Table of Contents
International Journal of Stochastic Analysis
Volume 2011, Article ID 247329, 89 pages
Research Article

Multiresolution Hilbert Approach to Multidimensional Gauss-Markov Processes

1Lewis-Sigler Institute, Princeton University, Carl Icahn Laboratory, Princeton, NJ 08544, USA
2Laboratory of Mathematical Physics, The Rockefeller University, New York, NY 10065, USA
3Mathematical Neuroscience Laboratory, Collège de France, CIRB, 11 Place Marcelin Berthelot, CNRS UMR 7241 and INSERM U 1050, Université Pierre et Marie Curie ED, 158 and Memolife PSL, 75005 Paris, France
4INRIA BANG Laboratory, Paris, France

Received 28 April 2011; Accepted 6 October 2011

Academic Editor: Agnès Sulem

Copyright © 2011 Thibaud Taillefumier and Jonathan Touboul. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 forṁ𝐗𝑡=𝑓𝐗𝑡,𝑡,𝝃𝑡,(1.1) where the perturbative term 𝝃𝑡 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 [37]). 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 functions𝐗𝑡=lim𝑁𝐗𝑁𝑡with𝐗𝑁𝑡=𝑁𝑛=0𝐟𝑛(𝑡)𝚵𝑛,(1.2) where 𝐟𝑛 is a deterministic function and Ξ𝑛 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 Ξ𝑛 are Gaussian independent and the elements 𝐟𝑛, called the Schauder elements and denoted by 𝐬𝑛, are obtained by time-dependent integration of the Haar basis elements: 𝐬0,0(𝑡)=𝑡𝐈𝑑 and 𝐬𝑛,𝑘(𝑡)=𝑠𝑛,𝑘(𝑡)𝐈𝑑, with for all 𝑛>0𝑠𝑛,𝑘(𝑡)=2(𝑛1)/2𝑡𝑙𝑛,𝑘,𝑘2𝑛+1𝑡(2𝑘+1)2𝑛,2(𝑛1)/2𝑟𝑛,𝑘𝑡,(2𝑘+1)2𝑛𝑡(𝑘+1)2𝑛+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 𝑓𝑛 depends only on the second-order statistics of the considered process and the coefficients 𝜉𝑛 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 𝐬𝑛 have compact supports that exhibit a nested structure: this fact entails that the finite sums 𝐖𝑁 are processes that interpolate the limit process 𝐖 on the endpoints of the supports, that is, on the dyadic points 𝑘2𝑁, 0𝑘2𝑁. 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:𝐗𝑡=𝐠(𝑡)𝑡0𝐟(𝑠)𝑑𝐖𝑠(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 𝑋𝑁 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 Ξ𝑛 are always independent and Gaussian (for convenience with law 𝒩(𝟎,𝐈𝑑), 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., [2022]).

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 [2325]. 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 [2932], 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,,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 (𝐖𝑡,𝑡,𝑡[0,1]) be an 𝑚-dimensional Wiener process, consider the continuous functions 𝜶[0,1]𝑑×𝑑Γ[0,1]𝑑×𝑚, and define the positive bounded continuous function Γ=ΓΓ𝑇[0,1]𝑑×𝑑. The 𝑑-dimensional Ornstein-Uhlenbeck process associated with these parameters is solution of the equation𝑑𝐗𝑡=𝜶(𝑡)𝐗𝑡𝑑𝑡+𝚪(𝑡)𝑑𝐖𝑡,(2.1) and with initial condition 𝐗𝑡0 in 𝑡0, it reads𝐗𝑡=𝐅𝑡0,𝑡𝐗𝑡0+𝐅𝑡0,𝑡𝑡𝑡0𝐅𝑠,𝑡0𝚪(𝑠)𝑑𝐖𝑠,(2.2) where 𝐅(𝑡0,𝑡) is the flow of the equation, namely, the solution in 𝑑×𝑑 of the linear equation:𝜕𝐅𝑡0,𝑡𝜕𝑡=𝜶(𝑡)𝐅(𝑡),𝐅𝑡0,𝑡0=𝐈𝑑.(2.3) Note that the flow 𝐅(𝑡0,𝑡) enjoys the chain rule property:𝐅𝑡0,𝑡=𝐅𝑡1,𝑡𝐅𝑡0,𝑡1.(2.4) For all 𝑡,𝑠 such that 𝑡0<𝑠,𝑡, the vectors 𝐗𝑡 and 𝐗𝑠 admit the covariance𝐂𝑡0(𝑠,𝑡)=𝐅𝑡0,𝑡𝑡𝑠𝑡0𝐅𝑤,𝑡0𝚪(𝑤)𝐅𝑤,𝑡0𝑇𝑑𝑤𝐅𝑡0,𝑠𝑇=𝐅𝑡0,𝑡𝐡𝑡0(𝑠,𝑡)𝐅𝑡0,𝑠𝑇,(2.5) where we further defined 𝐡𝑢(𝑠,𝑡) the function𝐡𝑢(𝑠,𝑡)=𝑡𝑠𝐅(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 𝐡𝑣(𝑠,𝑡)=𝐅(𝑣,𝑢)𝐡𝑢(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 𝐗𝑣 taking into account 𝐗𝑢 are nondeterministic random variables, which is equivalent to saying that the covariance matrix of 𝐗𝑣 taking into account 𝐗𝑢, denoted by 𝐂𝑢(𝑣,𝑣) is symmetric positive definite for any 𝑢𝑣. 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 form𝐗𝑡=𝐠(𝑡)𝑡0𝐟(𝑠)𝑑𝐖𝑠,(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 𝑡𝑥𝑡𝑧 two times in the interval [0,1]. For any 𝑡[𝑡𝑥,𝑡𝑧], the random variable 𝐗𝑡 conditioned on 𝐗𝑡𝑥=𝐱 and 𝐗𝑡𝑧=𝐳 is a Gaussian variable with covariance matrix 𝚺(𝑡) and mean vector 𝝁(𝑡) given by 𝚺𝑡;𝑡𝑥,𝑡𝑧=𝐡𝑡𝑡𝑥,𝑡𝐡𝑡(𝑡𝑥,𝑡𝑧)1𝐡𝑡𝑡,𝑡𝑧,𝝁(𝑡)=𝝁𝑙𝑡;𝑡𝑥,𝑡𝑧𝐱+𝝁𝑟𝑡;𝑡𝑥,𝑡𝑧𝐳,(2.9) where the continuous matrix functions 𝝁𝑙(;𝑡𝑥,𝑡𝑧) and 𝝁𝑟(;𝑡𝑥,𝑡𝑧) of 𝑑×𝑑 are given by 𝝁𝑙𝑡;𝑡𝑥,𝑡𝑧=𝐅𝑡𝑥,𝑡𝐡𝑡𝑥𝑡,𝑡𝑧𝐡𝑡𝑥𝑡𝑥,𝑡𝑧1,𝝁𝑟𝑡;𝑡𝑥,𝑡𝑧=𝐅𝑡𝑧,𝑡𝐡𝑡𝑧𝑡𝑥,𝑡𝐡𝑡𝑧(𝑡𝑥,𝑡𝑧)1.(2.10)

Note that the functions 𝜇𝑙 and 𝜇𝑟 have the property that 𝝁𝑙(𝑡𝑥;𝑡𝑥,𝑡𝑧)=𝝁𝑟(𝑡𝑧;𝑡𝑥,𝑡𝑧)=𝐈𝑑 and 𝝁𝑙(𝑡𝑧;𝑡𝑥,𝑡𝑧)=𝝁𝑟(𝑡𝑥;𝑡𝑥,𝑡𝑧)=𝟎 ensuring that the process is indeed equal to 𝐱 at time 𝑡𝑥 and 𝐳 at time 𝑡𝑧.

Proof. Let 𝑡𝑥,𝑡𝑧 be two times of the interval [0,1] such that 𝑡𝑥<𝑡𝑧, and let 𝑡[𝑡𝑥,𝑡𝑧]. We consider the Gaussian random variable 𝝃=(𝐗𝑡,𝐗𝑡𝑧) conditioned on the fact that 𝐗𝑡𝑥=𝐱. Its mean can be easily computed from expression (2.2) and reads 𝐦𝑡,𝐦𝑡𝑧=𝐅𝑡𝑥,𝑡𝐱,𝐅𝑡𝑥,𝑡𝑧𝐱=𝐠(𝑡)𝐠1𝑡𝑥𝐱,𝐠𝑡𝑧𝐠1𝑡𝑥𝐱,(2.11) and its covariance matrix, from (2.5), reads 𝐂𝑡,𝑡𝐂𝑡,𝑡𝑧𝐂𝑡𝑧,𝑡𝐂𝑡𝑧,𝑡𝑧=𝐅𝑡𝑥,𝑡𝐡𝑡𝑥𝑡𝑥,𝑡𝐅𝑡𝑥,𝑡𝑇𝐅𝑡𝑥,𝑡𝐡𝑡𝑥𝑡𝑥,𝑡𝐅𝑡𝑥,𝑡𝑧𝑇𝐅𝑡𝑥,𝑡𝑧𝐡𝑡𝑥𝑡𝑥,𝑡𝐅𝑡𝑥,𝑡𝑇𝐅𝑡𝑥,𝑡𝑧𝐡𝑡𝑥𝑡𝑥,𝑡𝑧𝐅𝑡𝑥,𝑡𝑧𝑇=𝐡𝑡𝑡𝑥,𝑡𝐡𝑡𝑡𝑥,𝑡𝐅𝑡,𝑡𝑧𝑇𝐅𝑡,𝑡𝑧𝐡𝑡𝑡𝑥,𝑡𝐅𝑡,𝑡𝑧𝐡𝑡𝑡𝑥,𝑡𝑧𝐅𝑡,𝑡𝑧𝑇.(2.12) From there, we apply the conditioning formula for the Gaussian vectors (see, e.g., [36]) to infer the law of 𝐗𝑡 conditioned on 𝐗𝑡𝑥=𝐱 and 𝐗𝑡𝑧=𝐳, that is the law 𝒩(𝝁(𝑡),𝚺(𝑡;𝑡𝑥,𝑡𝑧)) of 𝐁𝑡 where 𝐁 denotes the bridge process obtained by pinning 𝐗 in 𝑡𝑥 and 𝑡𝑧. The covariance matrix is given by 𝚺𝑡;𝑡𝑥,𝑡𝑧=𝐂𝑦,𝑦𝐂𝑦,𝑧𝐂1𝑧,𝑧𝐂𝑧,𝑦=𝐡𝑡𝑡𝑥,𝑡𝐡𝑡𝑡𝑥,𝑡𝐡𝑡𝑡𝑥,𝑡𝑧1𝐡𝑡𝑡𝑥,𝑡=𝐡𝑡𝑡𝑥,𝑡𝐡𝑡𝑡𝑥,𝑡𝑧1𝐡𝑡𝑡,𝑡𝑧,(2.13) and the mean reads 𝝁(𝑡)=𝐦𝑦+𝐂𝑦,𝑧𝐂1𝑧,𝑧𝐳𝐦𝑧=𝐅𝑡𝑥,𝑡𝐈𝑑𝐡𝑡𝑥𝑡𝑥,𝑡𝐡𝑡𝑥(𝑡𝑥,𝑡𝑧)1𝐱+𝐅𝑡𝑧,𝑡𝐡𝑡𝑧𝑡𝑥,𝑡𝐡𝑡𝑧(𝑡𝑥,𝑡𝑧)1𝐳,=𝐅(𝑡𝑥,𝑡)𝐡𝑡𝑥(𝑡,𝑡𝑧)𝐡𝑡𝑥(𝑡𝑥,𝑡𝑧)1𝝁𝑙(𝑡;𝑡𝑥,𝑡𝑧)𝐱+𝐅(𝑡𝑧,𝑡)𝐡𝑡𝑧(𝑡𝑥,𝑡)𝐡𝑡𝑧(𝑡𝑥,𝑡𝑧)1𝝁𝑟(𝑡;𝑡𝑥,𝑡𝑧)𝐳,(2.14) where we have used the fact that 𝐡𝑡𝑥(𝑡𝑥,𝑡𝑧)=𝐡𝑡𝑧(𝑡𝑥,𝑡)+𝐡𝑡𝑥(𝑡,𝑡𝑧). The regularity of the thus-defined functions 𝝁𝑥 and 𝝁𝑧 directly stems from the regularity of the flow operator 𝐅. Moreover, since for any 0𝑡,𝑢1, we observe that 𝐅(𝑡,𝑡)=𝐈𝑑 and 𝑢(𝑡,𝑡)=𝟎; we clearly have 𝝁𝑥(𝑡𝑥)=𝝁𝑦(𝑡)=𝐈𝑑 and 𝝁𝑥(𝑡)=𝝁𝑦(𝑡𝑥)=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 𝑆1,0=[𝑙1,0,𝑟1,0] with 𝑙1,0=0 and 𝑟1,0=1, we iteratively apply the following operation. Suppose that, at the 𝑛th step, the interval [0,1] is decomposed into 2𝑛1 intervals 𝑆𝑛,𝑘=[𝑙𝑛,𝑘,𝑟𝑛,𝑘], called supports, such that 𝑙𝑛,𝑘+1=𝑟𝑛,𝑘 for 0𝑘<2𝑛1. Each of these intervals is then subdivided into two child intervals, a left-child 𝑆𝑛+1,2𝑘 and a right-child 𝑆𝑛+1,2𝑘+1, and the subdivision point 𝑟𝑛+1,2𝑘=𝑙𝑛+1,2𝑘+1 is denoted by 𝑚𝑛,𝑘. Therefore, we have defined three sequences of real 𝑙𝑛,𝑘, 𝑚𝑛,𝑘, and 𝑟𝑛,𝑘 for 𝑛>0 and 0𝑘<2𝑛1 satisfying 𝑙0,0=0𝑙𝑛,𝑘<𝑚𝑛,𝑘<𝑟𝑛,𝑘𝑟0,0=1 and𝑙𝑛+1,2𝑘=𝑙𝑛,𝑘,𝑚𝑛,𝑘=𝑟𝑛+1,2𝑘=𝑙𝑛+1,2𝑘+1,𝑟𝑛+1,2𝑘+1=𝑟𝑛,𝑘(2.15) with the convention 𝑙0,0=0 and 𝑟0,0=1 and 𝑆0,0=[0,1]. The resulting sequence of supports {𝑆𝑛,𝑘;𝑛0,0𝑘<2𝑛1} clearly has a binary tree structure.

For the sake of compactness of notations, we define the set of indices=𝑛<𝑁𝑛with𝑁=(𝑛,𝑘)20<𝑛𝑁,0𝑘<2𝑛1,(2.16) and for 𝑁>0, we define 𝐷𝑁={𝑚𝑛,𝑘,(𝑛,𝑘)𝑁1}{0,1}, the set of endpoints of the intervals 𝑆𝑁,𝑘. We additionally require that there exists 𝜌(0,1) such that for all (𝑛,𝑘)max(𝑟𝑛,𝑘𝑚𝑛,𝑘,𝑚𝑛,𝑘𝑙𝑛,𝑘)<𝜌(𝑟𝑛,𝑘𝑙𝑛,𝑘) which in particular implies thatlim𝑛sup𝑘𝑟𝑛,𝑘𝑙𝑛,𝑘=0(2.17) and ensures that the set of endpoints 𝑁N𝐷𝑁 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) read𝑙𝑛,𝑘=𝑘2𝑛+1,𝑚𝑛,𝑘=(2𝑘+1)2𝑛,𝑟𝑛,𝑘=(𝑘+1)2𝑛+1,(2.18) in which case the endpoints are simply the dyadic points 𝑁𝐷𝑁={𝑘2𝑁|0𝑘2𝑁}. 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 𝐗𝑡, a multidimensional Gauss-Markov process, we call the multiresolution description of a process the sequence of conditional expectations on the nested sets of endpoints 𝐷𝑛. In detail, if we denote by 𝑁 the filtration generated by {𝐗𝑡;𝑡𝐷𝑁} given the values of the process at the endpoints 𝐷𝑁 of the partition, we introduce the sequence of the Gaussian processes (𝐙𝑁𝑡)𝑁1 defined by:𝐙𝑁𝑡=𝔼𝐗𝑡𝑁=𝔼𝑁𝐗𝑡.(2.19) These processes 𝐙𝑁 can be naturally viewed as an interpolation of the process 𝐗 sampled at the increasingly finer times 𝐷𝑁 since for all 𝑡𝐷𝑁 we have 𝐙𝑁𝑡=𝐗𝑁𝑡. The innovation process (𝜹𝑁𝑡,𝑡,𝑡[0,1]) is defined as the update transforming the process 𝐙𝑁𝑡 into 𝐙𝑁+1𝑡, that is, 𝜹𝑁𝑡=𝐙𝑁+1𝑡𝐙𝑁𝑡.(2.20) It corresponds to the difference the additional knowledge of the process at the points 𝑚𝑁,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 𝜹𝑁𝑡 is a centered Gaussian process independent of the processes 𝐙𝑛𝑡 for any 𝑛𝑁. For 𝑠𝑆𝑁,𝑘 and 𝑡𝑆𝑁,𝑝 with 𝑘,𝑝𝑁, the covariance of the innovation process reads 𝔼𝑁𝜹𝑁𝑡𝜹𝑁𝑠𝑇=𝝁𝑁,𝑘(𝑡)𝚺𝑁,𝑘𝝁𝑁,𝑘(𝑡)𝑇if𝑘=𝑝,𝟎if𝑘𝑝,(2.21) where 𝝁𝑁,𝑘(𝑡)=𝝁𝑟𝑡;𝑙𝑁,𝑘,𝑚𝑁,𝑘,𝑡𝑙𝑁,𝑘,𝑚𝑁,𝑘,𝝁𝑙𝑡;𝑚𝑁,𝑘,𝑟𝑁,𝑘,𝑡𝑚𝑁,𝑘,𝑟𝑁,𝑘.(2.22) with 𝝁𝑙, 𝝁𝑟 and 𝚺𝑁,𝑘=𝚺(𝑚𝑁,𝑘;𝑙𝑁,𝑘,𝑟𝑁,𝑘) as defined in Proposition 2.1.

Proof. Because of the Markovian property of the process 𝐗, the law of the process 𝐙𝑁 can be computed from the bridge formula derived in Proposition 2.1 and we have 𝐙𝑁𝑡=𝝁𝑙𝑡;𝑙𝑁,𝑘,𝑟𝑁,𝑘𝐗𝑙𝑁,𝑘+𝝁𝑟𝑡;𝑙𝑁,𝑘,𝑟𝑁,𝑘𝐗𝑟𝑁,𝑘,𝐙𝑁+1𝑡=𝝁𝑙𝑡;𝑙𝑁,𝑘,𝑚𝑁,𝑘𝐗𝑙𝑁,𝑘+𝝁𝑟𝑡;𝑙𝑁,𝑘,𝑚𝑁,𝑘𝐗𝑚𝑁,𝑘,for𝑡𝑙𝑁,𝑘,𝑚𝑁,𝑘,𝝁𝑙𝑡;𝑚𝑁,𝑘,𝑟𝑁,𝑘𝐗𝑚𝑁,𝑘+𝝁𝑟𝑡;𝑚𝑁,𝑘,𝑟𝑁,𝑘𝐗𝑟𝑁,𝑘,for𝑡𝑙𝑁,𝑘,𝑚𝑁,𝑘.(2.23) Therefore, the innovation process can be written for 𝑡𝑆𝑁,𝑘 as 𝜹𝑁𝑡=𝝁𝑁𝑁,𝑘(𝑡)𝐗𝑚𝑁,𝑘+𝝂𝑁(𝑡)𝐐𝑁𝑡,(2.24) where 𝐐𝑁𝑡 is a 𝑁 measurable process 𝝂𝑁(𝑡) a deterministic matrix function and 𝝁𝑁,𝑘(𝑡)=𝝁𝑟𝑡;𝑙𝑁,𝑘,𝑚𝑁,𝑘,𝑡𝑙𝑁,𝑘,𝑚𝑁,𝑘,𝝁𝑙𝑡;𝑚𝑁,𝑘,𝑟𝑁,𝑘,𝑡𝑚𝑁,𝑘,𝑟𝑁,𝑘.(2.25) The expressions of 𝝂 and 𝐐 are quite complex but are highly simplified when noting that 𝔼𝜹𝑁𝑡𝑁=𝔼𝐙𝑁+1𝑡𝑁𝐙𝑁𝑡=𝔼𝔼𝐙𝑡𝑁+1𝑁𝐙𝑁𝑡=𝟎(2.26) directly implying that 𝝂(𝑡)𝐐𝑁𝑡=𝝁𝑁(𝑡)𝐙𝑁𝑚𝑁,𝑘 and yielding the remarkably compact expression 𝜹𝑁𝑡=𝝁𝑁,𝑘(𝑡)𝐗𝑚𝑁,𝑘𝐙𝑁𝑚𝑁,𝑘.(2.27) This process is a centered Gaussian process. Moreover, observing that it is 𝑁-measurable, it can be written as 𝜹𝑁𝑡=𝝁𝑁,𝑘(𝑡)𝐗𝑚𝑁,𝑘𝑁𝐙𝑁𝑚𝑁,𝑘,(2.28) and the process {𝐗𝑚𝑁,𝑘𝑁} appears as the Gauss-Markov bridge conditioned at times 𝑙𝑁,𝑘 and 𝑟𝑁,𝑘, and whose covariance is given by Proposition 2.1 and that has the expression 𝚺𝑁,𝑘=𝚺𝑚𝑁,𝑘;𝑙𝑁,𝑘,𝑟𝑁,𝑘=𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑚𝑛,𝑘𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑟𝑛,𝑘1𝐡𝑚𝑛,𝑘𝑚𝑛,𝑘,𝑟𝑛,𝑘.(2.29) Let (𝑠,𝑡)[0,1]2, and assume that 𝑠𝑆𝑁,𝑘 and 𝑡𝑆𝑁,𝑝. If 𝑘𝑝, then, because of the Markov property of the process 𝐗, the two bridges are independent and therefore the covariance 𝔼𝑁[𝜹𝑁𝑡(𝜹𝑁𝑠)𝑇] is zero. If 𝑘=𝑝, we have 𝔼𝑁𝜹𝑁𝑡𝜹𝑁𝑠𝑇=𝝁𝑁,𝑘(𝑡)𝚺𝑁,𝑘𝝁𝑁,𝑘(𝑠)𝑇.(2.30) Eventually, the independence property stems from the simple properties of the conditional expectation. Indeed, let 𝑛𝑁. We have 𝔼𝐙𝑛𝑡𝜹𝑁𝑠𝑇=𝔼𝐙𝑛𝑡𝐙𝑁+1𝑠𝐙𝑁𝑠𝑇=𝔼𝔼𝐗𝑡𝑛𝔼𝐗𝑇𝑠𝑁+1𝔼𝐗𝑇𝑠𝑁=𝔼𝔼𝐗𝑡𝑛𝔼𝐗𝑇𝑠𝑁+1𝔼𝔼𝐗𝑡𝑛𝔼𝐗𝑇𝑠𝑁=𝔼𝐙𝑛𝑡𝐙𝑛𝑠𝑇𝔼𝐙𝑛𝑡𝐙𝑛𝑠𝑇=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, there exists a collection of 𝝍𝑁,𝑘[0,1]𝑑×𝑑 that are zero outside the subinterval 𝑆𝑁,𝑘 such that in distribution one has: 𝜹𝑁𝑡=𝑘𝑁𝝍𝑁,𝑘(𝑡)𝚵𝑁,𝑘,(2.32) where Ξ𝑁,𝑘 are independent 𝑑-dimensional standard normal random variables (i.e., of law 𝒩(0,𝐈𝑑)). This basis of functions is unique up to an orthogonal transformation.

Proof. The two processes 𝜹𝑁𝑡 and 𝐝𝑁𝑡def=𝑘𝑁𝝍𝑁,𝑘(𝑡)Ξ𝑁,𝑘 are two Gaussian processes of mean zero. Therefore, we are searching for functions 𝝍𝑁,𝑘 vanishing outside 𝑆𝑁,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 𝝍𝑁,𝑘(𝑡) functions that vanish outside the subinterval 𝑆𝑁,𝑘 and that ensure that the covariance of the process 𝐝𝑁 is equal to the covariance of 𝜹𝑁. Let (𝑠,𝑡)[0,1] such that 𝑠𝑆𝑁,𝑘 and 𝑡𝑆𝑁,𝑝. If 𝑘𝑝, the assumption fact that the functions 𝜓𝑁,𝑘 vanish outside 𝑆𝑁,𝑘 implies that 𝔼𝐝𝑁𝑡𝐝𝑁𝑠𝑇=𝟎.(2.33) If 𝑘=𝑝, the covariance reads 𝔼𝐝𝑁𝑡𝐝𝑁𝑠𝑇=𝔼𝝍𝑁,𝑘(𝑡)𝚵𝑁,𝑘𝚵𝑇𝑁,𝑘𝝍𝑁,𝑘(𝑠)𝑇=𝝍𝑁,𝑘(𝑡)𝝍𝑁,𝑘(𝑠)𝑇,(2.34) which needs to be equal to the covariance of 𝜹𝑁, namely, 𝝍𝑁,𝑘(𝑡)𝝍𝑁,𝑘(𝑠)𝑇=𝝁𝑁,𝑘(𝑡)𝚺𝑁,𝑘𝝁𝑁,𝑘(𝑠)𝑇.(2.35) Therefore, since 𝝁𝑁,𝑘(𝑚𝑁,𝑘)=𝐈𝑑, we have 𝝍𝑁,𝑘𝑚𝑁,𝑘𝝍𝑁,𝑘𝑚𝑁,𝑘𝑇=𝚺𝐍,𝐤.(2.36) We can hence now define 𝝍𝑁,𝑘(𝑚𝑁,𝑘) as a square root 𝝈𝑁,𝑘 of the symmetric positive matrix 𝚺𝐍,𝐤, by fixing 𝑠=𝑚𝑁,𝑘 in (2.35) 𝝍𝑁,𝑘(𝑡)𝝈𝑇𝑁,𝑘=𝝁(𝑡)𝝈𝑁,𝑘𝝈𝑇𝑁,𝑘.(2.37) Eventually, since by assumption we have that 𝚺𝑁,𝑘 is invertible, so is 𝝈𝑁,𝑘, and the functions 𝝍𝑁,𝑘 can be written as 𝝍𝑁,𝑘(𝑡)=𝝁𝑁,𝑘(𝑡)𝝈𝑁,𝑘(2.38) with 𝝈𝑁,𝑘 being a square root of 𝚺𝑁,𝑘. Square roots of positive symmetric matrices are uniquely defined up to an orthogonal transformation. Therefore, all square roots of 𝚺𝑁,𝑘 are related by orthogonal transformations 𝝈𝑁,𝑘=𝝈𝑁,𝑘𝐎𝑁,𝑘, where 𝐎𝑁,𝑘𝐎𝑇𝑁,𝑘=𝐈𝑑. This property immediately extends to the functions 𝝍𝑁,𝑘 we are studying: two different functions 𝝍𝑁,𝑘 and 𝝍𝑁,𝑘 satisfying the theorem differ from an orthogonal transformation 𝐎𝑁,𝑘. We proved that, for 𝝍𝑁,𝑘(𝑡)Ξ𝑁,𝑘 to have the same law as 𝜹𝑁(𝑡) in the interval 𝑆𝑁,𝑘, the functions 𝝍𝑁,𝑘 with support in 𝑆𝑁,𝑘 are necessarily of the form 𝝁𝑁,𝑘(𝑡)𝝈𝑁,𝑘. It is straightforward to show the sufficient condition that provided such a set of functions, the processes 𝜹𝑁𝑡 and 𝐝𝑁𝑡 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:𝝍𝑛,𝑘(𝑡)=𝐠(𝑡)𝐠1𝑚𝑛,𝑘𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑡𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑚𝑛,𝑘1𝝈𝑛,𝑘,for𝑙𝑛,𝑘𝑡𝑚𝑛,𝑘,𝐠(𝑡)𝐠1𝑚𝑛,𝑘𝐡𝑚𝑛,𝑘𝑡,𝑟𝑛,𝑘𝐡𝑚𝑛,𝑘𝑚𝑛,𝑘,𝑟𝑛,𝑘1𝝈𝑛,𝑘,for𝑚𝑛,𝑘𝑡𝑟𝑛,𝑘,(2.39) and 𝝈𝑛,𝑘 satisfies𝝈𝑛,𝑘𝝈𝑇𝑛,𝑘=𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑚𝑛,𝑘𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑟𝑛,𝑘1𝐡𝑚𝑛,𝑘𝑚𝑛,𝑘,𝑟𝑛,𝑘.(2.40) Note that 𝝈𝑛,𝑘 can be defined uniquely as the symmetric positive square root, or as the lower triangular matrix resulting from the Cholesky decomposition of 𝚺𝑛,𝑘.

Let us now define the function 𝝍0,0[0,1]𝑑×𝑑 such that the process 𝝍0,0(𝑡)Ξ0,0 has the same covariance as 𝐙0𝑡, which is computed using exactly the same technique as that developed in the proof of Theorem 2.4 and that has the expression𝝍0,0(𝑡)=𝐠(𝑡)𝐡0𝑙0,0,𝑡𝐡0(𝑙0,0,𝑟0,0)1𝐠1𝑟0,0𝝈0,0,(2.41) for 𝝈0,0, a square root of 𝐂𝑟0,0 the covariance matrix of 𝐗𝑟0,0 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 𝐙𝑁𝑡 is equal in law to the process 𝐗𝑁𝑡=𝑁1𝑛=0𝑘𝑛𝝍𝑛,𝑘(𝑡)𝚵𝑛,𝑘,(2.43) where Ξ𝑛,𝑘 are independent standard normal random variables 𝒩(0,𝐈𝑑).

Proof. We have 𝐙𝑁𝑡=𝐙𝑁𝑡𝐙𝑁1𝑡+𝐙𝑁1𝑡𝐙𝑁2𝑡++𝐙2𝑡𝐙1𝑡+𝐙1𝑡=𝑁1𝑛=1𝜹𝑛𝑡+𝐙1𝑡=𝑁1𝑛=1𝑘𝑛𝝍𝑛,𝑘(𝑡)𝚵𝑛,𝑘+𝝍0,0(𝑡)𝚵0,0=𝑁1𝑛=0𝑘𝑛𝝍𝑛,𝑘(𝑡)𝚵𝑛,𝑘.(2.44)

We therefore identified a collection of functions {𝝍𝑛,𝑘}(𝑛,𝑘) 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 𝐙𝑁𝑡 converges almost surely towards the Gauss-Markov process 𝐗𝑡 used to construct the basis, proving that these finite-dimensional continuous functions 𝐙𝑁𝑡 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 {𝝍𝑛,𝑘}(𝑛,𝑘) we now study in details. In particular, we enlighten the structure of the collection of functions 𝝍𝑛,𝑘 as a Schauder basis in a certain space 𝒳 of continuous functions from [0,1] to 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)𝝍𝑛,𝑘𝝃𝑛,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 𝝍𝑛,𝑘 as a Schauder basis for some given space. In (2.39), two constant matrices 𝑑×𝑑 appear that will have a particular importance in the sequel for (𝑛,𝑘) in with 𝑛0:𝐋𝑛,𝑘=𝐠𝑇𝑚𝑛,𝑘𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑚𝑛,𝑘1𝝈𝑛,𝑘=𝐡𝑙𝑛,𝑘,𝑚𝑛,𝑘1𝐠1𝑚𝑛,𝑘𝝈𝑛,𝑘,𝐑𝑛,𝑘=𝐠𝑇𝑚𝑛,𝑘𝐡𝑚𝑛,𝑘𝑚𝑛,𝑘,𝑟𝑛,𝑘1𝝈𝑛,𝑘=𝐡𝑚𝑛,𝑘,𝑟𝑛,𝑘1𝐠1𝑚𝑛,𝑘𝝈𝑛,𝑘,(3.2) where 𝐡 stands for 𝐡0. We further define the matrix𝐌𝑛,𝑘=𝐠𝑇𝑚𝑛,𝑘𝝈1𝑛,𝑘𝑇(3.3) and we recall that 𝝈𝑛,𝑘 is a square root of 𝚺𝑛,𝑘, the covariance matrix of 𝐗𝑚𝑛,𝑘, conditionally to 𝐗𝑙𝑛,𝑘 and 𝐗𝑟𝑛,𝑘, given in (2.29). We stress that the matrices 𝐋𝑛,𝑘,𝐑𝑛,𝑘, 𝐌𝑛,𝑘, and 𝚺𝑛,k are all invertible and satisfy the important following properties.

Proposition 3.1. For all (𝑛,𝑘) in , 𝑛0, one has:(i)𝐌𝑛,𝑘=𝐋𝑛,𝑘+𝐑𝑛,𝑘(ii)𝚺1𝑛,𝑘=(𝐡𝑚𝑛,𝑘(𝑙𝑛,𝑘,𝑚𝑛,𝑘))1+(𝐡𝑚𝑛,𝑘(𝑚𝑛,𝑘,𝑟𝑛,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 𝐺𝐿𝑛() such that 𝐶=𝐴+𝐵 is also invertible, if one defines 𝐷=𝐴𝐶1𝐵, one has the following properties:(i)𝐷=𝐴𝐶1𝐵=𝐵𝐶1𝐴(ii)𝐷1=𝐴1+𝐵1.

Proof. (i)𝐷=𝐴𝐶1𝐵=(𝐶𝐵)𝐶1𝐵=𝐵𝐵𝐶1𝐵=𝐵(𝐼𝐶1𝐵)=𝐵𝐶1(𝐶𝐵)=𝐵𝐶1𝐴.(ii)(𝐴1+𝐵1)𝐷=𝐴1𝐷+𝐵1𝐷=𝐴1𝐴𝐶1𝐵+𝐵1𝐵𝐶1𝐴=𝐶1(𝐵+𝐴)=𝐶1𝐶=𝐼.

Proof of Proposition 3.1. (ii) Directly stems from Lemma 3.2, item (ii) by posing 𝐴=𝐡𝑚𝑛,𝑘(𝑙𝑛,𝑘,𝑚𝑛,𝑘), 𝐵=𝐡𝑚𝑛,𝑘(𝑚𝑛,𝑘,𝑟𝑛,𝑘), and 𝐶=𝐴+𝐵=𝐡𝑚𝑛,𝑘(𝑙𝑛,𝑘,𝑟𝑛,𝑘). Indeed, the lemma implies that𝐷1=𝐴1𝐶𝐵1=𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑚𝑛,𝑘1𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑟𝑛,𝑘𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑚𝑛,𝑘1=𝚺1𝑛,𝑘.(3.4)(i)We have𝐋𝑛,𝑘+𝐑𝑛,𝑘=𝐠𝑚𝑛,𝑘𝑇𝐡𝑙𝑛,𝑘,𝑚𝑛,𝑘1+𝐡𝑚𝑛,𝑘,𝑟𝑛,𝑘1𝝈𝑛,𝑘=𝐠𝑚𝑛,𝑘𝑇𝚺1𝑛,𝑘𝝈𝑛,𝑘=𝐠𝑚𝑛,𝑘𝑇𝝈1𝑛,𝑘𝑇,(3.5) which ends the demonstration of the proposition.

Let us define 𝐋0,0=(𝐡(𝑙0,0,𝑟0,0))1𝐠1(𝑟0,0)𝝈0,0. With this notations we define the functions in a compact form as follows.

Definition 3.3. For every (𝑛,𝑘) in with 𝑛0, the continuous functions 𝝍𝑛,𝑘 are defined on their support 𝑆𝑛,𝑘 as 𝝍𝑛,𝑘(𝑡)=𝐠(𝑡)𝐡𝑙𝑛,𝑘,𝑡𝐋𝑛,𝑘,𝑙𝑛,𝑘𝑡𝑚𝑛,𝑘,𝐠(𝑡)𝐡𝑡,𝑟𝑛,𝑘𝐑𝑛,𝑘,𝑚𝑛,𝑘𝑡𝑟𝑛,𝑘,(3.6) and the basis element 𝝍0,0 is given on [0,1] by 𝝍0,0(𝑡)=𝐠(𝑡)𝐡𝑙0,0,𝑡𝐋0,0.(3.7)

The definition implies that 𝝍𝑛,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],𝑑×𝑑).

Before studying the property of the functions 𝝍𝑛,𝑘, it is worth remembering that their definitions include the choice of a square root 𝝈𝑛,𝑘 of 𝚺𝑛,𝑘. Properly speaking, there is thus a class of bases 𝝍𝑛,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 𝝈𝑛,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 𝜙𝑛,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 𝝍𝑛,𝑘 and Φ𝑛,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 with 𝑛0, we define a continuous function 𝜙𝑛,𝑘[0,1]𝑚×𝑑 which is zero outside its support 𝑆𝑛,𝑘 and has the expressions: 𝜙𝑛,𝑘(𝑡)=𝐟(𝑡)𝑇𝐋𝑛,𝑘if𝑙𝑛,𝑘𝑡<𝑚𝑛,𝑘,𝐟(𝑡)𝑇𝐑𝑛,𝑘if𝑚𝑛,𝑘𝑡<𝑟𝑛,𝑘.(3.8) The basis element 𝜙0,0 is defined on [0,1] by 𝜙0,0(𝑡)=𝐟(𝑡)𝑇𝐋0,0.(3.9)

Remark that the definitions make apparent the fact that these two families of functions are linked for all (𝑛,𝑘) in through the simple relation𝝍𝑛,𝑘=𝜶𝝍𝑛,𝑘+𝚪𝜙𝑛,𝑘.(3.10) Moreover, this collection of functions 𝜙𝑛,𝑘 constitutes an orthogonal basis of functions, in the following sense.

Proposition 3.5. Let 𝐿2𝐟 be the closure of 𝐮[0,1]𝑚𝐯𝐿2[0,1],𝑑,𝐮=𝐟𝑇𝐯,(3.11) equipped with the natural norm of 𝐿2([0,1],𝑚). It is a Hilbert space, and moreover, for all 0𝑗<𝑑, the family of functions 𝑐𝑗(𝜙𝑛,𝑘) defined as the columns of 𝜙𝑛,𝑘, namely 𝑐𝑗𝜙𝑛,𝑘=𝜙𝑛,𝑘𝑖,𝑗0𝑖<𝑚,(3.12) forms a complete orthonormal basis of 𝐿2𝐟.

Proof. The space 𝐿2𝐟 is clearly a Hilbert space as a closed subspace of the larger Hilbert space 𝐿2([0,1],𝑚) is equipped with the standard scalar product: 𝐮,𝐯𝐿2[0,1],𝑑,(𝐮,𝐯)=10𝐮(𝑡)𝑇𝐯(𝑡)𝑑𝑡.(3.13) We now proceed to demonstrate that the columns of 𝜙𝑛,𝑘 form an orthonormal family which generates a dense subspace of 𝐿2𝐟. To this end, we define 𝑀([0,1],𝑚×𝑑) as the space of functions 𝐀[0,1]𝑚×𝑑𝑗0𝑗<𝑑,𝑡𝐀𝑖,𝑗(𝑡)0𝑖<𝑚𝐿2([0,1],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],𝑚). This definition allows us to define the bilinear function 𝒫𝑀([0,1],𝑚×𝑑)×𝑀([0,1],𝑚×𝑑)𝑑×𝑑 as 𝒫(𝐀,𝐁)=10𝐀(𝑡)𝑇𝐁(𝑡)𝑑𝑡satisfying𝒫(𝐁,𝐀)=𝒫(𝐀,𝐁)𝑇,(3.15) and we observe that the columns of 𝜙𝑛,𝑘 form an orthonormal system if and only if ((𝑝,𝑞),(𝑛,𝑘))×,𝒫𝜙𝑛,𝑘,𝜙𝑝,𝑞=10𝜙𝑛,𝑘(𝑡)𝑇𝜙𝑝,q(𝑡)𝑑𝑡=𝛿𝑛,𝑘𝑝,𝑞𝐈𝑑,(3.16) where 𝛿𝑛,𝑘𝑝,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 𝜙𝑛,𝑘 are zero outside the interval 𝑆𝑛,𝑘, the matrix 𝒫(𝜙𝑛,𝑘,𝜙𝑝,𝑞) is nonzero only if 𝑆𝑛,𝑘𝑆𝑝,𝑞. In such cases, assuming that 𝑛𝑝 and, for example, that 𝑛<𝑝, we necessarily have 𝑆𝑛,𝑘 strictly included in 𝑆𝑝,𝑞: more precisely, 𝑆𝑛,𝑘 is either included in the left-child support 𝑆𝑝+1,2𝑞 or in the right-child support 𝑆𝑝+1,2𝑞+1 of 𝑆𝑝,𝑞. In both cases, writing the matrix 𝒫(𝜙𝑛,𝑘(𝑡),𝜙𝑝,𝑞) shows that it is expressed as a matrix product whose factors include 𝒫(𝜙𝑛,𝑘,𝐟𝑇). We then show that 𝒫𝜙𝑛,𝑘,𝐟𝑇=10𝜙𝑛,𝑘(𝑡)𝑇𝐟(𝑡)𝑇=𝐋𝑇𝑛,𝑘𝑚𝑛,𝑘𝑙𝑛,𝑘𝐟(𝑢)𝐟𝑇(𝑢)𝑑𝑢𝐑𝑇𝑛,𝑘𝑟𝑛,𝑘𝑚𝑛,𝑘𝐟(𝑢)𝐟𝑇(𝑢)𝑑𝑢,=𝐋𝑇𝑛,𝑘𝐡𝑙𝑛,𝑘,𝑚𝑛,𝑘𝐑𝑇𝑛,𝑘𝐡𝑚𝑛,𝑘,𝑟𝑛,𝑘=𝝈𝑇𝑛,𝑘𝐠1𝑚𝑛,𝑘𝑇𝝈𝑇𝑛,𝑘𝐠1𝑚𝑛,𝑘𝑇,(3.17) which entails that 𝒫(𝜙𝑛,𝑘,𝐟𝑇)=0 if 𝑛<𝑝. If 𝑛>𝑝, we remark that 𝒫(𝜙𝑛,𝑘,𝜙𝑝,𝑞)=𝒫(𝜙𝑝,𝑞,𝜙𝑛,𝑘)𝑇, and we conclude that 𝒫(𝜙𝑛,𝑘,𝜙𝑝,𝑞)=𝟎 from the preceding case. For 𝑛=𝑝, we directly compute for 𝑛>0 the only nonzero term 𝒫𝜙𝑛,𝑘,𝜙𝑛,𝑘=𝐋𝑇𝑛,𝑘𝑚𝑛,𝑘𝑙𝑛,𝑘𝐟(𝑢)𝐟𝑇(𝑢)𝑑𝑢𝐋𝑛,𝑘+𝐑𝑇𝑛,𝑘𝑟𝑛,𝑘𝑚𝑛,𝑘𝐟(𝑢)𝐟𝑇(𝑢)𝑑𝑢𝐑𝑛,𝑘,=𝝈𝑇𝑛,𝑘𝐠1𝑚𝑛,𝑘𝑇𝐡𝑙𝑛,𝑘,𝑚𝑛,𝑘1𝐠1𝑚𝑛,𝑘𝝈𝑛,𝑘+𝝈𝑇𝑛,𝑘𝐠1𝑚𝑛,𝑘𝑇𝐡𝑚𝑛,𝑘,𝑟𝑛,𝑘1𝐠1𝑚𝑛,𝑘𝝈𝑛,𝑘.(3.18) Using the passage relationship between the symmetric functions 𝐡 and 𝐡𝑚𝑛,𝑘 given in (2.7), we can then write 𝒫𝜙𝑛,𝑘,𝜙𝑛,𝑘=𝝈𝑇𝑛,𝑘𝐡𝑚𝑛,𝑘𝑙𝑛,𝑘,𝑚𝑛,𝑘1𝝈𝑛,𝑘+𝝈𝑇𝑛,𝑘𝐡𝑚𝑛,𝑘𝑚𝑛,𝑘,𝑟𝑛,𝑘1𝝈𝑛,𝑘.(3.19) Proposition 3.1 implies that 𝐡𝑚𝑛,𝑘(𝑙𝑛,𝑘,𝑚𝑛,𝑘)1+𝐡𝑚𝑛,𝑘(𝑚𝑛,𝑘,𝑟𝑛,𝑘)1=𝚺1𝑛,𝑘=(𝝈1𝑛,𝑘)𝑇𝝈1𝑛,𝑘 which directly implies that 𝒫(𝜙𝑛,𝑘,𝜙𝑇𝑛,𝑘)=𝐈𝑑. For 𝑛=0, a computation of the exact same flavor yields that 𝒫(𝜙0,0,𝜙0,0)=𝐈𝑑. Hence, we have proved that the collection of columns of 𝜙𝑛,𝑘 forms an orthonormal family of functions in 𝐿2𝐟 (the definition of 𝜙𝑛,𝑘 clearly states that its columns can be written in the form of elements of 𝐿2𝐟).
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 the functions 𝐏𝑛,𝑘[0,1]𝑑×𝑑 with support on 𝑆𝑛,𝑘 defined by 𝐏𝑛,𝑘(𝑡)=𝐋𝑛,𝑘if𝑙𝑛,𝑘𝑡<𝑚𝑛,𝑘,𝐑𝑛,𝑘if𝑚𝑛,𝑘𝑡<𝑟𝑛,𝑘,𝑛0,𝐏0,0(𝑡)=𝐋0,0.(3.20) Showing that the family of columns of 𝜙𝑛,𝑘 is dense in 𝐿2𝐟 is equivalent to show that the column vectors of the matrices 𝐏𝑛,𝑘 seen as a function of 𝑡 are dense in 𝐿2([0,1],𝑑). It is enough to show that the span of such functions contains the family of piecewise continuous 𝑑-valued functions that are to be constant on 𝑆𝑛,𝑘, (𝑛,𝑘) in (the density of the endpoints of the partition 𝑁𝐷𝑁 entails that the latter family generates 𝐿2([0,1],𝑑)).
In fact, we show that the span of functions 𝑉𝑁=span𝑡𝑐𝑗𝐏𝑛,𝑘(𝑡)0𝑗<𝑑,(𝑛,𝑘)𝑁(3.21) is exactly equal to the space 𝐾𝑁 of piecewise continuous functions from [0,1] to 𝑑 that are constant on the supports 𝑆𝑁+1,𝑘, for any (𝑁+1,𝑘) in . The fact that 𝑉𝑁 is included in 𝐾𝑁 is clear from the fact that the matrix-valued functions 𝐏𝑁,𝑘 are defined constant on the support 𝑆𝑁+1,𝑘, for (𝑁,𝑘) in 𝐼.
We prove that 𝐾𝑁 is included in 𝑉𝑁 by induction on 𝑁0. The property is clearly true at rank 𝑁=0 since 𝐏0,0 is then equal to the constant invertible matrix 𝐋0,0. Assuming that the proposition true at rank 𝑁1 for a given 𝑁>0, let us consider a piecewise continuous function 𝐜[0,1]𝑑 in 𝐾𝑁1. Remark that, for every (𝑁,𝑘) in , the function 𝐜 can only take two values on 𝑆𝑁,𝑘 and can have discontinuity jump in 𝑚𝑁,𝑘: let us denote these jumps as 𝐝𝑁,𝑘=𝐜𝑚+𝑁,𝑘𝐜𝑚𝑁,𝑘.(3.22) Now, remark that for every (𝑁,𝑘) in , the matrix-valued functions 𝐏𝑁,𝑘 take only two matrix values on 𝑆𝑁,𝑘, namely, 𝐋𝑁,𝑘 and 𝐑𝑁,𝑘. From Proposition 3.1, we know that 𝐋𝑁,𝑘+𝐑𝑁,𝑘=𝐌𝑁,𝑘 is invertible. This fact directly entails that there exist vectors 𝐚𝑁,𝑘, for any (𝑁,𝑘) in , such that 𝐝𝑁,𝑘=(𝐋𝑁,𝑘+𝐑𝑁,𝑘)(𝐚𝑁,𝑘). We then necessarily have that the function 𝐜=𝐜+𝐏𝑛,𝑘𝐚𝑛,𝑘 is piecewise constant on the supports 𝑆𝑁,𝑘, (𝑁,𝑘) in . By recurrence hypothesis, 𝐜 belongs to 𝑉𝑁1, so that 𝐜 belongs to 𝑉𝑁, and we have proved that 𝐾𝑁𝑉𝑁. Therefore, the space generated by the column vectors 𝑃𝑛,𝑘 is dense in 𝐿2[0,1], which completes the proof that the functions 𝑡[(𝜙𝑛,𝑘(𝑡))𝑖,𝑗]0𝑖<𝑚 form a complete orthonormal family of 𝐿2[0,1].

The fact that the column functions of 𝜙𝑛,𝑘 form a complete orthonormal system of 𝐿2𝐟 directly entails the following decomposition of the identity on 𝐿2𝐟.

Corollary 3.6. If 𝛿 is the real delta Dirac function, one has (𝑛,𝑘)𝜙𝑛,𝑘(𝑡)𝜙𝑇𝑛,𝑘(𝑠)=𝛿(𝑡𝑠)𝐼𝑑𝐿2𝐟.(3.23)

Proof. Indeed it easy to verify that, for all 𝐯 in 𝐿2𝐟, we have for all 𝑁>0𝑈(𝑛,𝑘)𝑁𝜙𝑛,𝑘(𝑡)𝜙𝑇𝑛,𝑘(𝑠)𝐯(𝑠)𝑑𝑠=(𝑛,𝑘)𝑁𝜙𝑛,𝑘(𝑡)𝒫𝜙𝑛,𝑘,𝐯=(𝑛,𝑘)𝑁𝑑1𝑝=0𝑐𝑝𝜙𝑛,𝑘𝑐𝑝𝜙𝑛,𝑘,𝐯,(3.24) where (𝑐𝑝(𝜙𝑛,𝑘),𝐯) denotes the inner product in 𝐿2𝐟 between 𝐯 and the 𝑝-column of 𝝍𝑛,𝑘. Therefore, by the Parseval identity, we have in the 𝐿2𝐟 sense 𝑈(𝑛,𝑘)𝜙𝑛,𝑘(𝑡)𝜙𝑇𝑛,𝑘(𝑠)𝐯(𝑠)𝑑𝑠=𝐯(𝑡).(3.25)

From now on, abusing language, we will say that the family of 𝑚×𝑑-valued functions 𝜙𝑛,𝑘 is an orthonormal family of functions to refer to the fact that the columns of such matrices form orthonormal set of 𝐿2𝐟. We now make explicit the relationship between this orthonormal basis and our functions (𝝍𝑛,𝑘) derived in our analysis of the multidimensional Gauss-Markov processes.

3.1.2. Generalized Dual Operators

The Integral Operator 𝓚
The basis 𝜙𝑛,𝑘 is of great interest in this paper for its relationship to the functions 𝝍𝑛,𝑘 that naturally arise in the decomposition of the Gauss-Markov processes. Indeed, the collection 𝝍𝑛,𝑘 can be generated from the orthonormal basis 𝜙𝑛,𝑘 through the action of the integral operator 𝒦 defined on 𝐿2([0,1],𝑚) into 𝐿2([0,1],𝑑) by 𝐮𝒦[𝐮]=𝑡𝐠(𝑡)𝑈𝟙[0,𝑡](𝑠)𝐟(𝑠)𝐮(𝑠)𝑑𝑠,(3.26) where 𝑈[0,1] is an open set and, for any set 𝐸𝑈,𝟙𝐸() denotes the indicator function of 𝐸. Indeed, realizing that 𝒦 acts on 𝑀([0,1],𝑚×𝑑) into 𝑀([0,1],𝑑×𝑑) through 𝐀𝑀[0,1],m×𝑑,𝒦[𝐀]=𝒦𝑐0(𝐀),,𝒦𝑐𝑑1(𝐀),(3.27) where 𝑐𝑗(𝐀) denotes the 𝑗th 𝑚-valued column function of 𝐀, we easily see that for all (𝑛,𝑘) in , 0𝑡1, 𝝍𝑛,𝑘(𝑡)=𝐠(𝑡)𝑡0𝐟(𝑠)𝜙𝑛,𝑘(𝑠)𝑑𝑠=𝒦𝜙𝑛,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 𝝍𝑛,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 𝜙𝑛,𝑘 will conveniently provide us with a dual basis to 𝝍𝑛,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 𝑑×𝑑, and that 𝐟 and 𝐟1 have continuous derivatives, which especially implies that 𝐿2𝐟=𝐿2(𝑑). In this setting, we define the space 𝐷0(𝑈,𝑑) of functions in 𝐶0(𝑈,𝑑) that are zero at zero and denote by 𝐷0(𝑈,𝑑) its dual in the space of distributions (or generalized functions). Under the assumptions just made, the operator 𝒦𝐷0(𝑈,𝑑)𝐷0(𝑈,𝑑) admits the differential operator 𝒟𝐷0(𝑈,𝑑)𝐷0(𝑈,𝑑) defined by 𝐮𝐷0𝑈,𝑑𝒟[𝐮]=𝑡𝐟1(𝑡)𝑑𝑑𝑡𝐠1(𝑡)𝐮(𝑡)(3.29) as its inverse, that is, when restricted to 𝐷0(𝑈,𝑑), we have 𝒟𝒦=𝒦𝒟=𝐼𝑑 on 𝐷0(𝑈,𝑑). The dual operators of 𝒦 and 𝒟 are expressed, for any 𝐮 in 𝐷0(𝑈,𝑑), as 𝒟[𝐮]=𝑡𝐠1(𝑡)𝑇𝑑𝑑𝑡𝐟1(𝑡)𝑇𝐮(𝑡),𝒦[𝐮]=𝑡𝐟(𝑡)𝑇𝑈𝟙[0,𝑡](𝑠)𝐠𝑇(𝑠)𝐮(𝑠)𝑑𝑠.(3.30) They satisfy (from the properties of 𝒦 and 𝒟) 𝒟𝒦=𝒦𝒟=𝐼𝑑 on 𝐷0(𝑈,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 𝐷0(𝑈,𝑑). In details, for any distribution 𝑇 in 𝐷0(𝑈,𝑑) and test function 𝐮 in 𝐷0(𝑈,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 𝜙𝑛,𝑘 to infer a dual family of the basis 𝝍𝑛,𝑘. For any function 𝐮 in 𝐿2(𝑈,𝑑), the generalized function 𝒦[𝐮] belongs to 𝐶0(𝑈,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,𝑑), the set of 𝑑-dimensional Radon measures with 𝑅0(𝑈,𝑑)𝐷0(𝑈,𝑑). Consequently, operating in the Gelfand triple 𝐶0𝑈,𝑑𝐿2𝑈,𝑑𝑅0𝑈,𝑑,(3.32) we can write, for any function 𝐮, 𝐯 in 𝐿2(𝑈,𝑑)𝑅0(𝑈,𝑑), (𝐮,𝐯)=((𝒟𝒦)[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 𝒟𝒦=𝐼𝑑 on 𝐿2(𝑈,𝑑). The dual pairing associated with the Gelfand triple (3.32) entails the second equality where 𝒟 is the generalized operator defined on 𝐷0(𝑈,𝑑) and where 𝒟[𝐯] is in 𝑅0(𝑈,𝑑).
As a consequence, defining the functions 𝜹𝑛,𝑘 in 𝑅0(𝑈,𝑑×𝑑), the 𝑑×𝑑-dimensional space of Radon measures, by 𝜹𝑛,𝑘=𝒟𝜙𝑛,𝑘=𝒟𝑐𝑖𝜙𝑛,𝑘,,𝒟𝑐𝑗𝜙𝑛,𝑘(3.34) provides us with a family of 𝑑×𝑑-generalized functions which are dual to the family 𝝍𝑛,𝑘 in the sense that, for all ((𝑛,𝑘),(𝑝,𝑞)) in ×, we have 𝒫𝜹𝑛,𝑘,𝝍𝑝,𝑞=𝛿𝑛,𝑘𝑝,𝑞𝐈𝑑,(3.35) where the definition of 𝒫 has been extended through dual pairing: given any 𝐀 in 𝑅0(𝑈,𝑚×𝑑) and any 𝐁 in 𝐶0(𝑈,𝑚×𝑑), we have 𝒫(𝐀,𝐁)=𝑐𝑖(𝐀),𝑐𝑗(𝐁)0𝑖,𝑗<𝑑(3.36) with (𝑐𝑖(𝐀),𝑐𝑗(𝐁)) denoting the dual pairing between the 𝑖th column of 𝐀 taking value in 𝑅0(𝑈,𝑑) and the 𝑗th column of 𝐁 taking value in 𝐶0(𝑈,𝑑). Under the favorable hypothesis of this section, the 𝑑×𝑑-generalized functions 𝜹𝑛,𝑘 can actually be easily computed since considering the definition of 𝜙𝑛,𝑘 shows that the functions (𝐟1)𝑇𝜙𝑛,𝑘 have support 𝑆𝑛,𝑘 and are constant on 𝑆𝑛+1,2𝑘 and S𝑛+1,2𝑘+1 in 𝑑×𝑑. Only the discontinuous jumps in 𝑙𝑛,𝑘, 𝑚𝑛,𝑘, and 𝑟𝑛,𝑘 intervene, leading to expressing for (𝑛,𝑘) in