involve the study of the multidimensional bridge formula for 𝐗_𝑡 and 𝐘_𝑡.

We eventually remark that this section provides us with a finite-dimensional counterpart of the Itô formula for discretized processes, which has important potential applications, and further assesses the suitability of using the finite resolution representation developed in this paper. Indeed, using the framework developed in the present paper allows considering finite-resolution processes and their transformation through nonlinear smooth transformation in a way that is concordant with the standard stochastic calculus processes, since the equation on the transformed process indeed converges towards its Itô representation as the resolution increases.

6.2. Girsanov Formula: A Geometric Viewpoint

In the framework we developed, transforming a process 𝑋 into a process 𝑌 is equivalent to substituting the Schauder construction basis related to 𝑌 for the basis related to 𝑋. Such an operation provides a pathwise mapping for each sample path of 𝑋 onto a sample path of 𝑌 having the same probability density in _𝜉Ω^. This fact sheds a new light on the geometry of multidimensional Gauss-Markov processes, since the relationship between two processes is seen as a linear change of basis. In our framework, this relationship between processes is straightforwardly studied in the finite rank approximations of the processes up to a certain resolution. Technical intricacy is nevertheless raised when dealing with the representation of the process itself in the infinite-dimensional Hilbert spaces. We solve these technical issues here and show that in the limit 𝑁→∞ one recovers Girsanov theorem as a limit of the linear transformations between the Gauss-Markov processes.

The general problem consists therefore in studying the relationship between two real Gauss-Markov processes 𝑋 and 𝑌 that are defined by𝑑𝑋_𝑡=𝛼_𝑋(𝑡)𝑋_𝑡𝑑𝑡+√Γ_𝑋(𝑡)𝑑𝑊_𝑡,𝑑𝑌_𝑡=𝛼_𝑌(𝑡)𝑋_𝑡𝑑𝑡+√Γ_𝑌(𝑡)𝑑𝑊_𝑡.(6.7)

We have noticed that the spaces _𝑥Ω^ are the same in the one-dimensional case as long as both Γ_𝑋 and Γ_𝑌 never vanish and therefore make this assumption here. In order to further simplify the problem, we assume that 𝛾_𝑋,𝑌=Γ_𝑋/Γ_𝑌 is continuously differentiable. This assumption allows us to introduce the process 𝑍_𝑡=𝛾_𝑋,𝑌(𝑡)𝑌_𝑡 that satisfies the stochastic differential equation𝑑𝑍_𝑡=𝑑𝑑𝑡𝛾_𝑋,𝑌(𝑡)𝑌_𝑡𝑑𝑡+𝛾_𝑋,𝑌(𝑡)𝑑𝑌_𝑡=𝑑𝑑𝑡𝛾_𝑋,𝑌(𝑡)+𝛾_𝑋,𝑌(𝑡)𝛼_𝑌(𝑡)𝑌_𝑡𝑑𝑡+√Γ_𝑋(𝑡)𝑑𝑊_𝑡=𝛼_𝑍(𝑡)𝑍_𝑡𝑑𝑡+√Γ_𝑋(𝑡)𝑑𝑊_𝑡,(6.8) with 𝛼_𝑍(𝑡)=(𝑑/𝑑𝑡)(𝛾_𝑋,𝑌(𝑡))𝛾_𝑋,𝑌(𝑡)⁻¹+𝛼_𝑌(𝑡). Moreover, if _𝑍𝜓_𝑛,𝑘 and _𝑌𝜓_𝑛,𝑘 are the bases of functions that describe the process 𝑍 and 𝑌, respectively, we have _𝑍𝜓_𝑛,𝑘=𝛾_𝑋,𝑌⋅_𝑌𝜓_𝑛,𝑘.

The previous remarks allow us to restrict without loss of generality our study to the processes defined for same function √Γ, thus reducing the parameterization of the Gauss-Markov processes to the linear coefficient 𝛼. Observe that in the classical stochastic calculus theory, it is well known that such hypotheses are necessary for the process 𝑋 to be absolutely continuous with respect to 𝑌 (through the use of the Girsanov theorem).

Let us now consider that 𝛼,𝛽, and √Γ three real Hölder continuous real functions, and introduce _𝛼𝑋 and _𝛽𝑋 solutions of the equations𝑑_𝛼𝑋_𝑡=𝛼(𝑡)_𝛼𝑋_𝑡𝑑𝑡+√Γ(𝑡)𝑑𝑊_𝑡,𝑑_𝛽𝑋_𝑡=𝛽(𝑡)_𝛽𝑋_𝑡𝑑𝑡+√Γ(𝑡)𝑑𝑊_𝑡.(6.9) All the functions and tools related to the process _𝛼𝑋 (resp., _𝛽𝑋) will be indexed by 𝛼 (𝛽) in the sequel.

6.2.1. Lift Operators

Depending on the space we are considering (either coefficients or trajectories), we define the two following operators mapping the process _𝛼𝑋_𝑡 on _𝛽𝑋_𝑡.(1)The coefficients lift operator _𝛼,𝛽𝐺 is the linear operator mapping in _𝜉Ω^ the process _𝛼𝑋 on the process _𝛽𝑋: _𝛼,𝛽𝐺=_𝛽Δ∘_𝛼Ψ∶_𝜉Ω^,ℬ_𝜉Ω^⟶_𝜉Ω^,ℬ_𝜉Ω^.(6.10) For any 𝜉∈_𝜉Ω^, the operator _𝛼,𝛽𝐺 maps a sample path of _𝛼𝑋 on a sample path of _𝛽𝑋.(2)The process lift operator _𝛼,𝛽𝐹 is the linear operator mapping in _𝑥Ω the process _𝛼𝑋 on the process _𝛽𝑋: _𝛼,𝛽𝐻=_𝛼Ψ∘_𝛼Δ∶_𝑥Ω,ℬ_𝑥Ω⟶_𝑥Ω,ℬ_𝑥Ω.(6.11) We summarize the properties of these operators now.

Proposition 6.3. The operators _𝛼,𝛽𝐺 and _𝛼,𝛽𝐻 satisfy the following properties.(i) They are linear measurable bijections(ii)For every 𝑁>0, the function _𝛼,𝛽𝐺_𝑁=𝑃_𝑁∘_𝛼,𝛽𝐺∘𝐼_𝑁∶_𝜉Ω^_𝑁→_𝜉Ω_𝑁 (resp., _𝛼,𝛽𝐻_𝑁=𝑃_𝑁∘_𝛼,𝛽𝐻∘𝐼_𝑁∶_𝜉Ω^_𝑁→_𝜉Ω_𝑁) is a finite-dimensional linear operator, whose matrix representation is triangular in the natural basis of _𝜉Ω_𝑁 (resp., _𝜉Ω^_𝑁) and whose eigenvalues _𝛼,𝛽𝜈_𝑛,𝑘 are given by _𝛼,𝛽𝜈_𝑛,𝑘=𝑔_𝛼𝑚_𝑛,𝑘𝑔_𝛽𝑚_𝑛,𝑘^𝛽_𝑛𝑀_,𝑘𝛼𝑀_𝑛,𝑘,0≤𝑛≤𝑁,0≤𝑘<2^𝑁−1(6.12) (resp., _𝛽,𝛼𝜈_𝑛,𝑘=(_𝛼,𝛽𝜈_𝑛,𝑘)⁻¹).(iii)_𝛼,𝛽𝐺 and _𝛼,𝛽𝐻 are bounded operators for the spectral norm with ‖‖_𝛼,𝛽𝐺‖‖₂=sup_𝑛sup_{𝑘𝛼,𝛽}𝜈_𝑛,𝑘≤sup𝑔_𝛼inf𝑔_𝛽sup𝑓²_𝛼inf𝑓²_𝛽<∞,(6.13) and ‖_𝛼,𝛽𝐻‖₂=‖_𝛽,𝛼𝐺‖₂<∞.(iv)The determinants of _𝛼,𝛽𝐺_𝑁 (denoted by _𝛼,𝛽𝐽_𝑁) and _𝛼,𝛽𝐻_𝑁 admit a limit when 𝑁 tends to infinity: _𝛼,𝛽𝐽=lim_{𝑁→∞𝛼,𝛽}𝐽_𝑁=exp12¹₀(𝛼(𝑡)−𝛽(𝑡))𝑑𝑡,lim_𝑁→∞det^𝛼,𝛽𝐻_𝑁=exp12¹₀(𝛽(𝑡)−𝛼(𝑡))𝑑𝑡=_𝛽,𝛼𝐽.(6.14)

The proof of these properties elementary stems from the analysis done on the functions Ψ and Δ that were previously performed, and these are detailed in Appendix C.

6.2.2. Radon-Nikodym Derivatives

From the properties proved on the lift operators, we are in position to further analyze the relationship between the probability distributions of _𝛼𝑋 and _𝛽𝑋. We first consider the finite-dimensional processes _𝛼𝑋^𝑁 and _𝛽𝑋^𝑁. We emphasize that, throughout this section, all equalities are true pathwise.

Lemma 6.4. Given the finite-dimensional measures 𝑃^𝑁_𝛼 and 𝑃^𝑁_𝛽, the Radon-Nikodym derivative of 𝑃^𝑁_𝛽 with respect to 𝑃^𝑁_𝛼 satisfies 𝑑𝑃^𝑁_𝛽𝑑𝑃^𝑁_𝛼(𝜔)=_𝛼,𝛽𝐽_𝑁⋅exp−12Ξ_𝑁(𝜔)^𝑇^𝛼,𝛽𝑆_𝑁−𝐼𝑑_𝜉Ω𝑁Ξ_𝑁(𝜔)(6.15) with _𝛼,𝛽𝑆_𝑁=_𝛼,𝛽𝐺_𝑁^𝑇⋅_𝛼,𝛽𝐺_𝑁 and the equality is true pathwise.

Proof. In the finite-dimensional case, for all 𝑁>0, we can write that 𝑃^𝑁_𝛼, 𝑃^𝑁_𝛽𝑝^𝑁_𝛼, and the Lebesgue measure _𝑥Ω^𝑁 are mutually absolutely continuous: we denote by 𝑝^𝑁_𝛼 and 𝑝^𝑁_𝛽 the Gaussian density of 𝑃^𝑁_𝛼 and 𝑃^𝑁_𝛽 with respect to the Lebesgue measure on _𝑥Ω^𝑁. Therefore, in the finite-dimensional case, the Radon-Nikodym derivative of 𝑃^𝑁_𝛽 with respect to 𝑃^𝑁_𝛼 is defined to be pathwise and is simply given by the quotient of the density of the vector {_𝛽𝑋^𝑁(𝑚_𝑖,𝑗)} with the density of the vector {_𝛼𝑋^𝑁(𝑚_𝑖,𝑗)} for 0≤𝑖≤𝑁, 0≤𝑗<2^𝑖−1, that is, 𝑑𝑃^𝑁_𝛽𝑑𝑃^𝑁_𝛼(𝜔)=𝑝^𝑁_𝛽_𝛼𝑋^𝑁(𝜔)𝑝^𝑁_𝛼_𝛼𝑋^𝑁(𝜔)=⎷det_𝛼Σ_𝑁det_𝛽Σ_𝑁⋅exp−12_𝛼𝑋^𝑁(𝜔)^𝑇_𝛽Σ⁻¹_𝑁−_𝛼Σ⁻¹_𝑁_𝛼𝑋^𝑁(𝜔).(6.16) We first make explicit det_𝛼Σ_𝑁det_𝛽Σ_𝑁=det_𝛼Σ_𝑁⋅_𝛽Σ⁻¹_𝑁=det_𝛼Ψ_𝑁⋅_𝛼Ψ_𝑁^𝑇⋅_𝛽Δ_𝑁^𝑇⋅_𝛽Δ_𝑁=det_𝛽Δ_𝑁⋅_𝛼Ψ_𝑁⋅_𝛼Ψ_𝑁^𝑇⋅_𝛽Δ_𝑁^𝑇=det_𝛼,𝛽𝐺_𝑁⋅_𝛼,𝛽𝐺_𝑁^𝑇=det_𝛼,𝛽𝐺_𝑁².(6.17) Then, we rearrange the exponent using the Cholesky decomposition: _𝛽Σ⁻¹_𝑁−_𝛼Σ⁻¹_𝑁=_𝛽Δ_𝑁^𝑇⋅_𝛽Δ_𝑁−_𝛼Δ_𝑁^𝑇⋅_𝛼Δ_𝑁,(6.18) so that we write the exponent of (6.16) as _𝛼𝑋_𝑁^𝑇^𝛽Δ_𝑁^𝑇⋅_𝛽Δ_𝑁−_𝛼Δ_𝑁^𝑇⋅_𝛼Δ_𝑁^𝛼𝑋_𝑁=Ξ_𝑁^𝑇⋅_𝛼Ψ_𝑁^𝑇^𝛽Δ_𝑁^𝑇⋅_𝛽Δ_𝑁−_𝛼Δ_𝑁^𝑇⋅_𝛼Δ_𝑁^𝛼Ψ_𝑁(𝜔)⋅Ξ_𝑁=Ξ_𝑁^𝑇^𝛼,𝛽𝐺_𝑁^𝑇⋅_𝛼,𝛽𝐺_𝑁−𝐼𝑑_𝜉Ω𝑁Ξ_𝑁.(6.19) We finally reformulate (6.16) as 𝑑𝑃^𝑁_𝛽𝑑𝑃^𝑁_𝛼(𝜔)=_𝛼,𝛽𝐽_𝑁⋅exp−12Ξ_𝑁(𝜔)^𝑇^𝛼,𝛽𝐺_𝑁^𝑇⋅_𝛼,𝛽𝐺_𝑁−𝐼𝑑_𝜉Ω𝑁Ξ_𝑁(𝜔).(6.20)

Let us now justify from a geometrical point of view why this formula is a direct consequence of the finite-dimensional change of variable formula on the model space _𝜉Ω_𝑁. If we introduce _𝛼Δ_𝑁, the coefficient application related to _𝛼𝑋^𝑁, we know that Ξ_𝑁=_𝛼Δ_𝑁(_𝛼𝑋^𝑁) follows a normal law 𝒩(𝟎,𝐈_𝜉Ω𝑁). We denote by 𝑝^𝑁_𝜉 its standard Gaussian density with respect to the Lebesgue measure on _𝜉Ω^𝑁. We also know that_𝛼Δ_𝑁_𝛽𝑋^𝑁=_𝛼Δ_𝑁∘_𝛼,𝛽𝐹_𝑁_𝛼𝑋^𝑁=_𝛽,𝛼𝐺_𝑁_𝛼Δ_𝑁_𝛼𝑋^𝑁=_𝛽,𝛼𝐺_𝑁Ξ_𝑁.(6.21)

Since _𝛽,𝛼𝐺_𝑁 is linear, the change of the variable formula directly entails that _𝛽,𝛼𝐺_𝑁(Ξ_𝑁) admits on _𝜉Ω_𝑁𝑝^𝑁_𝛽,𝛼𝜉_𝑁=||det_𝛼,𝛽𝐺||𝑝^𝑁_𝜉𝜉_𝑁^𝑇_𝛼,𝛽𝐺_𝑁^𝑇⋅_𝛼,𝛽𝐺_𝑁𝜉_𝑁(6.22) as density with respect to the Lebesgue measure. Consider now 𝐵 as a measurable set of (_𝑥Ω_𝑁,ℬ(_𝑥Ω_𝑁)); then we have𝑃^𝑁_𝛽(𝐵)=_{𝛼Δ𝑁(𝐵)}𝑝^𝑁_𝛽,𝛼𝜉_𝑁𝑑𝜉_𝑁=_{𝛼Δ𝑁(𝐵)}𝑝^𝑁_𝛽,𝛼𝜉_𝑁𝑝^𝑁_𝜉𝜉_𝑁𝑝^𝑁_𝜉𝜉_𝑁𝑑𝜉_𝑁=_𝐵𝑝^𝑁_𝛽,𝛼_𝛼Δ_𝑁𝑋^𝑁𝑝^𝑁_𝜉_𝛼Δ_𝑁𝑋^𝑁𝑑𝑃^𝑁_𝛼𝑋^𝑁,(6.23) from which we immediately conclude.

6.2.3. The Trace Class Operator

The pathwise expression of the Radon-Nikodym derivative extends to the infinite-dimensional representation of _𝛼𝑋 and _𝛽𝑋. This extension involves technical analysis on the infinite-dimensional Hilbert space 𝑙²(ℝ). We have shown in Proposition 6.3 that the application _𝛼,𝛽𝐺 was bounded for the spectral norm. Therefore, we have, for any 𝜉 in 𝑙²(ℝ), the inequality‖‖_𝛼,𝛽𝐺(𝜉)‖‖₂≤‖‖_𝛼,𝛽𝐺‖‖₂⋅‖𝜉‖₂(6.24) implying that _𝛼,𝛽𝐺 maps 𝑙²(ℝ) into 𝑙²(ℝ). We can then define the adjoint operator _𝛼,𝛽𝐺^𝑇 from 𝑙²(ℝ) to 𝑙²(ℝ), which is given by∀𝜉,𝜂∈𝑙²(ℝ),^𝛼,𝛽𝐺^𝑇(𝜂),𝜉=𝜂,_𝛼,𝛽𝐺(𝜉).(6.25) Let us now consider the self-adjoint operator _𝛼,𝛽𝑆=_𝛼,𝛽𝐺^𝑇∘_𝛼,𝛽𝐺∶𝑙²(ℝ)→𝑙²(ℝ). This operator is the infinite-dimensional counterpart of the matrix _𝛼,𝛽𝑆_𝑁.

Lemma 6.5. Considering the coefficients of the matrix representation of _𝛼,𝛽𝑆 in the natural basis 𝑒_𝑛,𝑘 of 𝑙²(ℝ) given as _𝛼,𝛽𝑆^𝑛,𝑘_𝑝,𝑞=(𝑒_𝑛,𝑘,_𝛼,𝛽𝑆(𝑒_𝑝,𝑞)), one has _𝛼,𝛽𝑆^𝑛,𝑘_𝑝,𝑞=¹₀⎛⎜⎝^𝛼𝜙_𝑛,𝑘(𝑡)+(𝛼(𝑡)−𝛽(𝑡))√Γ(𝑡)^𝛼𝜓_𝑛,𝑘(𝑡)⎞⎟⎠⎛⎜⎝^𝛼𝜙_𝑝,𝑞(𝑡)+(𝛼(𝑡)−𝛽(𝑡))√Γ(𝑡)^𝛼𝜓_𝑝,𝑞(𝑡)⎞⎟⎠𝑑𝑡.(6.26)

Proof. Assume that max(𝑛,𝑝)≤𝑁 and that (𝑛,𝑘) and (𝑝,𝑞)∈ℐ_𝑁. With the notations used previously with 𝑈, an open neighbourhood of [0,1], we have _𝛼,𝛽𝑆^𝑛,𝑘_𝑝,𝑞=_𝛼,𝛽𝐺𝑒_𝑛,𝑘,_𝛼,𝛽𝐺𝑒_𝑝,𝑞=_{(𝑖,𝑗)∈ℐ}_𝑈^𝛽𝛿_𝑖,𝑗(𝑡)_𝛼𝜓_𝑛,𝑘(𝑡)𝑑𝑡_𝑈^𝛽𝛿_𝑖,𝑗(𝑡)_𝛼𝜓_𝑝,𝑞(𝑡)𝑑𝑡.(6.27) Since we have by definition _𝛼𝜓_𝑛,𝑘=_𝛼𝒦[_𝛼𝜙_𝑛,𝑘], _𝛽𝛿_𝑖,𝑗=_𝛽𝒟[_𝛽𝜙_i,𝑗], _𝛼𝒦⁻¹=_𝛼𝒟^∗, _𝛽𝒟⁻¹=_𝛽𝒦^∗ on the space 𝐷^₀(𝑈), we have _𝛽𝜙_𝑖,𝑗(𝑡)=_𝛽𝒦^∗^𝛽𝛿_𝑖,𝑗(𝑡)=𝑓_𝛽(𝑡)_𝑈𝑔_𝛽(𝑠)_𝛽𝛿_𝑖,𝑗(𝑠)𝑑𝑠,(6.28)_𝛼𝜙_𝑛,𝑘(𝑡)=_𝛼𝒟^∗_𝛼𝜓_𝑛,𝑘(𝑡)=1𝑓_𝛼(𝑡)𝑑𝑑𝑡^𝛼𝜓_𝑛,𝑘(𝑡)𝑔_𝛼(𝑡).(6.29) From there using (6.28), we can write the integration by part formula in the sense of the generalized functions to get _𝑈^𝛽𝛿_𝑖,𝑗(𝑡)_𝛼𝜓_𝑛,𝑘(𝑡)𝑑𝑡=_𝑈1𝑓_𝛽(𝑡)𝑑𝑑𝑡^𝛼𝜓_𝑛,𝑘(𝑡)𝑔_𝛽(𝑡)^𝛽𝜙_𝑖,𝑗(𝑡)𝑑𝑡=_𝑈^𝛽𝒟^∗_𝛼𝜓_𝑛,𝑘(𝑡)_𝛽𝜙_𝑖,𝑗(𝑡)𝑑𝑡.(6.30) We now compute using (6.29): _𝛽𝒟^∗_𝛼𝜓_𝑛,𝑘(𝑡)=1𝑓_𝛽(𝑡)𝑑𝑑𝑡^𝛼𝜓_𝑛,𝑘(𝑡)𝑔_𝛼(𝑡)𝑔_𝛼(𝑡)𝑔_𝛽(𝑡)=𝑔_𝛼(𝑡)𝑓(𝑡)𝑔_𝛽(𝑡)𝑓_𝛽(𝑡)^𝛼𝜙(𝑡)−𝑑𝑑𝑡𝑔_𝛼(𝑡)𝑔_𝛽(𝑡)^𝛼𝜓_𝑛,𝑘(𝑡)𝑔_𝛽(𝑡)𝑓_𝛽(𝑡).(6.31) Specifying 𝑔_𝛼, 𝑔_𝛽, 𝑓_𝛼, and 𝑓_𝛽 and recalling the relations 𝑔_𝛼(𝑡)𝑓_𝛼(𝑡)=𝑔_𝛽(𝑡)𝑓_𝛽(𝑡)=√Γ(𝑡),𝑑𝑑𝑡𝑔_𝛼(𝑡)𝑔_𝛽(𝑡)=(𝛼(𝑡)−𝛽(𝑡))𝑔_𝛼(𝑡)𝑔_𝛽(𝑡),(6.32) we rewrite the function (6.31) in 𝐿²[0,1] as _𝛽𝒟^∗_𝛼𝜓_𝑛,𝑘(𝑡)=1𝑓_𝛽(𝑡)𝑑𝑑𝑡^𝛼𝜓_𝑛,𝑘(𝑡)𝑔_𝛽(𝑡)=_𝛼𝜙_𝑛,𝑘(𝑡)+(𝛼(𝑡)−𝛽(𝑡))√Γ(𝑡)^𝛼𝜓_𝑛,𝑘(𝑡).(6.33) Now, since the family _𝛽𝜙_𝑛,𝑘 forms a complete orthonormal system of 𝐿²[0,1] and is zero outside [0,1], by the Parseval identity, expression (6.27) can be written as the scalar product: _𝛼,𝛽𝑆^𝑛,𝑘_𝑝,𝑞=¹₀⎛⎜⎝^𝛼𝜙_𝑛,𝑘(𝑡)+(𝛼(𝑡)−𝛽(𝑡))√Γ(𝑡)^𝛼𝜓_𝑛,𝑘(𝑡)⎞⎟⎠⎛⎜⎝^𝛼𝜙_𝑝,𝑞(𝑡)+(𝛼(𝑡)−𝛽(𝑡))√Γ(𝑡)^𝛼𝜓_𝑝,𝑞(𝑡)⎞⎟⎠𝑑𝑡,(6.34) which ends the proof of the lemma.

Notice that we can further simplify expression (6.26):_𝛼,𝛽𝑆^𝑛,𝑘_𝑝,𝑞=𝛿^𝑛,𝑘_𝑝,𝑞+¹₀(𝛼(𝑡)−𝛽(𝑡))²Γ(𝑡)^𝛼𝜓_𝑛,𝑘(𝑡)_𝛼𝜓_𝑝,𝑞(𝑡)𝑑𝑡+¹₀(𝛼(𝑡)−𝛽(𝑡))√Γ(𝑡)^𝛼𝜙_𝑛,𝑘(𝑡)_𝛼𝜓_𝑝,𝑞(𝑡)+_𝛼𝜙_𝑝,𝑞(𝑡)_𝛼𝜓_𝑛,𝑘(𝑡)𝑑𝑡.(6.35) We are now in a position to show that the operator _𝛼,𝛽𝑆−𝐼𝑑 can be seen as the limit of the finite-dimensional operator _𝛼,𝛽𝑆_𝑁−𝐼𝑑_𝜉Ω𝑁, in the following sense.

Theorem 6.6. The operator _𝛼,𝛽𝑆−𝐼𝑑∶𝑙²(ℝ)→𝑙²(ℝ) is a trace class operator, whose trace is given by Tr^𝛼,𝛽𝑆−𝐼𝑑=¹₀(𝛼(𝑡)−𝛽(𝑡))𝑑𝑡+¹₀ℎ_𝛼(𝑡)𝑓_𝛼(𝑡)²(𝛼(𝑡)−𝛽(𝑡))²𝑑𝑡.(6.36)

We prove this essential point in Appendix F. The proof consists in showing that the operator _𝛼,𝛽𝑆−𝐼𝑑 is isometric to a Hilbert-Schmidt operator whose trace can be computed straightforwardly.

6.2.4. The Girsanov Theorem

We now proceed to prove the Girsanov theorem by extending the domain of the quadratic form associated with _𝛼,𝛽𝑆−𝐼𝑑 to the space _𝜉Ω^, which can only be done in law.

Theorem 6.7. In the infinite-dimensional case, the Radon-Nikodym derivative of 𝑃_𝛽=𝑃_𝛽𝑋−1 with respect to 𝑃_𝛼=𝑃_𝛼𝑋−1 reads 𝑑𝑃_𝛽(𝜔)𝑑𝑃_𝛼(𝜔)=exp−12¹₀𝛽(𝑡)−𝛼(𝑡)𝑑𝑡+Ξ(𝜔)^𝑇^𝛼,𝛽𝑆−𝐼𝑑_𝜉ΩΞ(𝜔),(6.37) which in terms of the Itô stochastic integral reads 𝑑𝑃_𝛽(𝜔)𝑑𝑃_𝛼(𝜔)=exp¹₀𝛽(𝑡)−𝛼(𝑡)𝑓_𝛼²(𝑡)^𝛼𝑋_𝑡(𝜔)𝑔_𝛼(𝑡)𝑑^𝛼𝑋_𝑡(𝜔)𝑔_𝛼(𝑡)−12¹₀(𝛽(𝑡)−𝛼(𝑡))²𝑓_𝛼²(𝑡)^𝛼𝑋_𝑡(𝜔)𝑔_𝛼(𝑡)²𝑑𝑡.(6.38)

In order to demonstrate the Girsanov theorem from our geometrical point of view, we need to establish the following result.

Lemma 6.8. The positive definite quadratic form on 𝑙²(ℝ)×𝑙²(ℝ) associated with operator _𝛼,𝛽𝑆−𝐼𝑑∶𝑙²(ℝ)→𝑙²(ℝ) is well defined on _𝜉Ω^. Moreover, for all _𝜉Ω^, 𝜉,^𝛼,𝛽𝑆−𝐼𝑑_𝜉Ω(𝜉)=2¹₀𝛼(𝑡)−𝛽(𝑡)𝑓_𝛼²(𝑡)^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)∘𝑑^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)+¹₀(𝛼(𝑡)−𝛽(𝑡))²𝑓_𝛼²(𝑡)^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)²𝑑𝑡,(6.39) where _𝛼𝑋_𝑡(𝜉)=_𝛼Φ(𝜉) and ∘ refers to the Stratonovich integral and the equality is true in law.

Proof of Theorem 6.7. We start by writing the finite-dimensional Radon-Nikodym derivative 𝑑𝑃_𝛽(𝜔)𝑑𝑃_𝛼(𝜔)=_𝛼,𝛽𝐽_𝑁⋅lim_𝑁→∞exp−12Ξ_𝑁(𝜔)^𝑇^𝛼,𝛽𝑆_𝑁−𝐼𝑑_𝜉Ω𝑁Ξ_𝑁(𝜔).(6.40) By Proposition 6.3, we have _𝛼,𝛽𝐽=lim_{𝑁→∞𝛼,𝛽}𝐽_𝑁=12¹₀(𝛼(𝑡)−𝛽(𝑡))𝑑𝑡.(6.41) If, as usual, Ξ denotes a recursively indexed infinite-dimensional vector of independent variables with law 𝒩(0,1) and Ξ_𝑁=_𝜉𝑃_𝑁∘Ξ, writing 𝜉_𝑛,𝑘=Ξ_𝑛,𝑘(𝜔), we have Ξ_𝑁(𝜔)^𝑇^𝛼,𝛽𝑆_𝑁−𝐼𝑑_𝜉Ω𝑁Ξ_𝑁(𝜔)=^𝑁_𝑛=0^𝑁_𝑝=0_{0≤𝑘<2𝑛−1}_{0≤𝑞<2𝑝−1}𝜉_𝑛,𝑘^𝛼,𝛽𝑆_𝑁−𝐼𝑑_𝜉Ω𝑁^𝑛,𝑘_𝑝,𝑞𝜉_𝑝,𝑞.(6.42) We know that 𝜉 is almost surely in _𝜉Ω^, and, by Lemma 6.8, we also know that on _𝜉Ω^×_𝜉Ω^lim_𝑁→∞𝜉,^𝛼,𝛽𝑆_𝑁−𝐼𝑑_𝜉Ω𝑁(𝜉)=𝜉,^𝛼,𝛽𝑆−𝐼𝑑_𝜉Ω^(𝜉),(6.43) so that we can effectively write the infinite-dimensional Radon-Nikodym derivative on _𝜉Ω^ as the point-wise limit of the finite-dimensional one on _𝜉Ω_𝑁 through the projectors _𝜉𝑃_𝑁: 𝑑𝑃_𝛽𝑑𝑃_𝛼(𝜔)=lim_𝑁→∞𝑑𝑃^𝑁_𝛽𝑑𝑃^𝑁_𝛼(𝜔),(6.44) which directly yields formula (6.37).
The derivation of the Girsanov formula (6.40) from (6.37) comes from the relationship between the Stratonovich and Itô formulas since the quadratic variation of _𝛼𝑋_𝑡/𝑔_𝛼(𝑡) and (𝛼(𝑡)−𝛽(𝑡))/𝑓²_𝛼(𝑡)×_𝛼𝑋_t/𝑔_𝛼(𝑡) reads ¹₀𝐸^𝑡₀𝑓_𝛼(𝑠)𝑑𝑊_𝑠,𝛼(𝑡)−𝛽(𝑡)𝑓_𝛼²(𝑡)^𝑡₀𝑓_𝛼(𝑠)𝑑𝑊_𝑠=¹₀(𝛼(𝑡)−𝛽(𝑡))𝑑𝑡.(6.45) Therefore, the expression of the Radon-Nikodym derivative in Lemma 6.8 can be written in terms of the Itô integrals as 𝜉,^𝛼,𝛽𝑆−𝐼𝑑_𝜉Ω(𝜉)=¹₀(𝛼(𝑡)−𝛽(𝑡))𝑑𝑡+2¹₀𝛼(𝑡)−𝛽(𝑡)𝑓²(𝑡)^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)𝑑^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)+¹₀(𝛼(𝑡)−𝛽(𝑡))²𝑓²(𝑡)^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)²𝑑𝑡(6.46) and the Radon-Nikodym derivative as 𝑑𝑃_𝛽𝑑𝑃_𝛼(𝜔)=exp¹₀𝛽(𝑡)−𝛼(𝑡)𝑓_𝛼²(𝑡)^𝛼𝑋_𝑡(𝜔)𝑔_𝛼(𝑡)𝑑^𝛼𝑋_𝑡(𝜔)𝑔_𝛼(𝑡)−12¹₀(𝛽(𝑡)−𝛼(𝑡))²𝑓_𝛼²(𝑡)^𝛼𝑋_𝑡(𝜔)𝑔_𝛼(𝑡)²𝑑𝑡.(6.47)

Observe that, if 𝛼(𝑡)=0, we recover the familiar expression𝑑𝑃_𝛽𝑑𝑃_𝛼(𝜔)=exp⎛⎜⎝¹₀𝛽(𝑡)√Γ(𝑡)𝑊_𝑡(𝜔)𝑑𝑊_𝑡(𝜔)−12¹₀𝛽(𝑡)²Γ(𝑡)𝑊(𝜔)²_𝑡𝑑𝑡⎞⎟⎠.(6.48)

Conclusion and Perspectives
The discrete construction we present displays both analytical and numerical interests for further applications. From the analysis viewpoint, even if the basis does not exhibit the same orthogonal properties as the Karhunen-Loève decomposition, it has the important advantage of saving the structure of sample paths through its property of strong pathwise convergence and of providing a multiscale representation of the processes, which contrasts with the convergence in the mean of the Karhunen-Loève decomposition. From the numerical viewpoint, three Haar-like properties make our decomposition particularly suitable for certain numerical computations: (i) all basis elements have compact support on an open interval that has the structure of dyadic rational endpoints, (ii) these intervals are nested and become smaller for larger indices of the basis element, and (iii) for any interval endpoint, only a finite number of basis elements are nonzero at that point. Thus the expansion in our basis, when evaluated at an interval endpoint (e.g., dyadic rational), terminates in a finite number of steps. Moreover, the very nature of the construction based on an increasingly refined description of the sample paths paves the way to coarse-graining approaches similar to wavelet decompositions in signal processing. In view of this, our framework offers promising applications.Dichotomic Search of First-Hitting Times
The first application we envisage concerns the problem of first-hitting times. Because of its manifold applications, finding the time when a process first exits a given region is a central question of stochastic calculus. However, closed-form theoretical results are scarce and one often has to resort to numerical algorithms [59]. In this regard, the multiresolution property suggests an exact scheme to simulate sample paths of a Gaussian Markov process 𝐗 in an iterative “top-down’’ fashion. Assuming the intervals are dyadic rational and that we have a conditional knowledge of a sample path on the dyadic points of 𝐷_𝑁={𝑘2^−𝑁∣0≤𝑘≤2^𝑁}, one can decide to further the simulation of this sample path at any time 𝑡 in 𝐷_𝑁+1 by drawing a point according to the conditional law of 𝐗_𝑡 given {𝐗_𝑡}_{𝑡∈𝐷𝑁}, which is simply expressed in the framework of our construction. This property can be used for great advantages in numerical computations such as dichotomic search algorithms for first passage times: the key element is to find an estimate of the true conditional probability that a hitting time has occurred when knowing the value of the process at two given times, one in the past and one in the future. With such an estimate, an efficient strategy to look for passage times consists in refining the sample path when and only when its trajectory is estimated likely to actually cross the barrier. Thus the sample path of the process is represented at poor temporal resolution when it is far from the boundary and at increasingly higher resolution closer to the boundary. Such an algorithmic principle achieves a high level of precision in the computation of the first-hitting time, while demanding far less operation than usual stochastic Runge-Kutta scheme. This approach has been successfully implemented for the one-dimensional case [58], see Figure 4. In that article, the precision of the algorithm is controlled as well as the probability to evaluate a first hitting time substantially different from the actual value. The approach proves to be extremely efficient compared to customary methods. The general multidimensional approach proposed in the present paper allows direct generalization of these results to the computation of exit times in any dimension and for general smooth sets [30–32].Gaussian Deformation Modes in Nonlinear Diffusions
The present study is developed for the Gauss-Markov systems. However, many models arising in applied science present nonlinearities, and in that case, the construction based on a sum of Gaussian random variables will not generalize. However, the Gaussian case treated here can nevertheless be applied to perturbation of nonlinear differential equations with small noise. Let 𝐅∶ℝ×ℝ^𝑑↦ℝ^𝑑 be a nonlinear time-varying vector field, and let us assume that 𝐗₀(𝑡) is a stable (attractive) solution of the dynamical system: 𝑑𝐗𝑑𝑡=𝐅(𝐭,𝐗).(6.49) This function 𝐗₀ can for instance be a fixed point (in which case it is a constant), a cycle (in which case it is periodic), or a general attractive orbit of the system. In the deterministic case, any solution having its initial condition in a given neighbourhood ℬ in ℝ×ℝ^𝑑 of the solution will asymptotically converge towards the solution, and therefore perturbations of the solutions are bounded. Let us now consider that the system is subject to a small amount of noise and define 𝐘∈ℝ^𝑑 as the solution of the stochastic nonlinear differential equation: 𝑑𝐘_𝑡=𝐅𝑡,𝐘_𝑡𝑑𝑡+𝜀𝚪𝑡,𝑌_𝑡𝑑𝐖_𝑡.(6.50) Assuming that the noise is small (i.e., 𝜀 is a small parameter), because of the attractivity of the solution 𝐗₀(𝑡), the function 𝐘(𝑡) will remain very close to 𝐗₀(𝑡) (at least in a bounded time interval). In this region, we define 𝜀𝐙_𝑡=𝐘_𝑡−𝐗₀(𝑡). This stochastic variable is the solution of the equation 𝑑𝐙_𝑡=1𝜀𝐅𝑡,𝐗₀(𝑡)+𝜀𝐙_𝑡−𝐅𝑡,𝐗₀(𝑡)+𝜀𝚪𝑡,𝐗₀(𝑡)+𝜀𝐙_𝑡𝑑𝐖_𝑡=∇_𝑥𝐅𝑡,𝐗₀(𝑡)𝐙_𝑡+𝚪𝑡,𝐗₀(𝑡)𝑑𝐖_𝑡+𝑂(𝜀).(6.51) The solution at the first order in 𝜀 is therefore the multidimensional Gaussian process with nonconstant coefficients: 𝑑𝐙_𝑡=∇_𝑥𝐅𝑡,𝐗₀(𝑡)𝐙_𝑡+𝚪𝑡,𝐗₀(𝑡)𝑑𝐖_𝑡,(6.52) and our theory describes the solutions in a multiresolution framework. Notice that, in that perspective, our basis functions 𝝍_𝑛,𝑘 can be seen as increasingly finer modes of deformation of a deterministic trajectory. This approach appears particularly relevant to the theory of weakly interconnected neural oscillators in computational neuroscience [60]. Indeed, one of the most popular approaches of this field, the phase model theory, formally consists in studying how perturbations are integrated in the neighborhood of an attracting cycle [61, 62].
All these instances are exemplary of how our multiresolution description of the Gauss-Markov processes offers a simple yet rigorous tool to broach a large number of open problems and promises fascinating applications both in theoretical and in applied science.

Figure 4 

First passage time for an Ornstein-Uhlenbeck process 𝑈 with a elastic coefficient 𝛼=1 and a noise intensity Γ=1, the barrier is constant Λ=1 and the initial condition is 𝑈0=0. The plots represent at different scales a realization 𝑈𝑡(𝜔) for which the algorithm of [58] returns a first passage time 𝜏(𝜔)=8.00469684 with a resolution 𝛿𝑡=221=5107 after 5 recursive calls. The whole sample path is represented in (a) and a series of zooms is carried out around 𝜏(𝜔) in (b), (c), and (d). The dilation coefficients are set according to the scale invariance of a Wiener process with the time scale being expanded by 20 and the distance scale by 4√5 during each dilation. The series (b), (c), and (c), (d) zoom on regions where the sample path gets close to the barrier. The simulation of the sample path has required 683 subdivisions and illustrates an unfavorable situation since the expected number of divisions is approximatively 284 for this particular setting.

Appendices

A. Formulae of the Basis for the Integrated Wiener Process

In the case of the primitive of the Wiener process, straightforward linear algebra computations lead to the two bases of functions ((𝜓_𝑛,𝑘)_1,1,(𝜓_𝑛,𝑘)_2,1) and ((𝜓_𝑛,𝑘)_1,2,(𝜓_𝑛,𝑘)_2,2) having the following expressions𝜓_𝑛,𝑘_1,1(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝜎_𝑛,𝑘_1,1𝑡−𝑙_𝑛,𝑘𝑚_𝑛,𝑘−𝑙_𝑛,𝑘²1+2𝑚_𝑛,𝑘−𝑡𝑚_𝑛,𝑘−𝑙_𝑛,𝑘,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,𝜎_𝑛,𝑘_1,1𝑟_𝑛,𝑘−𝑡𝑟_𝑛,𝑘−𝑚_𝑛,𝑘²1+2𝑡−𝑚_𝑛,𝑘𝑟_𝑛,𝑘−𝑚_𝑛,𝑘,𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘,𝜓_𝑛,𝑘_2,1(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝜎_𝑛,𝑘_1,16𝑡−𝑙_𝑛,𝑘𝑚_𝑛,𝑘−𝑡𝑚_𝑛,𝑘−𝑙_𝑛,𝑘³,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,−𝜎_𝑛,𝑘_1,16𝑟_𝑛,𝑘−𝑡𝑡−𝑚_𝑛,𝑘𝑟_𝑛,𝑘−𝑚_𝑛,𝑘³,𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘,𝜓_𝑛,𝑘_1,2(𝑡)=⎧⎪⎪⎨⎪⎪⎩−𝜎_𝑛,𝑘_2,2𝑚_𝑛,𝑘−𝑡𝑡−𝑙_𝑛,𝑘𝑚_𝑛,𝑘−𝑙_𝑛,𝑘²,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,𝜎_𝑛,𝑘_2,2𝑡−𝑚_𝑛,𝑘𝑟_𝑛,𝑘−𝑡𝑟_𝑛,𝑘−𝑚_𝑛,𝑘²,𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘,𝜓_𝑛,𝑘_2,2(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝜎_𝑛,𝑘_2,2𝑡−𝑙_𝑛,𝑘𝑚_𝑛,𝑘−𝑙_𝑛,𝑘²1−2𝑚_𝑛,𝑘−𝑡𝑡−𝑙_𝑛,𝑘,𝑙_𝑛,𝑘≤𝑡≤𝑚_𝑛,𝑘,𝜎_𝑛,𝑘_2,2𝑟_𝑛,𝑘−𝑡𝑟_𝑛,𝑘−𝑚_𝑛,𝑘²1−2𝑡−𝑚_𝑛,𝑘𝑟_𝑛,𝑘−𝑡,𝑚_𝑛,𝑘≤𝑡≤𝑟_𝑛,𝑘,(A.1) where𝜎_𝑛,𝑘_1,1=1196𝑟_𝑛,𝑘−𝑙_𝑛,𝑘³,𝜎_𝑛,𝑘_2,2=132𝑟_𝑛,𝑘−𝑙_𝑛,𝑘(A.2) are the diagonal components of the (diagonal) matrix 𝜎_𝑛,𝑘. As expected, we notice that the differential structure of the process is conserved at any finite rank since we have𝑑𝑑𝑡𝜓_𝑛,𝑘_1,1(𝑡)=𝜓_𝑛,𝑘_2,1(𝑡),𝑑𝑑𝑡𝜓_𝑛,𝑘_1,2(𝑡)=𝜓_𝑛,𝑘_2,2(𝑡).(A.3)

B. Formulae of the Basis for the Doubly-Integrated Wiener Process

For the doubly-integrated Wiener process, the construction of the three-dimensional process involves a family of three 3-dimensional functions, which constitutes the columns of a 3×3 matrix that we denote by 𝜓. This basis has again a simple expression when 𝑚_𝑛,𝑘 is the middle of the interval [𝑙_𝑛,𝑘,𝑟_𝑛,𝑘]:𝜓_𝑛,𝑘_1,1(𝑡)=𝜓_𝑛,𝑘_1,2(𝑡)⎧⎪⎪⎨⎪⎪⎩√560𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑡−𝑙_𝑛,𝑘³𝑙²_𝑛,𝑘−7𝑙_𝑛,𝑘𝑡+5𝑙_𝑛,𝑘𝑟_𝑛,𝑘−25𝑡𝑟_𝑛,𝑘+16𝑡²+10𝑟²_𝑛,𝑘,−√560𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑟_𝑛,𝑘−𝑡³𝑟²_𝑛,𝑘−7𝑟_𝑛,𝑘𝑡+5𝑙_𝑛,𝑘𝑟_𝑛,𝑘−25𝑡𝑙_𝑛,𝑘+16𝑡²+10𝑙²_𝑛,𝑘,𝜓_𝑛,𝑘_1,2(𝑡)=⎧⎪⎪⎨⎪⎪⎩√312𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑙_𝑛,𝑘−𝑡³2𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−3𝑡𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−2𝑡,√312𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑟_𝑛,𝑘−𝑡³𝑟_𝑛,𝑘+2𝑙_𝑛,𝑘−3𝑡𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−2𝑡,𝜓_𝑛,𝑘_1,3(𝑡)=⎧⎪⎨⎪⎩−16𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑡−𝑙_𝑛,𝑘³2𝑡−𝑟_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−3𝑡²𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−2𝑡,−16𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑟_𝑛,𝑘−𝑡³2𝑡−𝑟_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−3𝑡²𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−2𝑡,𝜓_𝑛,𝑘_2,1(𝑡)=⎧⎪⎪⎨⎪⎪⎩√16𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑡−𝑙_𝑛,𝑘²3𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−4∗𝑡𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−2𝑡,−√16𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑟_𝑛,𝑘−𝑡²3𝑙_𝑛,𝑘+𝑟_𝑛,𝑘−4∗𝑡𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−2𝑡,𝜓_𝑛,𝑘_2,2(𝑡)=⎧⎪⎪⎨⎪⎪⎩−√36𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑡−𝑙_𝑛,𝑘²3𝑟_𝑛,𝑘+2𝑙_𝑛,𝑘−5𝑡𝑟_𝑛,𝑘+2𝑙_𝑛,𝑘−3𝑡,−√36𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑟_𝑛,𝑘−𝑡²3𝑙_𝑛,𝑘+2𝑟_𝑛,𝑘−5𝑡𝑙_𝑛,𝑘+2𝑟_𝑛,𝑘−3𝑡,𝜓_𝑛,𝑘_2,3(𝑡)=⎧⎪⎨⎪⎩−16𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑡−𝑙_𝑛,𝑘³2𝑡−𝑟_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−3𝑡²𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−2𝑡,−16𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑟_𝑛,𝑘−𝑡³2𝑡−𝑟_𝑛,𝑘−𝑙_𝑛,𝑘𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−3𝑡²𝑟_𝑛,𝑘+𝑙_𝑛,𝑘−2𝑡,𝜓_𝑛,𝑘_3,1(𝑡)=⎧⎪⎪⎨⎪⎪⎩√53𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑡−𝑙_𝑛,𝑘4𝑙²_𝑛,𝑘−17𝑙_𝑛,𝑘𝑡+9𝑙_𝑛,𝑘𝑟_𝑛,𝑘−15𝑡𝑟_𝑛,𝑘+3𝑟²_𝑛,𝑘+16𝑡²,√53𝑟_𝑛,𝑘−𝑙_𝑛,𝑘^5/2𝑟_𝑛,𝑘−𝑡4𝑟²_𝑛,𝑘−17𝑡𝑟_𝑛,𝑘+9𝑙_𝑛,𝑘𝑟_𝑛,𝑘−15𝑡𝑙_𝑛,𝑘+3𝑙²_𝑛,𝑘+16𝑡²,𝜓_𝑛,𝑘_3,2(𝑡)=⎧⎪⎪⎨⎪⎪⎩√3(𝑟−𝑙)^5/2(𝑙−𝑡)(𝑟+𝑙−2𝑡)(𝑟+4𝑙−5𝑡),√3(𝑟−𝑙)^5/2(𝑟−𝑡)(𝑟+𝑙−2𝑡)(𝑙+4𝑟−5𝑡),𝜓_𝑛,𝑘_3,3(𝑡)=⎧⎪⎨⎪⎩13(𝑟−𝑙)^5/2(𝑙−𝑡)19𝑙²−56𝑙𝑡+18𝑙𝑟−24𝑡𝑟+3𝑟²+40𝑡²,13(𝑟−𝑙)^5/2(𝑟−𝑡)19𝑟²−56𝑟𝑡+18𝑙𝑟−24𝑡𝑙+3𝑙²+40𝑡².(B.1) Notice again that the basis functions satisfy the relationships𝑑𝑑𝑡𝜓_𝑛,𝑘_𝑖,𝑗(𝑡)=𝜓_𝑛,𝑘_𝑖+1,𝑗(𝑡)(B.2) for 𝑖∈{1,2} and 𝑗∈{1,2,3}. These functions also form a triorthogonal basis of functions, which makes it easy to simulate sample paths of the doubly-integrated Wiener process, as show in Figure 5.

(a)

(b)

Figure 5 

(a) Basis for the construction of the Doubly Integrated Wiener process (𝑑=3) and (b) 10-steps construction of the process.

C. Properties of the Lift Operators

This appendix is devoted to the proofs of the properties of the lift operator enumerated in Proposition 6.3. The proposition is split into three lemmas for the sake of clarity.

Lemma C.1. The operator _𝛼,𝛽𝐺 is a linear measurable bijection. Moreover, for every 𝑁>0, the function _𝛼,𝛽𝐺_𝑁=𝑃_𝑁∘_𝛼,𝛽𝐺∘𝐼_𝑁∶_𝜉Ω^_𝑁→_𝜉Ω_𝑁 is a finite-dimensional linear operator, whose matrix representation is triangular in the natural basis of _𝜉Ω_𝑁 and whose eigenvalues _𝛼,𝛽𝜈_𝑛,𝑘 are given by _𝛼,𝛽𝜈_𝑛,𝑘=𝑔_𝛼𝑚_𝑛,𝑘𝑔_𝛽𝑚_𝑛,𝑘^𝛽_𝑛𝑀_,𝑘𝛼𝑀_𝑛,𝑘,0≤𝑛≤𝑁,0≤𝑘<2^𝑁−1.(C.1)
Eventually, _𝛼,𝛽𝐺 is abounded operator for the spectral norm with ‖‖_𝛼,𝛽𝐺‖‖₂=sup_𝑛sup_{𝑘𝛼,𝛽}𝜈_𝑛,𝑘≤sup𝑔_𝛼inf𝑔_𝛽sup𝑓_𝛽inf𝑓sup𝑓²_𝛼inf𝑓²_𝛽<∞,(C.2) and the determinant of _𝛼,𝛽𝐺_𝑁 denoted by _𝛼,𝛽𝐽_𝑁 admits a limit when 𝑁 tends to infinity: lim_𝑁→∞det_𝛼,𝛽𝐺_𝑁=lim_{𝑁→∞𝛼,𝛽}𝐽_𝑁=exp12¹₀(𝛼(𝑡)−𝛽(𝑡))𝑑𝑡=_𝛼,𝛽𝐽.(C.3)

Proof. All these properties are deduced from the properties of the functions Δ and Ψ derived previously.(i)_𝛼,𝛽𝐺=_𝛽Δ∘_𝛼Ψ is a linear measurable bijection of _𝜉Ω^ due to the composed application of two linear bijective measurable functions _𝛼Δ∶_𝑥Ω^→_𝜉Ω^ and _𝛼Ψ∶_𝜉Ω^→_𝑥Ω^.(ii)Since we have the expressions of the matrices of the finite-dimensional linear transformations, it is easy to write the linear transformation of _𝛼,𝛽𝐺_𝑁 on the natural basis 𝑒_𝑛,𝑘 as _𝛼,𝛽𝐺_𝑁(𝜉)_𝑛,𝑘=_𝑈^𝛽𝛿_𝑛,𝑘(𝑡)_𝛼Ψ(𝜉)(𝑡)𝑑𝑡=_{(𝑝,𝑞)∈ℐ𝑁}_𝑈^𝛽𝛿_𝑛,𝑘(𝑡)_𝛼𝜓_𝑝,𝑞(𝑡)𝑑𝑡𝜉_𝑝,𝑞,(C.4) leading to the coefficient expression _𝛼,𝛽𝐺^𝑛,𝑘_𝑝,𝑞=_𝑈^𝛽𝛿_𝑛,𝑘(𝑡)_𝛼𝜓_𝑝,𝑞(𝑡)𝑑𝑡=_{𝑖,𝑗∈ℐ𝑁}^𝛽Δ^𝑛,𝑘_𝑖,𝑗⋅_𝛼Ψ^𝑖,𝑗_𝑝,𝑞,(C.5) where we have dropped the index 𝑁 since the expression of the coefficients does not depend on it. We deduce from the form of the matrices _𝛽Δ_𝑁 and _𝛼Ψ_𝑁 that the application _𝛼,𝛽𝐺_𝑁 has a matrix representation in the basis 𝑒_𝑛,𝑘 of the form _𝛼,𝛽𝐺_𝑁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣^𝛼,𝛽0,0𝐺_0,0𝛼,𝛽𝐺^0,0_1,0𝛼,𝛽𝐺^1,0_1,0𝛼,𝛽𝐺^0,0_2,0𝛼,𝛽𝐺^1,0_2,0𝛼,𝛽𝐺^2,0_2,0𝛼,𝛽𝐺^0,0_2,1𝛼,𝛽𝐺^1,0_2,1𝛼,𝛽𝐺^2,1_2,1𝛼,𝛽𝐺^0,0_3,0𝛼,𝛽𝐺^1,0_3,0𝛼,𝛽𝐺^2,0_3,0𝛼,𝛽𝐺^3,0_3,0𝛼,𝛽𝐺^0,0_3,1𝛼,𝛽𝐺^1,0_3,1𝛼,𝛽𝐺^2,0_3,1𝛼,𝛽𝐺^3,1_3,1𝛼,𝛽𝐺^0,0_3,2𝛼,𝛽𝐺^1,0_3,2𝛼,𝛽𝐺^2,1_3,2𝛼,𝛽𝐺^3,2_3,2𝛼,𝛽𝐺^0,0_3,3𝛼,𝛽𝐺^1,0_3,3𝛼,𝛽𝐺^2,1_3,3𝛼,𝛽𝐺^3,3_3,3⋮⋱⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(C.6) where we only represent the nonzero terms.The eigenvalues of the operator are therefore the diagonal elements _𝛼,𝛽𝐺^𝑛,𝑘_𝑛,𝑘 that are easily computed from the expressions of the general term of the matrix: _𝛼,𝛽𝐺^𝑛,𝑘_𝑛,𝑘=_𝛽Δ^𝑛,𝑘_{𝑛,𝑘𝛼}𝜓^𝑛,𝑘_𝑛,𝑘=^𝛽𝑀_𝑛,𝑘𝑔_𝛽𝑚_𝑛,𝑘𝑔_𝛼𝑚_𝑛,𝑘_𝛼𝑀_𝑛,𝑘=𝑔_𝛼𝑚_𝑛,𝑘𝑔_𝛽𝑚_𝑛,𝑘^𝛽_𝑛𝑀_,𝑘𝛼𝑀_𝑛,𝑘.(C.7)(iii)From the expression of _𝛼𝑀_𝑛,𝑘=√(ℎ_𝛼(𝑟)−ℎ_𝛼(𝑚))(ℎ_𝛼(𝑚)−ℎ_𝛼(𝑙))/(ℎ_𝛼(𝑟)−ℎ_𝛼(𝑙)), we deduce the inequalities sup𝑓inf𝑓²_𝛼2^𝑛+1≤_𝛼𝑀_𝑛,𝑘≤sup𝑓inf𝑓²_𝛼2^𝑛+1,(C.8) from which follows the given upper-bound to the singular values.(iv)_𝛼,𝛽𝐺_𝑁 is a finite-dimensional triangular linear matrix in the basis {𝑒_𝑛,𝑘}. Its determinant is simply given as the product _𝛼,𝛽𝐽_𝑁=^𝑁_𝑛=0_{0≤𝑘<2𝑛−1}^𝛼,𝛽𝐺^𝑛,𝑘_𝑛,𝑘=^𝑁_𝑛=0_{0≤𝑘<2𝑛−1}^𝛼,𝛽𝜈_𝑛,𝑘,(C.9) where we noticed that the eigenvalues _𝛼,𝛽𝜈_𝑛,𝑘 are of the form _𝛼,𝛽𝜈_𝑛,𝑘=𝑔_𝛼𝑚_𝑛,𝑘𝑔_𝛽𝑚_𝑛,𝑘^𝛽_𝑛𝑀_,𝑘𝛼𝑀_𝑛,𝑘.(C.10) Since, for every 𝑁>0, we have _𝛼,𝛽𝐺_𝑁=_𝜁,𝛽𝐺_𝑁∘_𝛼,𝜁𝐺_𝑁, which entails that _𝛼,𝛽𝐽_𝑁=_𝜁,𝛽𝐽_𝑁⋅(_𝜁,𝛼𝐽_𝑁)⁻¹, it is enough to show that we have lim_{𝑁→∞𝛼,0}𝐽_𝑁=exp12¹₀𝛼(𝑡)𝑑𝑡.(C.11) Now, writing for every 0≤𝑠<𝑡≤1 the quantity _𝛼𝒱_𝑡,𝑠=^𝑡_𝑠Γ(𝑢)𝑒^{2∫𝑡𝑢𝛼(𝑣)𝑑𝑣}𝑑𝑢,(C.12) we have ⎛⎜⎝𝑔_𝛼𝑚_𝑛,𝑘_𝛼𝑀_𝑛,𝑘⎞⎟⎠²=^𝛼𝒱_{𝑙𝑛,𝑘,𝑚𝑛,𝑘}⋅_𝛼𝒱_{𝑚𝑛,𝑘,𝑟𝑛,𝑘𝛼}𝒱_{𝑙𝑛,𝑘,𝑟𝑛,𝑘},(C.13) so that _𝛼,0𝐽_𝑁 is a telescoping product that can be written as ^𝛼,0𝐽_𝑁²=^2𝑁_𝑘=0^𝛼_𝑘𝒱_{2−𝑁,(𝑘+1)2−𝑁0}𝒱_{𝑘2−𝑁,(𝑘+1)2−𝑁}.(C.14) If 𝛼 is Hölder continuous, there exist 𝛿>0 and 𝐶>0 such that sup_{0≤𝑠,𝑡≤1}||𝛼(𝑡)−𝛼(𝑠)|||𝑡−𝑠|^𝛿<𝐶,(C.15) and introducing, for any 0≤𝑠<𝑡≤1, the quantity 𝒬_𝑡,𝑠, 𝒬_𝑡,𝑠=𝑒^{−∫𝑡𝑠𝛼(𝑣)𝑑𝑣}⋅^𝛼_𝑡𝒱_,𝑠0𝒱_𝑡,𝑠=|||||∫^𝑡_𝑠Γ(𝑢)𝑒^{∫𝑡𝑢𝛼(𝑣)𝑑𝑣−∫𝑢𝑠𝛼(𝑣)𝑑𝑣}𝑑𝑢∫^𝑡_𝑠Γ(𝑢)𝑑𝑢|||||,(C.16) we have that 𝒬_𝑡,𝑠≤𝒬_𝑡,𝑠≤𝒬_𝑡,𝑠 with 𝒬_𝑡,𝑠=∫^𝑡_𝑠Γ(𝑢)𝑒^{−(𝐶/(1+𝛿))((𝑡−𝑢)1+𝛿+(𝑢−𝑠)1+𝛿)}𝑑𝑢∫^𝑡_𝑠Γ(𝑢)𝑑𝑢,𝒬_𝑡,𝑠=∫^𝑡_𝑠Γ(𝑢)𝑒^{(𝐶/(1+𝛿))((𝑡−𝑢)1+𝛿+(𝑢−𝑠)1+𝛿)}𝑑𝑢∫^𝑡_𝑠Γ(𝑢)𝑑𝑢.(C.17) After expanding the exponential in the preceding definitions, we have 𝒬_𝑡,𝑠≥1−2𝐶sup_0≤𝑡≤1Γ(𝑡)inf_0≤𝑡≤1Γ(𝑡)(𝑡−𝑠)^(1+𝛿)(1+𝛿)(2+𝛿)+𝑜(𝑡−𝑠)^(1+𝛿),𝒬_𝑡,𝑠≤1+2𝐶sup_0≤𝑡≤1Γ(𝑡)inf_0≤𝑡≤1Γ(𝑡)(𝑡−𝑠)^(1+𝛿)(1+𝛿)(2+𝛿)+𝑜(𝑡−𝑠)^(1+𝛿),(C.18) now, from ^2𝑁_𝑘=0𝒬_{𝑘2−𝑁,(𝑘+1)2−𝑁}=1+𝑜2^−𝑁,(C.19) we can directly conclude that lim_{𝑁→∞𝛼,0}𝐽_𝑁=𝑒^{(1/2)∫10𝛼(𝑡)𝑑𝑡}⎷lim_𝑁→∞^2𝑁_𝑘=0𝒬_{𝑘2−𝑁,(𝑘+1)2−𝑁}=𝑒^{(1/2)∫10𝛼(𝑡)𝑑𝑡}.(C.20)

Notice that, if 𝛼=𝛽, _𝛼,𝛼𝐺 is the identity and _𝛼,𝛼𝐽=1 as expected.

Similar properties are now proved for the process lift operator _𝛼,𝛽𝐻.

Lemma C.2. The function _𝛼,𝛽𝐻 is a linear measurable bijection.
Moreover, for every 𝑁>0, the function _𝛼,𝛽𝐻_𝑁=𝑃_𝑁∘_𝛼,𝛽𝐻∘𝐼_𝑁∶_𝑥Ω^_𝑁→_𝑥Ω_𝑁 is a finite-dimensional linear operator, whose matrix representation is triangular in the natural basis of _𝑥Ω_𝑁 and whose eigenvalues are given by _𝛽,𝛼𝜈_𝑛,𝑘=_𝛼,𝛽𝜈_𝑛,𝑘⁻¹=𝑔_𝛽𝑚_𝑛,𝑘𝑔_𝛼𝑚_𝑛,𝑘^𝛼_𝑛𝑀_,𝑘𝛽𝑀_𝑛,𝑘.(C.21) Eventually, _𝛼,𝛽𝐻 is a bounded operator for the spectral norm with ‖‖_𝛼,𝛽𝐻‖‖₂=‖‖_𝛽,𝛼𝐺‖‖₂=sup_𝑛sup_{𝑘𝛽,𝛼}𝜈_𝑛,𝑘≤sup𝑔_𝛽inf𝑔_𝛼sup𝑓inf𝑓_𝛽sup𝑓²_𝛽inf𝑓²_𝛼<∞,(C.22) and the determinant of _𝛼,𝛽𝐻_𝑁 admits a limit when 𝑁 tends to infinity: lim_𝑁→∞det_𝛼,𝛽𝐻=exp12¹₀(𝛽(𝑡)−𝛼(𝑡))𝑑𝑡=_𝛽,𝛼𝐽.(C.23)

Proof. (i)The function _𝛼,𝛽𝐻=_𝛽Ψ∘_𝛼Δ is a linear measurable bijection of _𝑥Ω^ onto _𝑥Ω^ because _𝛼Δ∶_𝑥Ω^→_𝜉Ω^ and _𝛼Ψ∶_𝜉Ω^→_𝑥Ω^ are linear bijective measurable functions.(ii)We write the linear transformation of _𝛼,𝛽𝐻 for 𝑥 in _𝑥Ω^ as _𝛼,𝛽𝐻[𝑥](𝑡)=_{(𝑛,𝑘)∈ℐ}^𝛽𝜓_𝑛,𝑘(𝑡)_𝑈^𝛼𝛿_𝑛,𝑘(𝑠)𝑥(𝑠)𝑑𝑠=_𝑈⎛⎜⎝_(𝑛,_𝑘)∈ℐ^𝛽𝜓_𝑛,𝑘(𝑡)_𝛼𝛿_𝑛,𝑘(𝑡)⎞⎟⎠𝑥(𝑡)𝑑𝑡.(C.24) If we denote the class of 𝑥 in _𝑥Ω^ by 𝑥={𝑥_𝑖,𝑗}={𝑥(𝑚_𝑖,𝑗)}, (𝑖,𝑗)∈ℐ_𝑁, we can write (C.24) as _𝛼,𝛽𝐻_𝑁(𝑥)_𝑖,𝑗=_{(𝑘,𝑙)∈ℐ𝑁}⎛⎜⎝_(𝑝,𝑞_)∈ℐ𝑁^𝛽Ψ^𝑖,𝑗_𝑝,𝑞⋅_𝛼Δ^𝑝,𝑞_𝑘,𝑙⎞⎟⎠𝑥_𝑘,𝑙,(C.25) from which we deduce the expression of the coefficients of the matrix _𝛼,𝛽𝐻_𝑁: _𝛼,𝛽𝐻^𝑖,𝑗_𝑘,𝑙=_{(𝑝,𝑞)∈ℐ𝑁}^𝛽Ψ^𝑖,𝑗_𝑝,𝑞⋅_𝛼Δ^𝑝,𝑞_𝑘,𝑙,(C.26) where as usual we drop the index 𝑁. Because of the the form of the matrices _𝛼Δ_𝑁 and _𝛽Ψ_𝑁, the matrix _𝛼,𝛽𝐻_𝑁 in the basis 𝑓_𝑖,𝑗 has the following triangular form: _𝛼,𝛽𝐻_𝑁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣^𝛼,𝛽0,0𝐻_0,0𝛼,𝛽𝐻^0,0_1,0𝛼,𝛽𝐻^1,0_1,0𝛼,𝛽𝐻^0,0_2,0𝛼,𝛽𝐻^1,0_2,0𝛼,𝛽𝐻^2,0_2,0𝛼,𝛽𝐻^0,0_2,1𝛼,𝛽𝐻^1,0_2,1𝛼,𝛽𝐻^2,1_2,1𝛼,𝛽𝐻^0,0_3,0𝛼,𝛽𝐻^1,0_3,0𝛼,𝛽𝐻^2,0_3,0𝛼,𝛽𝐻^3,0_3,0𝛼,𝛽𝐻^0,0_3,1𝛼,𝛽𝐻^1,0_3,1𝛼,𝛽𝐻^2,0_3,1𝛼,𝛽𝐻^3,1_3,1𝛼,𝛽𝐻^0,0_3,2𝛼,𝛽𝐻^1,0_3,2𝛼,𝛽𝐻^2,1_3,2𝛼,𝛽𝐻^3,2_3,2𝛼,𝛽𝐻^0,0_3,3𝛼,𝛽𝐻^1,0_3,3𝛼,𝛽𝐻^2,1_3,3𝛼,𝛽𝐻^3,3_3,3⋮⋱⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(C.27) From the matrix representations _𝛼Δ and _𝛽Ψ, the diagonal terms of _𝛼,𝛽𝐻 read _𝛼,𝛽𝐻^𝑖,𝑗_𝑖,𝑗=_𝛽Ψ^𝑖,𝑗_𝑖,𝑗⋅_𝛼Δ^𝑖,𝑗_𝑖,𝑗=_𝛽𝜓_𝑖,𝑗𝑚_𝑖,𝑗^𝛼𝑀_𝑖,𝑗𝑔_𝛼𝑚_𝑖,𝑗=𝑔_𝛽𝑚_𝑖,𝑗𝑔_𝛼𝑚_𝑖,𝑗^𝛼_𝑖𝑀_,𝑗𝛽𝑀_𝑖,𝑗=_𝛽,𝛼𝜈_𝑖,𝑗=_𝛼,𝛽𝜈_𝑖,𝑗⁻¹.(C.28)(iii)The upper bound directly follows from the fact that _𝛽,𝛼𝜈_𝑖,𝑗=(_𝛼,𝛽𝜈_𝑖,𝑗)⁻¹.(iv) Since _𝛽,𝛼𝜈_𝑖,𝑗=(_𝛼,𝛽𝜈_𝑖,𝑗)⁻¹, the value of the determinant of _𝛼,𝛽𝐻_𝑁 is clearly the inverse of the determinant of _𝛼,𝛽𝐺_𝑁, so that lim_𝑁→∞det(_𝛼,𝛽𝐻_𝑁)=(_𝛼,𝛽𝐽)⁻¹=_𝛽,𝛼𝐽.

Note that Lemma C.2 directly follows from the fact that _𝛼Ψ and _𝛼Δ are inverse of each other and admit a triangular matrix representation. More precisely, when restricted to the finite-dimensional case, we have set the following properties.

Properties C.3. We have the following set of properties in terms of matrix operations(i)_𝛼,𝛽𝐻_𝑁=_𝛽Ψ_𝑁⋅_𝛼Δ_𝑁 and _𝛼,𝛽𝐺_𝑁=_𝛽Δ_𝑁⋅_𝛼Ψ_𝑁,(ii)_𝛼,𝛽𝐻_𝑁=_𝜁,𝛽𝐻_𝑁⋅_𝛼,𝜁𝐻_𝑁 and _𝛼,𝛽𝐺_𝑁=_𝜁,𝛽𝐺_𝑁⋅_𝛼,𝜁𝐺_𝑁,(iii)_𝛼,𝛽𝐻⁻¹_𝑁=_𝛽,𝛼𝐻_𝑁 and _𝛼,𝛽𝐺⁻¹_𝑁=_𝛽,𝛼𝐺_𝑁,(iv)_𝛼,𝛽𝐻_𝑁⋅_𝛼Ψ_𝑁=_𝛽Ψ_𝑁 and _𝛼,𝛽𝐺_𝑁⋅_𝛼Δ_𝑁=_𝛽Δ_𝑁,(v)_𝛼,𝛽𝐻_𝑁^𝑇⋅_𝛽Δ_𝑁^𝑇=_𝛼Δ_𝑁^𝑇 and _𝛼,𝛽𝐺_𝑁^𝑇⋅_𝛽Φ_𝑁^𝑇=_𝛼Φ_𝑁^𝑇.

Proof. (i)Let us write _𝜉𝑉_𝑁=⊕^𝑁_𝑛=0vect({𝑒_𝑛,𝑘}_{0≤𝑘<2𝑛−1}) and _𝑥𝑉_𝑁=⊕^𝑁_𝑛=0vect({𝑓_𝑖,𝑗}_{0≤𝑗<2𝑖−1}). Since _𝛼Ψ and _𝛽Ψ project the flag _𝜉𝑉₀⊂_𝜉𝑉₁⊂⋯⊂_𝜉𝑉_𝑁 onto the flag _𝑥𝑉₀⊂_𝑥𝑉₁⊂⋯⊂_𝑥𝑉_𝑁, and since conversely _𝛼Δ and _𝛼Δ project the flag _𝑥𝑉₀⊂_𝑥𝑉₁⊂…⊂_𝑥𝑉_𝑁 onto the flag _𝜉𝑉₀⊂_𝜉𝑉₁⊂…⊂_𝜉𝑉_𝑁, we can write_𝛼,𝛽𝐻_𝑁=_𝑥𝑃_𝑁∘_𝛽Ψ∘_𝛼Δ∘_𝑥𝐼_𝑁=^𝑥𝑃_𝑁∘_𝛽Ψ∘_𝑥𝐼_𝑁∘_𝑥𝑃_𝑁𝛼Δ∘_𝑥𝐼_𝑁=_𝛽Ψ_𝑁∘_𝛼Δ_𝑁,_𝛼,𝛽𝐺_𝑁=_𝜉𝑃_𝑁∘_𝛽Δ∘_𝛼Ψ∘_𝜉𝐼_𝑁=^𝜉𝑃_𝑁∘_𝛽Δ∘_𝜉𝐼_𝑁∘^𝜉𝑃_𝑁𝛼Ψ∘_𝜉𝐼_𝑁=_𝛽Δ_𝑁∘_𝛼Ψ_𝑁.(C.29)(ii)We have_𝜁,𝛽𝐻_𝑁∘_𝛼,𝜁𝐻_𝑁=_𝛽Ψ_𝑁∘_𝜁Δ_𝑁∘_𝜁Ψ_𝑁∘_𝛼Δ_𝑁=_𝛽Ψ_𝑁∘𝐼𝑑_𝜉Ω𝑁∘_𝛼Δ_𝑁=_𝛼,𝛽𝐻_𝑁,_𝜁,𝛽𝐺_𝑁∘_𝛼,𝜁𝐺_𝑁=_𝛽Δ_𝑁∘^𝜁Ψ_𝑁∘_𝜁Δ_𝑁∘_𝛼Ψ_𝑁=_𝛽Δ_𝑁∘𝐼𝑑_𝑥Ω𝑁∘_𝛼Ψ_𝑁.=_𝛼,𝛽𝐻_𝑁.(C.30)(iii)We have_𝛼,𝛽𝐻_𝑁∘_𝛽,𝛼𝐻_𝑁=_𝛼,𝛼𝐻_𝑁=𝐼𝑑_𝜉Ω𝑁,_𝛼,𝛽𝐺_𝑁∘_𝛽,𝛼𝐺_𝑁=_𝛼,𝛼𝐺_𝑁=𝐼𝑑_𝜉Ω𝑁.(C.31)(iv)We have_𝛼,𝛽𝐻_𝑁⋅_𝛼Ψ_𝑁=_𝛽Ψ_𝑁∘_𝛼Δ_𝑁∘_𝛼Ψ_𝑁=_𝛽Ψ_𝑁∘𝐼𝑑_𝜉Ω𝑁=_𝛽Ψ_𝑁,_𝛼,𝛽𝐺_𝑁⋅_𝛼Δ_𝑁=_𝛽Δ_𝑁∘_𝛼Ψ_𝑁∘_𝛼Δ_𝑁=_𝛽Δ_𝑁∘𝐼𝑑_𝑥Ω𝑁=_𝛽Δ_𝑁.(C.32)(v)We have_𝛼,𝛽𝐻_𝑁^𝑇⋅_𝛽Δ_𝑁^𝑇=_𝛼Δ_𝑁^𝑇∘^𝛽Ψ_𝑁^𝑇∘_𝛽Δ_𝑁^𝑇=_𝛼Δ_𝑁^𝑇∘𝐼𝑑_𝑥Ω𝑁=_𝛼Δ_𝑁^𝑇,_𝛼,𝛽𝐺_𝑁^𝑇⋅_𝛽Ψ_𝑁^𝑇=_𝛼Ψ_𝑁^𝑇∘^𝛽Δ_𝑁^𝑇∘_𝛽Ψ_𝑁^𝑇=_𝛼Ψ_𝑁^𝑇∘𝐼𝑑_𝜉Ω𝑁=_𝛼Ψ_𝑁^𝑇.(C.33)

D. Construction and Coefficient Applications

In this appendix, we provide the proofs of the main properties used in the paper regarding the construction and the coefficient applications.

D.1. The Construction Application

We start by addressing the case of the construction application introduced in Section 3.2.1.

We start by proving Proposition 3.11.

Proof. For the sake of simplicity, we will denotes for any function 𝐀∶[0,1]→ℝ^𝑚×𝑑, the uniform norm as |𝐀|_∞=sup_0≤𝑡≤1|𝐀(𝑡)|, where |𝐀(𝑡)|=sup_{0≤𝑖<𝑚}(∑^𝑑−1₀|𝐴_𝑖,𝑗(𝑡)|) is the operator norm induced by the uniform norms. We will also denote the 𝑖th line of 𝐀 by 𝑙_𝑖(𝐀) (it is a ℝ^𝑑-valued function) and the 𝑗th column of 𝐀 by 𝑐_𝑗(𝐀).
Let 𝝃∈_𝜉Ω^ be fixed. These coefficients induce a sequence of continuous functions 𝝍^𝑁(𝝃) through the action of the sequence of the partial construction applications. To prove that this sequence converges towards a continuous function, we show that it uniformly converges, which implies the result of the proposition using the fact that a uniform limit of continuous functions is a continuous function. Moreover, since the functions take values in ℝ^𝑑, which is a complete space, we show that for any sequence of coefficients 𝜉∈_𝜉Ω^, the sequence of functions 𝜓^𝑁(𝑡) constitutes a Cauchy sequence for the uniform norm.
By definition of _𝜉Ω^, for every 𝝃 in _𝜉Ω^, there exist 𝛿<1 and 𝑛_𝜉 such that, for every 𝑛>𝑛_𝜉, we have sup_{0≤𝑘<2𝑛−1}||𝝃_𝑛,𝑘||<2^𝑛𝛿/2,(D.1) which implies that for, 𝑁>𝑛_𝜉, we have ||𝚿^𝑁(𝝃)(𝑡)−𝚿^𝑛𝜉(𝝃)(𝑡)||≤_{(𝑛,𝑘)∈ℐ𝑁⧵ℐ𝑛𝜉}||𝝍_𝑛,𝑘(𝑡)⋅𝝃_𝑛,𝑘||≤^∞_{𝑛=𝑛𝜉}2^𝑛𝛿/2||𝝍_𝑛,𝑘||.(D.2)
We therefore need to upperbound the uniform norm of the function 𝝍_𝑛,𝑘. To this purpose, we use the definition of 𝝍_𝑛,𝑘 given by (3.28): 𝝍_𝑛,𝑘(𝑡)=𝑔(𝑡)⋅^𝑡₀𝑓(𝑠)⋅𝚽_𝑛,𝑘(𝑠)𝑑𝑠.(D.3) The coefficient in position (𝑖,𝑗) of the integral term in the right-hand side of the previous inequality can be written as a function of the lines and columns of 𝑓 and Φ_𝑛,𝑘 and can be upperbounded using the Cauchy-Schwarz inequality on 𝐿²([0,1],ℝ^𝑑) as follows: 𝟙_[0,𝑡]⋅𝑐_𝑖𝐟^𝑇,𝑐_𝑗𝜙_𝑛,𝑘=_𝑈𝟙_{[0,𝑡]∩𝑆n,𝑘}(𝑠)𝑙_𝑖(𝐟)(𝑠)⋅𝑐_𝑗𝜙_𝑛,𝑘(𝑠)𝑑𝑠≤‖‖𝟙_{[0,𝑡]∩𝑆𝑛,𝑘}𝑙_𝑖(𝐟)‖‖₂‖‖𝑐_𝑗(𝜙_𝑛,𝑘)‖‖₂.(D.4) Since the columns of Φ_𝑛,𝑘 form an orthogonal basis of functions for the standard scalar product in 𝐿²([0,1],ℝ^𝑑) (see Proposition 3.5), we have ‖𝑐_𝑗(𝜙_𝑛,𝑘)‖₂=1. Moreover, since 𝐟 is bounded continuous on [0,1], we can define constants 𝐾_𝑖=sup_0≤𝑡≤1‖𝑙_𝑖(𝐟)(𝑡)‖<∞ and write ‖‖𝟙_{[0,𝑡]∩𝑆𝑛,𝑘}𝑙_𝑖(𝐟)‖‖²=_𝑈𝟙_{[0,𝑡]∩𝑆𝑛,𝑘}(𝑠)𝑙_𝑖(𝐟)(𝑠)^𝑇⋅𝑙_𝑖(𝐟)(𝑠)𝑑𝑠≤_𝑈𝟙_{[0,𝑡]∩𝑆𝑛,𝑘}(𝑠)𝐾²_𝑖𝑑𝑠=2^−𝑛+1𝐾²_𝑖<∞.(D.5) Setting 𝐾=max_{0≤𝑖<𝑑}𝐾_𝑖, for all (𝑛,𝑘) in ℐ, the ℝ^𝑑×𝑑-valued functions 𝜿_𝑛,𝑘(𝑡)=_𝑈𝟙_[0,𝑡](𝑠)𝐟(𝑠)𝜙_𝑛,𝑘(𝑠)𝑑𝑠(D.6) satisfy ‖𝜿_𝑛,𝑘‖_∞=sup_0≤𝑡≤1|𝜿_𝑛,𝑘(𝑡)|≤𝐾2^{−(𝑛+1)/2}.
Moreover, since 𝐠 is also bounded continuous on [0,1], there exists 𝐿 such that ‖𝐠‖_∞=sup_0≤𝑡≤1|𝐠(𝑡)|≤𝐿, and we finally have, for all 0≤𝑡≤1, ‖‖𝝍_𝑛,𝑘‖‖_∞≤‖𝐠‖_∞‖‖𝜿_𝑛,𝑘‖‖_∞≤𝐿𝐾2^{−(𝑛+1)/2}.(D.7)
Now using this bound and (D.2), we have ||𝚿^𝑁(𝝃)(𝑡)−𝚿^𝑛𝜉(𝝃)(𝑡)||≤_{(𝑛,𝑘)∈ℐ𝑁⧵ℐ𝑛𝜉}||𝝍_𝑛,𝑘(𝑡)⋅𝝃_𝑛,𝑘||≤𝐿𝐾√2^∞_𝑛_=𝑛𝜉2^(𝛿−1)/2^𝑛,(D.8) and since 𝛿<1, for the continuous functions 𝑡↦Ψ^𝑁_𝑡(𝝃) forms a uniformly convergent sequence of functions for the 𝑑-dimensional uniform norm. This sequence therefore converges towards a continuous function, and Ψ is well defined on _𝜉Ω^ and takes values in 𝐶₀([0,1],ℝ^𝑑).

This proposition being proved, we dispose of the map Ψ=lim_𝑁→∞Ψ^𝑁. We now turn to prove different useful properties on this function. We denote by ℬ(𝐶₀([0,1],ℝ^𝑑)) the Borelian sets of the 𝑑-dimensional Wiener space 𝐶₀([0,1],ℝ^𝑑).

Lemma D.2. The function Ψ∶(_𝜉Ω^,ℬ(_𝜉Ω^))→(𝐶₀([0,1],ℝ^𝑑),ℬ(𝐶₀([0,1],ℝ^𝑑))) is a linear injection.

Proof. The application Ψ is clearly linear. The injective property simply results from the existence of the dual family of distributions 𝜹_𝑛,𝑘. Indeed, for every 𝝃,𝝃^ in _𝜉Ω^, we have that Ψ(𝜉)=Ψ(𝜉^) entails, that for all 𝑛,𝑘,𝝃_𝑛,𝑘=𝒫(𝜹_𝑛,𝑘,Ψ(𝜉))=𝒫(𝜹_𝑛,𝑘,Ψ(𝜉^))=𝝃^_𝑛,𝑘.

In the one-dimensional case, as mentioned in the main text, because the uniform convergence of the sample paths is preserved as long as 𝛼 is continuous and Γ is nonzero through (D.8), the definition _𝑥Ω^ does not depend on 𝛼 or Γ and the space _𝑥Ω^ is large enough to contain reasonably regular functions.

Proposition D.3. In the one-dimensional case, the space _𝑥Ω^ contains the space of uniformly Hölder continuous functions 𝐻 defined as 𝐻=𝑥∈𝐶[0,1]∣∃𝛿>0,sup_{0≤𝑠,𝑡≤1}||𝑥(𝑡)−𝑥(𝑠)|||𝑡−𝑠|^𝛿<+∞.(D.9)

Remark D.3. This point can be seen as a direct consequence of the characterization of the local Hölder exponent of a continuous real function in terms of the asymptotic behavior of its coefficients in the decomposition on the Schauder basis [63].

Proof. To underline that we place ourselves in the one-dimensional case, we drop the bold notations that indicate multidimensional quantities. Supposing that 𝑥 is uniformly Hölder continuous for a given 𝛿>0, there always exists 𝜉 such that Ψ^𝑁(𝜉) coincides with 𝑥 on 𝐷_𝑁: it is enough to take 𝜉 such that, for all (𝑛,𝑘) in ℐ_𝑁, 𝜉_𝑛,𝑘=(𝛿_𝑛,𝑘,𝑥). We can further write for 𝑛>0𝑥,𝛿_𝑛,𝑘=𝑀_𝑛,𝑘𝑥𝑚_𝑛,𝑘𝑔𝑚_𝑛,𝑘−⎛⎜⎝𝐿_𝑛,𝑘𝑥𝑙_𝑛,𝑘𝑔_𝛼𝑙_𝑛,𝑘+𝑅_𝑛,𝑘𝑥𝑟_𝑛,𝑘𝑔𝑟_𝑛,𝑘⎞⎟⎠,=𝐿_𝑛,𝑘⎛⎜⎝𝑥𝑚_𝑛,𝑘𝑔𝑚_𝑛,𝑘−𝑥𝑙_𝑛,𝑘𝑔𝑙_𝑛,𝑘⎞⎟⎠+𝑅_𝑛,𝑘⎛⎜⎝𝑥𝑚_𝑛,𝑘𝑔𝑚_𝑛,𝑘−𝑥𝑟_𝑛,𝑘𝑔𝑟_𝑛,𝑘⎞⎟⎠.(D.10) For a given function 𝛼, posing 𝑁_𝛼=sup_0≤𝑡≤1𝑓_𝛼(𝑡)/inf_0≤𝑡≤1𝑓²_𝛼(𝑡), we have _𝛼𝑀_𝑛,𝑘≤𝑁_𝛼2^(𝑛+1)/2,_𝛼𝐿_𝑛,𝑘≤𝑁_𝛼2^(𝑛−1)/2,_𝛼𝑅_𝑛,𝑘≤𝑁_𝛼2^(𝑛−1)/2.(D.11) Moreover, if 𝛼 is in 𝐻, it is straightforward to see that 𝑔_𝛼 has a continuous derivative. Then, since 𝑥 is 𝛿-Hölder, for any 𝜖≥0, there exists 𝐶>0 such that |𝑡−𝑠|≤𝜖 entails that ||||𝑥(𝑡)𝑔(𝑡)−𝑥(𝑠)𝑔(𝑠)||||≤𝐶𝜖^𝛿,(D.12) from which we directly deduce ||𝜉_𝑛,𝑘||≤𝑁_𝛼𝐶√22^{𝑛((1/2)−2𝛿)}.(D.13) This demonstrates that {𝜉_𝑛,𝑘} belongs to _𝜉Ω^ and ends the proof.

We equip the space _𝑥Ω^ with the topology induced by the uniform norm on 𝐶₀([0,1],ℝ^𝑑). As usual, we denote by ℬ(_𝑥Ω^) the corresponding Borelian sets. We now show Proposition 3.12.

Proposition D.4. The function Ψ∶(_𝜉Ω^,ℬ(_𝜉Ω^))→(_𝑥Ω^,ℬ(_𝑥Ω^)) is a bounded continuous bijection.

Proof. Consider an open ball _𝑥𝐵(𝑥,𝜖) of _𝑥Ω^ of radius 𝜖. If we take 𝑀=𝐿𝐾/√2 as defined in (D.8), we can choose a real 𝛿>0 such that 𝛿<𝜖𝑀⎛⎜⎝^∞_𝑛=02^−𝑛/2⎞⎟⎠⁻¹.(D.14) Let us consider 𝝃 in _𝜉Ω^ such that Ψ(𝜉)=𝑥. Then, by (D.8), we immediately have that, for all 𝝃^ in the ball of radius _𝜉𝐵(𝜉,𝛿) of _𝜉Ω,‖Ψ(𝝃−𝝃^)‖_∞≤𝜖. This shows that Ψ⁻¹(_𝑥𝐵(𝑥,𝜖)) is open and that Ψ is continuous for the 𝑑-dimensional uniform norm topology.

D.2. The Coefficient Application

In this section of the appendix, we show some useful properties of the coefficient application introduced in Section 3.2.2.

Lemma D.5. The function Δ∶(𝐶₀([0,1],ℝ^𝑑),ℬ(𝐶₀([0,1],ℝ^𝑑)))→(_𝜉Ω,ℬ(_𝜉Ω)) is a measurable linear injection.

Proof. (i)The function Δ is clearly linear.(ii)To prove that Δ is injective, we show that for 𝐱 and 𝐲 in 𝐶₀([0,1],ℝ^𝑑), 𝐱≠𝐲 implies that Δ(𝐱)≠Δ(𝑦). To this end, we fix 𝐱≠𝐲 in 𝐶₀([0,1],ℝ^𝑑) equipped with the uniform norm and consider the continuous function 𝐝_𝑁(𝑡)=_{(𝑛,𝑘)∈ℐ𝑁}𝝍_𝑛,𝑘(𝑡)𝚫(𝐱)_𝑛,𝑘−𝚫(𝐲)_𝑛,𝑘.(D.15) This function coincides with 𝐱−𝐲 on every dyadic number in 𝐷_𝑁 and has zero value if Δ(𝐱)=Δ(𝐲). Since 𝐱≠𝐲, there exists 𝑠 in]0,1[such that 𝐱(𝑠)≠𝐲(𝑠), and by continuity of 𝐱−𝐲, there exists an 𝜀>0 such that 𝐱≠𝐲 on the ball]𝑠−𝜀,𝑠+𝜀[. But, for 𝑁 large enough, there exists 𝑘, 0≤k<2^𝑁−1 such that |𝑠−𝑘2^−𝑁|<𝜀. We then necessarily have that Δ(𝑓)≠Δ(𝑔); otherwise, we would have 𝑑_𝑁(𝑘2^−𝑁)=(𝐱−𝐲)(𝑘2^−𝑁)=0, which would contradict the choice of 𝜀.(iii)Before proving the measurability of Δ, we need the following observation. Consider for 𝑁>0, the finite-dimensional linear function Δ_𝑁𝐶₀[0,1],ℝ^𝑑⟶ℝ^𝑑^2𝑁−1,𝐱⟼𝚫_𝑁(𝑥)=𝚫(𝑥)_𝑁,𝑘_{(𝑁,𝑘)∈ℐ𝑁}.(D.16) Since for all (𝑁,𝑘), the matrices 𝐌_𝑁,𝑘, 𝐑_𝑁,𝑘, 𝐋_𝑁,𝑘 are all bounded, the function Δ_𝑁∶(𝐶₀([0,1],ℝ^𝑑),ℬ(𝐶₀([0,1],ℝ^𝑑)))→((ℝ^𝑑)^2𝑁−1,ℬ((ℝ^𝑑)^2𝑁−1)) is a continuous linear application. To show that the function Δ is measurable, it is enough to show that the pre-image by Δ of the generative cylinder sets of ℬ(_𝜉Ω) belongs to ℬ(𝐶₀([0,1],ℝ^𝑑)).For any 𝑁≥0, take an arbitrary Borel set 𝐵=_(𝑛,_{𝑘)∈ℐ𝑁}𝐵_𝑛,𝑘∈ℬℝ^𝑑^ℐ𝑁(D.17) and define the cylinder set 𝒞_𝑁(𝐵) as 𝒞_𝑁(𝐵)=𝝃∈_𝜉Ω∣∀(𝑛,𝑘)∈𝐼_𝑁,𝝃_𝑛,𝑘∈𝐵_𝑛,𝑘,(D.18) and we write the collection of cylinder sets 𝐶 as 𝐶=_𝑛≥0𝐶_𝑁with𝐶_𝑁=_𝐵∈_{ℝ𝑑ℐ𝑁}𝒞_𝑁(𝐵).(D.19) We proceed by induction of 𝑁 to show that the preimage by Δ of any cylinder set in 𝐶 is in ℬ(𝐶₀([0,1],ℝ^𝑑)). For 𝑁=0, a cylinder set of 𝐶₀ is of the form 𝐵_0,0 in ℬ(ℝ^𝑑), Δ⁻¹(𝐵)={𝐱∈𝐶₀([0,1],ℝ^𝑑)∣𝐱(1)∈𝐋^𝑇_0,0𝐠⁻¹(𝑟_0,0)(𝐵_0,0)}, which is measurable for being a cylinder set of ℬ(𝐶₀([0,1],ℝ^𝑑)). Suppose now that, for 𝑁>0, for any set 𝐴 in 𝐶_𝑁−1, the set Δ⁻¹(𝐴) is measurable. Then, considering a set 𝐴 in 𝐶_𝑁, there exists 𝐵 in ℬ((ℝ^𝑑)^ℐ𝑁) such that 𝐴=𝒞_𝑁(𝐵). Define 𝐴′ in 𝐶_𝑁 such that 𝐴′=𝒞_𝑁−1(𝐵′), where 𝐵^=_(𝑛,𝑘_{)∈ℐ𝑁−1}𝐵_𝑛,𝑘,(D.20) and remark that 𝐴=𝒞_𝑁(𝐵)⊂𝐴^=𝒞_𝑁(𝐵^). Clearly, we have that 𝐴=𝐴′∩𝐷, where we have defined the cylinder set 𝐷 as 𝐷=𝒞_𝐼𝑁⎛⎜⎝_(𝑁,𝑘_{)∈ℐ𝑁,𝑘}𝐵_𝑁,𝑘⎞⎟⎠.(D.21) Having defined the function Δ_𝑁, we now have Δ⁻¹(𝐴)=Δ⁻¹(𝐴^∩𝐷)=Δ⁻¹(𝐴^)∩Δ⁻¹(𝐷)=Δ⁻¹(𝐴^)∩Δ⁻¹_𝑁(𝐷). Because of the continuity of Δ_𝑁, Δ⁻¹_𝑁(𝐷) is a Borel set of ℬ(𝐶₀([0,1],ℝ^𝑑)). Since, by hypothesis of recurrence, Δ⁻¹(𝐴′) is in ℬ(𝐶₀([0,1],ℝ^𝑑)), Δ⁻¹(𝐴) is also in ℬ(𝐶₀([0,1],ℝ^𝑑)) as the intersection of two Borel sets. The proof of the measurability of Δ is complete.

We now demonstrate Theorem 3.14.

Proposition D.6. The function Δ∶(_𝑥Ω^,ℬ(_𝑥Ω^))→(_𝜉Ω^,ℬ(_𝜉Ω^)) is a measurable linear bijection whose inverse is Ψ=Δ⁻¹.

Proof. Let 𝑥∈_𝑥Ω^ be a continuous function. We have 𝚿(𝚫(𝑥))(𝑡)=_{(𝑛,𝑘)∈ℐ}𝝍_𝑛,𝑘(𝑡)⋅𝚫_𝑛,𝑘=_{(𝑛,𝑘)∈ℐ}𝝍_𝑛,𝑘(𝑡)⋅𝒫𝜹_𝑛,𝑘,𝑥.(D.22) This function is equal to 𝑥(𝑡) for any 𝑡∈𝐷, the set of dyadic numbers. Since 𝐷 is dense in [0,1] and both 𝑥 and Ψ(Δ(𝑥)) are continuous, the two functions, coinciding on the dyadic numbers, are equal for the uniform distance, and hence Ψ(Δ(𝑥))=𝑥.

E. Itô Formula

In this section, we provide rigorous proofs of Proposition 6.1 and Theorem 6.2 related to the Itô formula.

Proposition E.1 (integration by parts). Let (𝑋_𝑡) and (𝑌_𝑡) be two one-dimensional Gauss-Markov processes starting from zero. Then one has the following equality in law: 𝑋_𝑡𝑌_𝑡=^𝑡₀𝑋_𝑠∘𝑑𝑌_𝑠+^𝑡₀𝑌_𝑠∘𝑑𝑋_𝑠,(E.1) where ∫^𝑡₀𝐴_𝑠∘𝑑𝐵_𝑠 two stochastic processes denotes for 𝐴_𝑡 and 𝐵_𝑡 the Stratonovich integral. In terms of the Itô integral, this formula is written as 𝑋_𝑡𝑌_𝑡=^𝑡₀𝑋_𝑠𝑑𝑌_𝑠+^𝑡₀𝑌_𝑠𝑑𝑋_𝑠+⟨𝑋,𝑌⟩_𝑡,(E.2) where the brackets denote the mean quadratic variation.

Proof. We assume that 𝑋 and 𝑌 satisfy the equations: 𝑑𝑋_𝑡=𝛼_𝑋(𝑡)𝑋_𝑡+√Γ_𝑋(𝑡)𝑑𝑊_𝑡,𝑑𝑌_𝑡=𝛼_𝑌(𝑡)𝑋_𝑡+√Γ_𝑌(𝑡)𝑑𝑊_𝑡,(E.3) and we introduce the functions 𝑓_𝑋, 𝑓_𝑌, 𝑔_𝑋, and 𝑔_𝑌 such that 𝑋_𝑡=𝑔_𝑋(𝑡)∫^𝑡₀𝑓_𝑋(𝑠)and 𝑌_𝑡=𝑔_𝑌(𝑡)∫^𝑡₀𝑓_𝑌(𝑠).
We define (_𝑋𝜓_𝑛,𝑘)_{(𝑛,𝑘)∈ℐ} and (_𝑌𝜓_𝑛,𝑘)_{(𝑛,𝑘)∈ℐ}, the construction bases of the processes 𝑋 and 𝑌. Therefore, using Theorem 4.5, there exist (_𝑋Ξ_𝑛,𝑘)_{(𝑛,𝑘)∈ℐ} and (_𝑌Ξ_𝑝,𝑞)_{(𝑝,𝑞)∈ℐ} standard normal independent variables such that 𝑋=∑_{(𝑛,𝑘)∈ℐ𝑋}𝜓_𝑛,𝑘⋅_𝑋Ξ_𝑛,𝑘 and 𝑌=∑_{(𝑝,𝑞)∈ℐ𝑌}𝜓_𝑛,𝑘⋅_𝑌Ξ_𝑛,𝑘 and we know that the processes 𝑋 and 𝑌 are almost-surely uniform limits when 𝑁→∞ of the processes 𝑋^𝑁 and 𝑌^𝑁 defined as the partial sums: 𝑋^𝑁=_{(𝑛,𝑘)∈ℐ𝑁}^𝑋𝜓_𝑛,𝑘⋅_𝑋Ξ_𝑛,𝑘,𝑌^𝑁=_{(𝑝,𝑞)∈ℐ𝑁}^𝑌𝜓_𝑛,𝑘⋅_𝑌Ξ_𝑛,𝑘.(E.4) Using the fact that the functions _𝑋𝜓_𝑛,𝑘 and _𝑌𝜓_𝑛,𝑘 have piecewise continuous derivatives, we have 𝑋^𝑁_𝑡𝑌^𝑁_𝑡=_{(𝑛,𝑘)∈𝐼𝑁}_{(𝑝,𝑞)∈𝐼𝑁}^𝑋𝜓_𝑛,𝑘(𝑡)_𝑌𝜓_𝑝,𝑞(𝑡)_𝑋Ξ_{𝑛,𝑘𝑌}Ξ_𝑝,𝑞=_{(𝑛,𝑘)∈𝐼𝑁}_{(𝑝,𝑞)∈𝐼𝑁}^𝑋Ξ_{𝑛,𝑘𝑌}Ξ_𝑝,𝑞^𝑡₀𝑑𝑑𝑠^𝑋𝜓_𝑛,𝑘(𝑠)_𝑌𝜓_𝑝,𝑞(𝑠)𝑑𝑡=_{(𝑛,𝑘)∈𝐼𝑁}_{(𝑝,𝑞)∈𝐼𝑁}^𝑋Ξ_{𝑛,𝑘𝑌}Ξ_𝑝,𝑞^𝑡₀^𝑋𝜓^_𝑛,𝑘(𝑠)_𝑌𝜓_𝑝,𝑞(𝑠)+_𝑋𝜓_𝑛,𝑘(𝑠)_𝑌𝜓^_𝑝,𝑞(𝑠)𝑑𝑠.(E.5) Therefore, we need to evaluate the piecewise derivative of the functions _𝑋𝜓_𝑛,𝑘 and _𝑌𝜓_𝑛,𝑘. We know that 1𝑓_𝑋^𝑋𝜓_𝑛,𝑘𝑔_𝑋^(𝑡)=_𝑋𝜙_𝑛,𝑘(𝑡),(E.6) which entails that _𝑋𝜓^_𝑛,𝑘=𝛼_𝑋𝑋𝜓_𝑛,𝑘+𝑔_𝑋𝑓_𝑋𝑋𝜙_𝑛,𝑘=𝛼_𝑋𝑋𝜓_𝑛,𝑘+√Γ_𝑋𝑋𝜙_𝑛,𝑘(E.7) and similarly so for the process 𝑌. Therefore, we have 𝑋^𝑁𝑌^𝑁=𝐴^𝑛,𝑘_𝑝,𝑞+𝐵^𝑛,𝑘_𝑝,𝑞+𝐶^𝑛,𝑘_𝑝,𝑞+𝐷^𝑛,𝑘_𝑝,𝑞,(E.8) with 𝐴_𝑡=_{(𝑛,𝑘)∈𝐼𝑁}_{(𝑝,𝑞)∈𝐼𝑁}^𝑡₀𝛼_𝑋(𝑠)_𝑋𝜓_𝑛,𝑘(𝑠)_𝑌𝜓_𝑝,𝑞(𝑠)𝑑𝑠^𝑋Ξ_{𝑛,𝑘𝑌}Ξ_𝑝,𝑞,𝐵_𝑡=_{(𝑛,𝑘)∈𝐼𝑁}_{(𝑝,𝑞)∈𝐼𝑁}^𝑡₀√Γ_𝑋(𝑠)_𝑋𝜙_𝑛,𝑘(𝑠)_𝑌𝜓_𝑝,𝑞(𝑠)𝑑𝑠^𝑋Ξ_{𝑛,𝑘𝑌}Ξ_𝑝,𝑞,𝐶_𝑡=_{(𝑛,𝑘)∈𝐼𝑁}_{(𝑝,𝑞)∈𝐼𝑁}^𝑡₀𝛼_𝑌(𝑠)_𝑋𝜓_𝑛,𝑘(𝑠)_𝑌𝜓_𝑝,𝑞(𝑠)𝑑𝑠^𝑋Ξ_{𝑛,𝑘𝑌}Ξ_𝑝,𝑞,𝐷_𝑡=_{(𝑛,𝑘)∈𝐼𝑁}_{(𝑝,𝑞)∈𝐼𝑁}^𝑡₀√Γ_𝑌(𝑠)_𝑌𝜙_𝑛,𝑘(𝑠)_𝑋𝜓_𝑝,𝑞(𝑠)𝑑𝑠_𝑋Ξ_{𝑛,𝑘𝑌}Ξ_𝑝,𝑞.(E.9) We easily compute 𝐴_𝑡+𝐶_𝑡=^𝑡₀𝛼_𝑋(𝑠)+𝛼_𝑌(𝑠)𝑋^𝑁(𝑠)𝑌^𝑁(𝑠)𝑑𝑠.(E.10) For 𝑡∈[0,1], as it is our case, 𝑋^𝑁(𝑠) and 𝑌^𝑁(𝑠) are both almost surely finite for all 𝑡 in [0,1]. For almost all _𝑌𝜉 and _𝑌𝜉 drawn with respect to the law of the Gaussian infinite vector Ξ, we therefore have, by the Lebesgue dominated convergence theorem, that this integral converges almost surely towards ^𝑡₀𝛼_𝑋(𝑠)+𝛼_𝑌(𝑠)𝑋(𝑠)𝑌(𝑠)𝑑𝑠.(E.11) The other two terms 𝐵_𝑡 and 𝐷_𝑡 necessitate a more thorough analysis, and we treat them as follows. Let us start by considering the first one of this term: 𝐵_𝑡=¹₀𝟙_[0,𝑡](𝑠)√Γ_𝑋(𝑠)_{(𝑛,𝑘)∈𝐼𝑁}_{(𝑝,𝑞)∈𝐼𝑁}^𝑋𝜙_𝑛,𝑘(𝑠)_𝑌𝜓_𝑝,𝑞(s)_𝑋Ξ_{𝑛,𝑘𝑌}Ξ_𝑝,𝑞𝑑𝑠=_{𝑡𝑖∈𝒟𝑁⧵{1}}^𝑡𝑖+1_𝑡𝑖𝟙_[0,𝑡](𝑠)√Γ_𝑋(𝑠)⎛⎜⎝_(𝑛,𝑘_)∈ℐ𝑁^𝑋𝜙_𝑛,𝑘(𝑠)⋅_𝑋Ξ_𝑛,𝑘⎞⎟⎠𝑌^𝑁(𝑠)𝑑𝑠=_{𝑡𝑖∈𝒟𝑁⧵{1}}^𝑡𝑖+1_𝑡𝑖𝟙_[0,𝑡](𝑠)√Γ_𝑋(𝑠)1𝑓_𝑋(𝑠)𝑋^𝑁𝑔_𝑋(𝑠)^𝑌^𝑁(𝑠)𝑑𝑠.(E.12) Let us now have a closer look at the process 𝑋^𝑁_𝑡 for 𝑡∈[𝑡_𝑖,𝑡_𝑖+1] where [𝑡_𝑖,𝑡_𝑖+1]=𝑆_𝑁,𝑖 for 𝑖 such that (𝑁,𝑖)∈ℐ. Because of the structure of our construction, we have 𝑌^𝑁(𝑡)=𝑔_𝑌(𝑡)𝑔_𝑌𝑡_𝑖ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌(𝑡)ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌𝑡_𝑖⋅𝑌_𝑡𝑖+𝑔_𝑌(𝑡)𝑔_𝑌𝑡_𝑖+1ℎ_𝑌(𝑡)−ℎ_𝑌𝑡_𝑖ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌𝑡_𝑖⋅𝑌_𝑡𝑖+1,(E.13)1𝑓_𝑋(𝑡)𝑋^𝑁𝑔_𝑋^(𝑡)=𝑓_𝑋(𝑡)ℎ_𝑋𝑡_𝑖+1−ℎ_𝑋𝑡_𝑖⎛⎜⎝𝑋_𝑡𝑖+1𝑔_𝑋𝑡_𝑖+1−𝑋_𝑡𝑖𝑔_𝑋𝑡_𝑖⎞⎟⎠.(E.14) We therefore have ^𝑡𝑖+1_𝑡𝑖⎛⎜⎝1_[0,𝑡](𝑠)√Γ_𝑋(𝑠)𝑓_𝑋(𝑠)𝑑𝑑𝑠𝑋^𝑁(𝑠)𝑔_𝑋(𝑠)⎞⎟⎠𝑌^𝑁(𝑠)𝑑𝑠=^𝑡𝑖+1_𝑡𝑖1_[0,𝑡](𝑠)√Γ_𝑋(𝑠)𝑓_𝑋(𝑠)ℎ_𝑋𝑡_𝑖+1−ℎ_𝑋𝑡_𝑖𝑔_𝑌(𝑠)×⎡⎢⎣ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌(𝑠)ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌𝑡_𝑖𝑌_𝑡𝑖𝑔_𝑌𝑡_𝑖+ℎ_𝑌(𝑠)−ℎ_𝑌𝑡_𝑖ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌𝑡_𝑖𝑌_t𝑖+1𝑔_𝑌𝑡_𝑖+1⎤⎥⎦𝑑𝑠⎛⎜⎝𝑋_𝑡𝑖+1𝑔_𝑋𝑡_𝑖+1−𝑋_𝑡𝑖𝑔_𝑋𝑡_𝑖⎞⎟⎠=𝑣_𝑖(𝑡)^𝑁𝑌^𝑁_𝑡𝑖+𝑤_𝑖(𝑡)^𝑁𝑌^𝑁_𝑡𝑖+1⎛⎜⎝𝑋_𝑡𝑖+1𝑔_𝑋𝑡_𝑖+1−𝑋_𝑡𝑖𝑔_𝑋𝑡_𝑖⎞⎟⎠(E.15) with 𝑣^𝑁_𝑖(𝑡)=^𝑡𝑖+1_𝑡𝑖𝟙_[0,𝑡](𝑠)√Γ_𝑋(𝑠)𝑓_𝑋(𝑠)ℎ_𝑋𝑡_𝑖+1−ℎ_𝑋𝑡_𝑖𝑔_𝑌(𝑠)g_𝑌𝑡_𝑖ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌(𝑠)ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌𝑡_𝑖𝑑𝑠,𝑤^𝑁_𝑖(𝑡)=^𝑡𝑖+1_𝑡𝑖𝟙_[0,𝑡](𝑠)√Γ_𝑋(𝑠)𝑓_𝑋(𝑠)ℎ_𝑋𝑡_𝑖+1−ℎ_𝑋𝑡_𝑖𝑔_𝑌(𝑠)𝑔_𝑌𝑡_𝑖+1ℎ_𝑌(𝑠)−ℎ_𝑌𝑡_𝑖ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌𝑡_𝑖𝑑𝑠.(E.16) Let us denote by 𝛿_𝑁 the time step of the partition 𝛿_𝑁=max_{𝑡𝑖∈𝒟𝑁⧵{1}}(𝑡_𝑖+1−𝑡_𝑖), which is smaller than 𝜌^𝑁 with 𝜌∈(0,1) from the assumption made in Section 3.2.1. Moreover, we know that the functions 𝑔_𝑋=𝑔_𝑌, ℎ_𝑋 and ℎ_𝑌 are continuously differentiable, and since √Γ_𝑋 and √Γ_𝑌 are 𝛿-Hölder, so are 𝑓_𝑋 and 𝑓_𝑌. When 𝑁→∞ (i.e., when 𝛿_𝑁→0), using the Taylor and Hölder expansions for the differential functions, we can further evaluate the integrals we are considering. Let us first assume that 𝑡>𝑡_𝑖+1. We have 𝑣_𝑖(𝑡)^𝑁=^𝑡𝑖+1_𝑡𝑖√Γ_𝑋(𝑠)𝑓_𝑋(𝑠)ℎ_𝑋𝑡_𝑖+1−ℎ_𝑋𝑡_𝑖𝑔_𝑌(𝑠)𝑔_𝑌𝑡_𝑖ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌(𝑠)ℎ_𝑌𝑡_𝑖+1−ℎ_𝑌𝑡_𝑖𝑑𝑠=√Γ_𝑋𝑡_𝑖1+𝑂𝛿^𝛿_𝑁𝑓_𝑋𝑡_𝑖1+𝑂𝛿^𝛿_𝑁𝑓_𝑋𝑡_𝑖²𝑡_𝑖+1−𝑡_𝑖1+𝑂𝛿_𝑁×𝑔_𝑌𝑡_𝑖1+𝑂𝛿_𝑁𝑔_𝑌𝑡_𝑖^𝑡𝑖+1_𝑡𝑖𝑓_𝑌𝑡_𝑖²𝑡_i+1−𝑠1+𝑂𝛿_𝑁𝑓_𝑌𝑡_𝑖²𝑡_𝑖+1−𝑡_𝑖1+𝑂𝛿_𝑁𝑑𝑠=√Γ_𝑋𝑡_𝑖𝑓_𝑋𝑡_𝑖𝑡_𝑖+1−𝑡_𝑖^𝑡𝑖+1_𝑡𝑖𝑡_𝑖+1−𝑠𝑡_𝑖+1−𝑡_𝑖𝑑𝑠1+𝑂𝛿_𝑁+𝑂𝛿^𝛿_𝑁=12𝑔_𝑋𝑡_𝑖+𝑂𝛿_𝑁+𝛿^𝛿_𝑁.(E.17) Similarly, we show that 𝑤^𝑁_𝑖(𝑡)=(1/2)𝑔_𝑋(𝑡_𝑖)+𝑂(𝛿_𝑁+𝛿^𝛿_𝑁) when 𝑁→∞. If 𝑡<𝑡_𝑖, we have 𝑣^𝑁_𝑖(𝑡)=𝑤^𝑁_𝑖(𝑡)=0 and for 𝑡 in [𝑡_𝑖0,𝑡_𝑖0+1) we have 𝑣^𝑁_𝑖0(𝑡)=𝑔_𝑋𝑡_𝑖02⎛⎜⎝𝑡_𝑖0+1−𝑡𝑡_𝑖0+1−𝑡_𝑖0⎞⎟⎠²+𝑂𝛿_𝑁+𝛿^𝛿_𝑁,𝑤^𝑁_𝑖0(𝑡)=𝑔_𝑋𝑡_𝑖02⎛⎜⎝𝑡−𝑡_𝑖0𝑡_𝑖0+1−𝑡_𝑖0⎞⎟⎠²+𝑂𝛿_𝑁+𝛿^𝛿_𝑁=𝑣^𝑁_𝑖0(𝑡)+𝑂𝛿_𝑁+𝛿^𝛿_𝑁.(E.18) We then finally have 𝐵_𝑡=_{𝑡𝑖∈𝒟𝑁;𝑡𝑖+1≤𝑡}𝑔_𝑋𝑡_𝑖2𝑌^𝑁_𝑡𝑖+𝑌^𝑁_𝑡𝑖+1⎛⎜⎝𝑋^𝑁_𝑡𝑖+1𝑔_𝑋𝑡_𝑖+1−𝑋^𝑁_𝑡𝑖𝑔_𝑋𝑡_𝑖⎞⎟⎠+𝑔_𝑋𝑡_𝑖02⎛⎜⎝𝑡_𝑖0+1−𝑡𝑡_𝑖0+1−𝑡_𝑖0⎞⎟⎠²𝑌^𝑁_𝑡𝑖0+𝑌^𝑁_𝑡⎛⎜⎜⎝𝑋^𝑁_𝑡𝑖+1𝑔_𝑋(𝑡)−𝑋^𝑁_𝑡𝑖0𝑔_𝑋𝑡_𝑖0⎞⎟⎟⎠+𝑂𝛿_𝑁+𝛿^𝛿_𝑁.(E.19) Moreover, we observe that the process 𝑋_𝑡/𝑔_𝑋(𝑡)=∫^𝑡𝑖+1_𝑡𝑖𝑓_𝑋(𝑠)𝑑𝑊_𝑠 is a martingale, and by definition of the Stratonovich integral for martingale processes, we have 𝐵_𝑡⟶_𝑁→∞^𝑡₀𝑔_𝑋(𝑠)𝑌_𝑠∘𝑑𝑋_𝑠𝑔_𝑋(𝑠)=^𝑡₀√Γ_𝑋(𝑠)𝑌_𝑠∘𝑑𝑊(𝑠),(E.20) where ∘ is used to denote the Stratonovich stochastic integral and the limit is taken in distribution. Notice that the fact that the sum converges towards the Stratonovich integral does not depend on the type of sequence of partition chosen which can be different from the dyadic partition. Putting all these results together, we obtain the equality in law: 𝑋_𝑡𝑌_𝑡=^𝑡₀𝛼_𝑋(𝑠)𝑋_𝑠𝑌_𝑠𝑑𝑠+^𝑡₀√Γ_𝑋(𝑠)𝑌_𝑠∘𝑑𝑊_𝑠+^𝑡₀𝛼_𝑌(𝑠)𝑋_𝑠𝑌_𝑠𝑑𝑠+^𝑡₀√Γ_𝑌(𝑠)𝑋_𝑠∘𝑑𝑊_𝑠,(E.21) which is exactly the integration by parts formula we were searching for. The integration by parts formula for the Itô stochastic integral directly comes from the relationship between the Stratonovich and Itô stochastic integrals.

Theorem E.1 (Itô). Let 𝑋 be a Gauss-Markov process and 𝑓 in 𝐶²(ℝ). The process 𝑓(𝑋_𝑡) is a Markov process and satisfies the relation 𝑓𝑋_𝑡=𝑓𝑋₀+^𝑡₀𝑓^𝑋_𝑠𝑑𝑋_𝑠+12^𝑡₀𝑓^𝑋_𝑠𝑑⟨𝑋⟩_𝑠.(E.22)

Proof. The integration by parts formula directly implies the Itô formula through a density argument as follows. Let 𝒜 be the set of functions 𝑓∈𝐶²([0,1],ℝ) such that (E.22) is true. It is clear that 𝒜 is a vector space. Moreover, because of the result of Proposition 6.1, the space 𝒜 is an algebra. Since all constant functions and the identity function 𝑓(𝑥)=𝑥 trivially belong to 𝒜, the algebra 𝒜 contains all polynomial functions.
Let now 𝑓∈𝐶²([0,1],ℝ). There exists a sequence of polynomials 𝑃_𝑘 such that 𝑃_𝑘 (resp., 𝑃_𝑘,𝑃^_𝑘) uniformly converges towards 𝑓 (resp., 𝑓^,𝑓^). Let us denote by 𝑈_𝑛 the sequence of stopping times: 𝑈_𝑛=inf𝑡∈[0,1];||𝑋_𝑡||>𝑛.(E.23) This sequence grows towards infinity. We have 𝑃_𝑘𝑋_{𝑡∧𝑈𝑛}−𝑃_𝑘𝑋₀=^𝑡₀𝑃^_𝑘𝑋_𝑠𝟙_[0,𝑈𝑛](𝑠)𝑑𝑋_𝑠+12^𝑡₀𝑃^_𝑘𝑋_𝑠𝟙_[0,𝑈𝑛](𝑠)𝑑⟨𝑋⟩_𝑠.(E.24) On the interval [0,𝑈_𝑛], we have 𝑋_𝑡≤𝑛, which allows to use the Lebesgue dominated convergence theorem on each term of the equality. We have 𝔼⎡⎢⎣||||^𝑡₀𝑃^_𝑘𝑋_𝑠𝟙_[0,𝑈𝑛](𝑠)𝑑𝑋_𝑠−^𝑡₀𝐹^𝑋_𝑠𝟙_[0,𝑈𝑛](𝑠)𝑑𝑋_𝑠||||²⎤⎥⎦=𝔼^𝑡₀||𝑃^_𝑘(𝑋_𝑠)−𝐹^𝑋_𝑠||²𝟙_[0,𝑈𝑛](𝑠)𝑑⟨𝑋⟩_𝑠,(E.25) which converges towards zero because of the Lebesgue theorem for the Steljes integration. The same argument directly applies to the other term. Therefore, letting 𝑘→∞, we proved Itô formula for 𝑋_{𝑡∧𝑈𝑛}, and eventually letting 𝑛→∞, we obtain the desired formula.

F. Trace Class Operator

In this section, we demonstrate Theorem 6.6, which proves the instrumental to extend the finite-dimensional change of variable formula to the infinite-dimensional case. The proof relies on the following lemma.

Lemma F.1. The operator _𝛼,𝛽𝑆−𝐼𝑑∶𝑙²(ℝ)→𝑙²(ℝ) is isometric to the operator _𝛼,𝛽𝑅∶𝑙²(ℝ)→𝑙²(ℝ) defined by _𝛼,𝛽𝑅[𝑥]=¹₀^𝛼,𝛽𝑅(𝑡,𝑠)𝑥(𝑠)𝑑𝑠,(F.1) with the kernel _𝛼,𝛽𝑅(𝑡,𝑠)=(𝛼(𝑡∨𝑠)−𝛽(𝑡∨𝑠))𝑓_𝛼(𝑡∧𝑠)𝑓_𝛼(𝑡∨𝑠)+𝑓_𝛼(𝑡)¹_𝑡∨𝑠(𝛼(𝑢)−𝛽(𝑢))²𝑓_𝛼²(𝑢)𝑑𝑢𝑓_𝛼(𝑠).(F.2)

Proof. Notice first that (𝛼(𝑡∨𝑠)−𝛽(𝑡∨𝑠))𝑓_𝛼(𝑡∧𝑠)𝑓_𝛼(𝑡∨𝑠)=𝟏_{𝑠<𝑡}(𝛼(𝑡)−𝛽(𝑡))𝑓_𝛼(𝑠)𝑓_𝛼(𝑡)+𝟏_{{𝑠≥𝑡}}(𝛼(𝑠)−𝛽(𝑠))𝑓_𝛼(𝑡)𝑓_𝛼(𝑠),(F.3) which leads to writing in 𝐿²[0,1], for any (𝑛,𝑘) and (𝑝,𝑞) in ℐ^𝛼𝜙_𝑛,𝑘,_𝛼,𝛽𝑅^𝛼𝜙_𝑝,𝑞=¹₀^𝛼,𝛽𝑅(𝑡,𝑠)_𝛼𝜙_𝑛,𝑘(𝑡)_𝛼𝜙_𝑝,𝑞(𝑠)𝑑𝑡𝑑𝑠=𝐴^𝑛,𝑘_𝑝,𝑞+𝐵^𝑛,𝑘_𝑝,𝑞+𝐶^𝑛,𝑘_𝑝,𝑞,(F.4) with 𝐴^𝑛,𝑘_𝑝,𝑞=¹₀𝛼(𝑡)−𝛽(𝑡)𝑓_𝛼(𝑡)^𝛼𝜙_𝑛,𝑘(𝑡)^𝑡₀𝑓_𝛼(𝑠)_𝛼𝜙_𝑝,𝑞(𝑠)𝑑𝑠𝑑𝑡,=¹₀𝛼(𝑡)−𝛽(𝑡)√Γ(𝑡)^𝛼𝜙_𝑛,𝑘(𝑡)_𝛼𝜓_𝑝,𝑞(𝑡)𝑑𝑡,𝐵^𝑛,𝑘_𝑝,𝑞=¹₀^𝛼𝑓(𝑡)𝜙_𝑛,𝑘(𝑡)¹_𝑡𝛼(𝑠)−𝛽(𝑠)𝑓_𝛼(𝑠)^𝛼𝜙_𝑝,𝑞(𝑠)𝑑𝑠𝑑𝑡,=¹₀𝛼(𝑡)−𝛽(𝑡)√Γ(𝑡)^𝛼𝜓_𝑛,𝑘(𝑡)_𝛼𝜙_𝑝,𝑞(𝑡)𝑑𝑡,𝐶^𝑛,𝑘_𝑝,𝑞=¹₀^𝛼𝜙_𝑛,𝑘(𝑡)𝑓_𝛼(𝑡)_𝑠,𝑡(𝛼(𝑢)−𝛽(𝑢))²𝑓_𝛼²(𝑢)𝑑𝑢𝑓(𝑠)_𝛼𝜙_𝑝,𝑞(𝑠)𝑑𝑡𝑑𝑠,=¹₀(𝛼(𝑢)−𝛽(𝑢))²𝑓_𝛼²(𝑢)×𝟏_[0,𝑢](𝑡)𝑓_𝛼(𝑡)_𝛼𝜙_𝑝,𝑞(𝑡)𝟏_[0,𝑢](𝑠)𝑓_𝛼(𝑠)_𝛼𝜙_𝑝,𝑞(𝑠)𝑑𝑡𝑑𝑠𝑑𝑢,=¹₀(𝛼(𝑢)−𝛽(𝑢))²Γ(𝑢)^𝛼𝜓_𝑛,𝑘(𝑡)_𝛼𝜓_𝑝,𝑞(𝑡)𝑑𝑡𝑑𝑠.(F.5) This proves that (_𝛼𝜙_𝑛,𝑘,_𝛼,𝛽𝑅[_𝛼𝜙_𝑝,𝑞])=[_𝛼,𝛽𝑆−𝐼𝑑]^𝑛,𝑘_𝑝,𝑞. Therefore, if we denote the isometric linear operator _𝛼Φ∶𝑙²(ℝ)⟶𝐿²(ℝ),𝜉⟼_𝛼Φ[𝜉]=^∞_𝑛=0_{0≤𝑘<2𝑛−1}^𝛼𝜙_𝑛,𝑘⋅𝜉_𝑛,𝑘,(F.6) we clearly have _𝛼Φ^𝑇∘_𝛼,𝛽𝑅∘_𝛼Φ=_𝛼,𝛽𝑆−𝐼𝑑 with _𝛼Φ^𝑇=_𝛼Φ⁻¹.

We now proceed to demonstrate that _𝛼,𝛽𝑆−𝐼𝑑 is a trace class operator.

Proof of Theorem 6.6. Since the kernel _𝛼,𝛽𝑅(𝑡,𝑠) is integrable in 𝐿²([0,1]×[0,1]), the integral operator _𝛼,𝛽𝑅∶𝐿²[0,1]→𝐿²[0,1] is a Hilbert-Schmidt operator and thus is compact. Moreover, it is a trace class operator since we have Tr_𝛼,𝛽𝑅=¹₀(𝛼(𝑡)−𝛽(𝑡))𝑑𝑡+¹₀𝑓_𝛼²(𝑡)¹_𝑡(𝛼(𝑡)−𝛽(𝑡))²𝑓_𝛼²(𝑢)𝑑𝑠𝑑𝑡=¹₀(𝛼(𝑡)−𝛽(𝑡))𝑑𝑡+¹₀ℎ_𝛼(𝑡)𝑓_𝛼(𝑡)²(𝛼(𝑡)−𝛽(𝑡))²𝑑𝑡.(F.7) Since _𝛼,𝛽𝑆−𝐼𝑑 and _𝛼,𝛽𝑅 are isometric through _𝛼Φ, the compactness of _𝛼,𝛽𝑆−𝐼𝑑 is equivalent to the compactness of _𝛼,𝛽𝑅. Moreover, the traces of both operators coincide: ^∞_𝑛=0_{0≤𝑘<2𝑛−1}^𝛼,𝛽𝑆^𝑛,𝑘_𝑛,𝑘=^∞_𝑛=0_{0≤𝑘<2𝑛−1}¹₀^𝛼𝜙_𝑛,𝑘(𝑡)_𝛼𝜙_𝑛,𝑘(𝑠)_𝛼,𝛽𝑅(𝑡,𝑠)𝑑𝑠𝑑𝑡,=¹₀⎛⎜⎝^∞_𝑛=0_{0≤𝑘<2𝑛−1}^𝛼𝜙_𝑛,𝑘(𝑡)_𝛼𝜙_𝑛,𝑘(𝑠)⎞⎟⎠^𝛼,𝛽𝑅(𝑡,𝑠)𝑑𝑠𝑑𝑡,=¹₀^𝛼,𝛽𝑅(𝑡,𝑡)𝑑𝑠𝑑𝑡,(F.8) using the result of Corollary 3.6.

G. Girsanov Formula

In this section we provide the quite technical proof of Lemma 6.8 which is useful in proving the Girsanov formula.

Lemma G.1. The positive definite quadratic form on 𝑙²(ℝ)×𝑙²(ℝ) associated with operator _𝛼,𝛽𝑆−𝐼𝑑∶𝑙²(ℝ)→𝑙²(ℝ) is well defined on _𝜉Ω^. Moreover for all _𝜉Ω^, 𝜉,^𝛼,𝛽𝑆−𝐼𝑑_𝜉Ω(𝜉)=2¹₀𝛼(𝑡)−𝛽(𝑡)𝑓²(𝑡)^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)∘𝑑^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)+¹₀(𝛼(𝑡)−𝛽(𝑡))²𝑓²(𝑡)^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)²𝑑𝑡,(G.1) where _𝛼𝑋_𝑡(𝜉)=_𝛼Ψ(𝜉) and ∘ refers to the Stratonovich integral and the equality is true in law.

Proof. The proof of this lemma uses quite similar materials to those used in the proof of the Itô theorem. However, since this result is central for giving insight on the way our geometric considerations relate to the Girsanov theorem, we provide the detailed proof here.
Consider 𝜉 in _𝜉Ω^, denote 𝜉_𝑁=_𝜉𝑃_𝑁(𝜉), and write 𝜉_𝑁,^𝛼,𝛽𝑆_𝑁−𝐼𝑑_𝜉Ω𝑁𝜉_𝑁=_{(𝑛,𝑘)∈ℐ𝑁}_{(𝑝,𝑞)∈ℐ𝑁}𝐴^𝑛,𝑘_𝑝,𝑞+𝐵^𝑛,𝑘_𝑝,𝑞𝜉_𝑛,𝑘𝜉_𝑝,𝑞,(G.2) where we have posited 𝐴^𝑛,𝑘_𝑝,𝑞=2¹₀𝛼(𝑡)−𝛽(𝑡)√Γ(𝑡)^𝛼𝜙_𝑛,𝑘(𝑡)_𝛼𝜓_𝑝,𝑞(𝑡)+_𝛼𝜙_𝑝,𝑞(𝑡)_𝛼𝜓_𝑛,𝑘(𝑡)𝑑𝑡,𝐵^𝑛,𝑘_𝑝,𝑞=¹₀(𝛼(𝑡)−𝛽(𝑡))²Γ(𝑡)^𝛼𝜓_𝑛,𝑘(𝑡)_𝛼𝜓_𝑝,𝑞(𝑡)𝑑𝑡.(G.3) It is easy to see, using similar arguments to those in the proof of the integration by parts formula, Proposition 6.1: 𝐴^𝑁(𝜉)=_{(𝑛,𝑘)∈ℐ𝑁}_{(𝑝,𝑞)∈ℐ𝑁}𝐴^𝑛,𝑘_𝑝,𝑞𝜉_𝑛,𝑘𝜉_𝑝,𝑞=2¹₀𝛼(𝑡)−𝛽(𝑡)√Γ(𝑡)^𝛼𝑋^𝑁_𝑡(𝜉)𝑓_𝛼(𝑡)𝑑𝑑𝑡^𝛼𝑋^𝑁_𝑡(𝜉)𝑔_𝛼(𝑡)𝑑𝑡,_(𝑛,𝑘_)∈ℐ𝑁_{(p,𝑞)∈ℐ𝑁}𝐵^𝑛,𝑘_𝑝,𝑞𝜉_𝑛,𝑘𝜉_𝑝,𝑞=¹₀(𝛼(𝑡)−𝛽(𝑡))²Γ(𝑡)^𝛼𝑋^𝑁_𝑡(𝜉)²𝑑𝑡.(G.4) Because of the uniform convergence property of 𝑋^𝑁 towards 𝑋 and the fact that it has almost surely bounded sample paths, the latter sum converges towards ¹₀(𝛼(𝑡)−𝛽(𝑡))²𝑓²(𝑡)^𝛼𝑋_𝑡(𝜉)𝑔_𝛼(𝑡)²𝑑𝑡.(G.5) Now writing quantity 𝐴^𝑁(𝜉) as the sum of elementary integrals between the points of discontinuity 𝑡_𝑖=𝑖2^−𝑁, 0≤𝑖≤2^𝑁, 𝐴^𝑁(𝜉)=−2^2𝑁−1_𝑖=0^𝑡𝑖+1_𝑡𝑖𝛼(𝑡)−𝛽(𝑡)√Γ(𝑡)^𝛼𝑋^𝑁_𝑡(𝜉)𝑓(𝑡)𝑑𝑑𝑡^𝛼𝑋^𝑁_𝑡(𝜉)𝑔_𝛼(𝑡)𝑑𝑡(G.6) and using the identities of (E.13) and (E.14), we then have 𝐴^𝑁(𝜉)=−2^2𝑁−1_𝑖=0⎛⎜⎝𝑤^𝑁_𝑖^𝛼𝑋_𝑡𝑖(𝜉)𝑔_𝛼𝑡_𝑖+𝑤^𝑁_𝑖+1^𝛼𝑋_𝑡𝑖+1(𝜉)𝑔_𝛼𝑡_𝑖+1⎞⎟⎠⎛⎜⎝^𝛼𝑋_𝑡𝑖+1(𝜉)g_𝛼𝑡_𝑖+1−^𝛼𝑋_𝑡𝑖(𝜉)𝑔_𝛼𝑡_𝑖⎞⎟⎠,(G.7) where we denote 𝑤^𝑁_𝑖=∫^𝑡𝑖+1_𝑡𝑖(𝛼(𝑡)−𝛽(𝑡))ℎ𝑡_𝑖+1−ℎ(𝑡)𝑑𝑡ℎ𝑡_𝑖+1−ℎ𝑡_𝑖²,𝑤^𝑁_𝑖+1=∫^𝑡𝑖+1_𝑡𝑖(𝛼(𝑡)−𝛽(𝑡))ℎ𝑡_𝑖+1−ℎ(𝑡)𝑑𝑡ℎ𝑡_𝑖+1−ℎ𝑡_𝑖².(G.8) Let us define the function 𝑤 in 𝐶[0,1] by 𝑤(𝑡)=𝛼(𝑡)−𝛽(𝑡)𝑓²(𝑡).(G.9) If 𝛼 and 𝛽 are uniformly 𝛿-Hölder continuous, so is 𝑤. Therefore, there exist an integer 𝑁^>0 and a real 𝑀>0 such that if 𝑁>𝑁^, for all 0≤𝑖<2^𝑁, we have ||𝑤^𝑁_𝑖−𝑤𝑡_𝑖||=||||||^𝑡𝑖+1_𝑡𝑖𝑤(𝑡)−𝑤𝑡_𝑖(𝑑/𝑑𝑡)ℎ𝑡_𝑖+1−ℎ(𝑡)²ℎ𝑡_𝑖+1−ℎ𝑡_𝑖²𝑑𝑡||||||≤𝑀||||||^𝑡𝑖+1_𝑡𝑖𝑡−𝑡_𝑖^𝛿(𝑑/𝑑𝑡)ℎ𝑡_𝑖+1−ℎ(𝑡)²ℎ𝑡_𝑖+1−ℎ𝑡_𝑖²𝑑𝑡||||||≤𝑀⎛⎜⎝𝑡−𝑡_𝑖^𝛿+12(𝛿+1)⎞⎟⎠+𝑀|||||^𝑡𝑖+1_𝑡𝑖𝑡−𝑡_𝑖^𝛿+1𝛿+1ℎ𝑡_𝑖+1−ℎ(𝑡)²ℎ𝑡_𝑖+1−ℎ𝑡_𝑖²𝑑𝑡|||||≤𝑀⎛⎜⎝𝑡−𝑡_𝑖^𝛿+12(𝛿+1)⎞⎟⎠+𝑀⎛⎜⎝𝑡−𝑡_𝑖^𝛿+22(𝛿+1)(𝛿+2)⎞⎟⎠,(G.10) which shows that |𝑤^𝑁_𝑖−𝑤(𝑡_𝑖)|=𝑂(2^{−𝑁(1+𝛿)}), and similarly |𝑤^𝑁_𝑖+1−𝑤(𝑡_𝑖+1)|=𝑂(2^{−𝑁(1+𝛿)}) as well. As a consequence, expression Lemma G.1 converges when 𝑁 tends to infinity toward the desired Stratonovich integral.

Acknowledgments

The authors wish to thank Professor Marcelo Magnasco for many illuminating discussions. This work was partially supported by NSF Grant EF-0928723 and ERC Grant NERVI-227747.