A Class of Bridges of Iterated Integrals of Brownian Motion Related to Various Boundary Value Problems Involving the One-Dimensional Polyharmonic Operator

Abstract

Let (𝐵(𝑡))_𝑡∈[0,1] be the linear Brownian motion and (𝑋_𝑛(𝑡))_𝑡∈[0,1] the (𝑛−1)-fold integral of Brownian motion, with 𝑛 being a positive integer: 𝑋_𝑛(𝑡)=∫^𝑡₀((𝑡−𝑠)^𝑛−1/(𝑛−1)!)d𝐵(𝑠) for any 𝑡∈[0,1]. In this paper we construct several bridges between times 0 and 1 of the process (𝑋_𝑛(𝑡))_𝑡∈[0,1] involving conditions on the successive derivatives of 𝑋_𝑛 at times 0 and 1. For this family of bridges, we make a correspondence with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical problem of prediction. Our results involve various Hermite interpolation polynomials.

1. Introduction

Throughout the paper, we will denote, for any enough differentiable function 𝑓, its 𝑖th derivative by 𝑓^(𝑖) or d^𝑖𝑓/d𝑡^𝑖.

Let (𝐵(𝑡))_𝑡∈[0,1] be the linear Brownian motion started at 0 and (𝛽(𝑡))_𝑡∈[0,1] be the linear Brownian bridge within the time interval [0,1]: (𝛽(𝑡))_𝑡∈[0,1]=(𝐵(𝑡)∣𝐵(0)=𝐵(1)=0)_𝑡∈[0,1]. These processes are Gaussian processes with covariance functions 𝑐_𝐵(𝑠,𝑡)=𝑠∧𝑡,𝑐_𝛽(𝑠,𝑡)=𝑠∧𝑡−𝑠𝑡. (1.1) For a given continuous function 𝑢, the functions 𝑣_𝐵 and 𝑣_𝛽, respectively, defined on [0,1] by 𝑣_𝐵(𝑡)=¹₀𝑐_𝐵(𝑠,𝑡)𝑢(𝑠)d𝑠,𝑣_𝛽(𝑡)=¹₀𝑐_𝛽(𝑠,𝑡)𝑢(𝑠)d𝑠 (1.2) are the solutions of the respective boundary value problems on [0,1]: 𝑣_𝐵^′′𝑣=−𝑢,_𝐵(0)=𝑣^′_𝐵(1)𝑣=0,𝑣^𝛽′′=−𝑢,_𝛽(0)=𝑣_𝛽(1)=0. (1.3) Observe that the differential equations are the same in both cases. Only the boundary conditions differ. They are Dirichlet-type boundary conditions for Brownian bridge while they are Dirichlet/Neumann-type boundary conditions for Brownian motion.

These well-known connections can be extended to the polyharmonic operator d^2𝑛/d𝑡^2𝑛, where 𝑛 is a positive integer. This latter is associated with the (𝑛−1)-fold integral of Brownian motion (𝑋_𝑛(𝑡))_𝑡∈[0,1]: 𝑋_𝑛(𝑡)=^𝑡₀(𝑡−𝑠)^𝑛−1()!(𝑠)[].𝑛−1d𝐵forany𝑡∈0,1 (1.4) (Notice that all of the derivatives at time 0 naturally vanish: 𝑋_𝑛(0)=𝑋_𝑛−1(0)=⋯=𝑋₂(0)=𝑋₁(0)=0.) Indeed, the following facts, for instance, are known (see, e.g., [1, 2]).(i)The covariance function of the process (𝑋_𝑛(𝑡))_𝑡∈[0,1] coincides with the Green function of the boundary value problem 𝑣^(2𝑛)=(−1)^𝑛[],𝑣(0)𝑢on0,1=𝑣^′(0)=⋯=𝑣^(𝑛−1)(0)𝑣=0,_(𝑛)(1)=𝑣^(𝑛+1)(1)=⋯=𝑣^(2𝑛−1)(1)=0. (1.5)(ii)The covariance fonction of the bridge (𝑋_𝑛(𝑡)∣𝑋_𝑛(1)=0)_𝑡∈[0,1] coincides with the Green function of the boundary value problem 𝑣^(2𝑛)=(−1)^𝑛[],𝑣(0)𝑢on0,1=𝑣^′(0)=⋯=𝑣^(𝑛−1)(0)𝑣=0,_(𝑛−1)(1)=𝑣^(𝑛+1)(1)=⋯=𝑣^(2𝑛−1)(1)=0. (1.6)(iii)The covariance fonction of the bridge (𝑋_𝑛(𝑡)∣𝑋_𝑛(1)=𝑋_𝑛−1(1)=⋯=𝑋₁(1)=0)_𝑡∈[0,1] coincides with the Green function of the boundary value problem 𝑣^(2𝑛)=(−1)^𝑛[],𝑣(0)𝑢on0,1=𝑣^′(0)=⋯=𝑣^(𝑛−1)(0)𝑣(1)=0,=𝑣^′(1)=⋯=𝑣^(𝑛−1)(1)=0. (1.7)

Observe that the differential equations and the boundary conditions at 0 are the same in all cases. Only the boundary conditions at 1 differ. Other boundary value problems can be found in [3, 4].

We refer the reader to [5] for a pioneering work dealing with the connections between general Gaussian processes and Green functions; see also [6]. We also refer to [7–13] and the references therein for various properties, namely, asymptotical study, of the iterated integrals of Brownian motion and more general Gaussian processes as well as to [3, 4, 14, 15] for interesting applications of these processes to statistics.

The aim of this work is to examine all the possible conditioned processes of (𝑋_𝑛(𝑡))_𝑡∈[0,1] involving different events at time 1: 𝑋_𝑛(𝑡)∣𝑋_𝑗1(1)=𝑋_𝑗2(1)=⋯=𝑋_𝑗𝑚(1)=0_𝑡∈[0,1] (1.8) for a certain number 𝑚 of events, 1≤𝑚≤𝑛, and certain indices 𝑗₁,𝑗₂,…,𝑗_𝑚 such that 1≤𝑗₁<𝑗₂<⋯<𝑗_𝑚≤𝑛, and to make the connection with the boundary value problems:𝑣^(2𝑛)=(−1)^𝑛[],𝑣(0)𝑢on0,1=𝑣^′(0)=⋯=𝑣^(𝑛−1)(0)𝑣=0,_(𝑖1)(1)=𝑣^(𝑖2)(1)=⋯=𝑣^(𝑖𝑛)(1)=0 (BVP) for certain indices 𝑖₁,𝑖₂,…,𝑖_𝑛 such that 0≤𝑖₁<𝑖₂<⋯<𝑖_𝑛≤2𝑛−1. Actually, we will see that this connection does not recover all the possible boundary value problems, and we will characterize those sets of indices for which such a connection exists.

The paper is organized as follows. In Section 2, we exhibit the relationships between general Gaussian processes and Green functions of certain boundary value problems. In Section 3, we consider the iterated integrals of Brownian motion. In Section 4, we construct several bridges associated with the foregoing processes and depict explicitly their connections with the polyharmonic operator together with various boundary conditions. One of the main results is Theorem 4.4. Moreover, we exhibit several interesting properties of the bridges (Theorems 4.1 and 4.3) and solve the prediction problem (Theorem 4.6). In Section 5, we illustrate the previous results on the case 𝑛=2 related to integrated Brownian motion. Finally, in Section 6, we give a characterization for the Green function of the boundary value problem (BVP) to be a covariance function. Another one of the main results is Theorem 6.3.

2. Gaussian Processes and Green Functions

We consider an 𝑛-Markov Gaussian process (𝑋(𝑡))_𝑡∈[0,1] evolving on the real line ℝ. By “𝑛-Markov,” it is understood that the trajectory 𝑡↦𝑋(𝑡) is 𝑛 times differentiable and the 𝑛-dimensional process (𝑋(𝑡),𝑋′(𝑡),…,𝑋^(𝑛−1)(𝑡))_𝑡∈[0,1] is a Markov process. Let us introduce the covariance function of (𝑋(𝑡))_𝑡∈[0,1]: for 𝑠,𝑡∈[0,1], 𝑐_𝑋(𝑠,𝑡)=𝔼[𝑋(𝑠)𝑋(𝑡)]. It is known (see [6]) that the function 𝑐_𝑋 admits the following representation:𝑐_𝑋(𝑠,𝑡)=^𝑛−1_𝑘=0𝜑_𝑘(𝑠∧𝑡)𝜓_𝑘(𝑠∨𝑡), (2.1) where 𝜑_𝑘,𝜓_𝑘, 𝑘∈{0,1,…,𝑛−1}, are certain functions.

Let 𝒟₀,𝒟₁ be linear differential operators of order less than 𝑝, and let 𝒟 be a linear differential operator of order 𝑝 defined by 𝒟=^𝑝_𝑖=0𝛼_𝑖d^𝑖d𝑡^𝑖, (2.2) where 𝛼₀,𝛼₁,…,𝛼_𝑝 are continuous functions on [0,1]. More precisely, we have for any 𝑝 times differentiable function 𝑓 and any 𝑡∈[0,1], 𝒟𝑓(𝑡)=^𝑝_𝑖=0𝛼_𝑖(𝑡)𝑓^(𝑖)(𝑡). (2.3)

Theorem 2.1. Assume that the functions 𝜑_𝑘,𝜓_𝑘, 𝑘∈{0,1,…,𝑛−1}, are 𝑝 times differentiable and satisfy the following conditions, for a certain constant 𝜅: ^𝑛−1_𝑘=0𝜑_k𝜓_𝑘^(𝑖)−𝜑_𝑘^(𝑖)𝜓_𝑘=0𝑖𝑓0≤𝑖≤𝑝−2,𝜅𝑖𝑓𝑖=𝑝−1,𝒟𝜑_𝑘=𝒟𝜓_𝑘=0,𝒟₀𝜑_𝑘(0)=0,𝒟₁𝜓_𝑘(1)=0. (2.4)(2.5) Then, for any continuous function 𝑢 on [0,1], the function 𝑣 defined on [0,1] by 𝑣(𝑡)=¹₀𝑐_𝑋(𝑠,𝑡)𝑢(𝑠)d𝑠 (2.6) solves the boundary value problem 𝒟𝑣=𝛼_𝑝𝒟𝑢,₀𝑣(0)=𝒟₁𝑣(1)=0. (2.7)

Remark 2.2. If the problem (2.7) is determining, that is, if it has a unique solution, then the covariance function 𝑐_𝑋 is exactly the Green function of the boundary value problem (2.7).

Proof. In view of (2.1), the function 𝑣 can be written as 𝑣(𝑡)=^𝑛−1_𝑘=0𝜓_𝑘(𝑡)^𝑡₀𝜑_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜑_𝑘(𝑡)¹_𝑡𝜓_𝑘(𝑠)𝑢(𝑠)d𝑠. (2.8) The derivative of 𝑣 is given by 𝑣^′(𝑡)=^𝑛−1_𝑘=0𝜓^′_𝑘(𝑡)^𝑡₀𝜑_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜑^′_𝑘(𝑡)¹_𝑡𝜓_𝑘(𝑠)𝑢(𝑠)d𝑠 (2.9) and its second-order derivative, since ∑^𝑛−1_𝑘=0[𝜑_𝑘𝜓^′_𝑘−𝜑^′_𝑘𝜓_𝑘]=0, by 𝑣^′′(𝑡)=^𝑛−1_𝑘=0𝜓_𝑘^′′(𝑡)^𝑡₀𝜑_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜑_𝑘^′′(𝑡)¹_𝑡𝜓_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜑_𝑘(𝑡)𝜓^′_𝑘(𝑡)−𝜑^′_𝑘(𝑡)𝜓_𝑘(𝑡𝑢(𝑡=))_{𝑛−1𝑘=0}𝜓_𝑘^′′(𝑡)^𝑡₀𝜑_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜑_𝑘^′′(𝑡)¹_𝑡𝜓_𝑘(𝑠)𝑢(𝑠)d𝑠. (2.10) More generally, because of the assumptions (2.4), we easily see that, for 𝑖∈{0,1,…,𝑝−1}, 𝑣^(𝑖)(𝑡)=^𝑛−1_𝑘=0𝜓_𝑘^(𝑖)(𝑡)^𝑡₀𝜑_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜑_𝑘^(𝑖)(𝑡)¹_𝑡𝜓_𝑘(𝑠)𝑢(𝑠)d𝑠, (2.11) and the 𝑝th order derivative of 𝑣 is given by 𝑣^(𝑝)(𝑡)=^𝑛−1_𝑘=0𝜓_𝑘^(𝑝)(𝑡)^𝑡₀𝜑_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜑_𝑘^(𝑝)(𝑡)¹_𝑡𝜓_𝑘(𝑠)𝑢(𝑠)+𝜑d𝑠_𝑘(𝑡)𝜓_𝑘^(𝑝−1)(𝑡)−𝜑_𝑘^(𝑝−1)(𝑡)𝜓(𝑡𝑢(𝑡=))_{𝑛−1𝑘=0}𝜓_𝑘^(𝑝)(𝑡)^𝑡₀𝜑_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜑_𝑘^(𝑝)(𝑡)¹_𝑡𝜓_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜅𝑢(𝑡). (2.12) Actually, we have proved that, for 𝑖∈{0,1,…,𝑝−1}, 𝑣^(𝑖)(𝑡)=^𝑡₀𝜕^𝑖𝑐_𝑋𝜕𝑡^𝑖(𝑠,𝑡)𝑢(𝑠)d𝑠. (2.13) Finally, due to (2.5), 𝒟𝑣(𝑡)=^𝑛−1_𝑘=0𝒟𝜓_𝑘(𝑡)^𝑡₀𝜑_𝑘(𝑠)𝑢(𝑠)d𝑠+𝒟𝜑_𝑘(𝑡)¹_𝑡𝜓_𝑘(𝑠)𝑢(𝑠)d𝑠+𝜅𝛼_𝑝𝑢(𝑡)=𝜅𝛼_𝑝𝑢(𝑡). (2.14)
Concerning the boundary value conditions, referring to (2.5), we similarly have 𝒟₀𝑣(0)=^𝑛−1_𝑘=0𝒟₀𝜑_𝑘(0)¹₀𝜓_𝑘(𝑠)𝑢(𝑠)𝒟d𝑠=0,₁𝑣(1)=^𝑛−1_𝑘=0𝒟₁𝜓_𝑘(1)¹₀𝜑_𝑘(𝑠)𝑢(𝑠)d𝑠=0. (2.15) The proof of Theorem 2.1 is finished.

In the two next sections, we construct processes connected to the equation 𝒟𝑣=𝑢 subject to the boundary value conditions at 0: (𝒟^𝑖₀𝑣)(0)=0 for 𝑖∈{0,1,…,𝑛−1} and others at 1 that will be discussed subsequently, where 𝒟 and 𝒟^𝑖₀ are the differential operators (𝒟 being of order 𝑝=2𝑛) defined by 𝒟=(−1)^𝑛d^2𝑛d𝑡^2𝑛,𝒟^𝑖₀=d^𝑖d𝑡^𝑖. (2.16)

3. The (𝑛−1)-Fold Integral of Brownian Motion

Let (𝐵(𝑡))_𝑡∈[0,1] be the linear Brownian motion limited to the time interval [0,1] and started at 0. We introduce the (𝑛−1)-fold integral of Brownian motion: for any 𝑡∈[0,1], 𝑋_𝑛(𝑡)=^𝑡₀(𝑡−𝑠)^𝑛−1()!(𝑠).𝑛−1d𝐵 (3.1) In particular, 𝑋₁=𝐵. The trajectories of (𝑋_𝑛(𝑡))_𝑡∈[0,1] are 𝑛 times differentiable and we have 𝑋_𝑛^(𝑖)=𝑋_𝑛−𝑖 for 0≤𝑖≤𝑛−1. Moreover, we have at time 0 the equalities 𝑋_𝑛(0)=𝑋_𝑛−1(0)=⋯=𝑋₂(0)=𝑋₁(0)=0. The process (𝑋_𝑛(𝑡))_𝑡∈[0,1] is an 𝑛-Markov Gaussian process since the 𝑛-dimensional process (𝑋_𝑛(𝑡),𝑋_𝑛−1(𝑡),…,𝑋₁(𝑡))_𝑡∈[0,1] is Markovian. The covariance function of the Gaussian process (𝑋_𝑛(𝑡))_𝑡∈[0,1] is given by 𝑐_𝑋𝑛(𝑠,𝑡)=₀^𝑠∧𝑡(𝑠−𝑢)^𝑛−1()!()𝑛−1𝑡−𝑢^𝑛−1(𝑛−1)!d𝑢. (3.2) In order to apply Theorem 2.1, we decompose 𝑐_𝑋𝑛 into the form (2.1). We have for, e.g., 𝑠≤𝑡, 𝑐_𝑋𝑛(𝑠,𝑡)=^𝑠₀(𝑠−𝑢)^𝑛−1(𝑛−1)!^2𝑛−1_𝑘=0𝑘()𝑛−1−𝑢^{𝑛−1−𝑘}𝑡^𝑘=d𝑢_{𝑛−1𝑘=0}(−1)^{𝑛−1−𝑘}𝑡^𝑘𝑘!^𝑠₀(𝑠−𝑢)^𝑛−1()!𝑢𝑛−1(^{𝑛−1−𝑘})!=𝑛−1−𝑘d𝑢_{𝑛−1𝑘=0}(−1)^{𝑛−1−𝑘}𝑠^{2𝑛−1−𝑘}()!𝑡2𝑛−1−𝑘^𝑘.𝑘! (3.3) We then obtain the following representation: 𝑐_𝑋𝑛(𝑠,𝑡)=^𝑛−1_𝑘=0𝜑_𝑘(𝑠)𝜓_𝑘(𝑡) (3.4) with, for any 𝑘∈{0,1,…,𝑛−1}, 𝜑_𝑘(𝑠)=(−1)^{𝑛−1−𝑘}𝑠^{2𝑛−1−𝑘}(2𝑛−1−𝑘)!,𝜓_𝑘(𝑡)=𝑡^𝑘.𝑘! (3.5) We state below a result of [2] that we revisit here by using Theorem 2.1.

Theorem 3.1. Let 𝑢 be a fixed continuous function on [0,1]. The function 𝑣 defined on [0,1] by 𝑣(𝑡)=¹₀𝑐_𝑋𝑛(𝑠,𝑡)𝑢(𝑠)d𝑠 (3.6) is the solution of the boundary value problem 𝑣^(2𝑛)=(−1)^𝑛[],𝑣𝑢𝑜𝑛0,1_(𝑖)(0){},𝑣=0𝑓𝑜𝑟𝑖∈0,1,…,𝑛−1_(𝑖)(1)=0𝑓𝑜𝑟𝑖∈{𝑛,𝑛+1,…,2𝑛−1}. (3.7)

Proof. Let us check that conditions (2.4) and (2.5) of Theorem 2.1 are fulfilled. First, we have ^𝑛−1_𝑘=0𝜑_𝑘(𝑡)𝜓_𝑘^(𝑖)(𝑡)−𝜑_𝑘^(𝑖)(𝑡)𝜓_𝑘(𝑡=_{𝑛−1𝑘=0})(−1)^{𝑛−1−𝑘}𝟙_{{𝑘≥𝑖}}𝑡^{2𝑛−1−𝑘}()!𝑡2𝑛−1−𝑘(𝑘−𝑖^𝑘−𝑖)!−𝟙_{{k≤2𝑛−1−𝑖}}𝑡^{2𝑛−1−𝑖−𝑘}()!𝑡2𝑛−1−𝑖−𝑘^𝑘=()𝑘!−1^𝑛−1𝑡^{2𝑛−1−𝑖}()!𝟙2𝑛−1−𝑖^{{𝑖≤𝑛−1}𝑛−1}_𝑘=𝑖(−1)^𝑘−2𝑛−1−𝑖𝑘−𝑖^{(2𝑛−1−𝑖)∧(𝑛−1)𝑘=0}(−1)^𝑘𝑘=()2𝑛−1−𝑖−1^𝑛−1𝑡^{2𝑛−1−𝑖}()!𝟙2𝑛−1−𝑖^{{𝑖≤𝑛−1}𝑛−1−𝑖}_𝑘=0(−1)^𝑖+𝑘𝑘−2𝑛−1−𝑖^𝑛−1_𝑘=0(−1)^𝑘𝑘2𝑛−1−𝑖−𝟙_{{𝑖≥𝑛}}^{2𝑛−1−𝑖}_𝑘=0(−1)^𝑘𝑘.2𝑛−1−𝑖 (3.8) Performing the transformation 𝑘↦2𝑛−1−𝑖−𝑘 in the first sum lying within the last equality, we get ^{𝑛−1−𝑖}_𝑘=0(−1)^𝑖+𝑘𝑘−2𝑛−1−𝑖^𝑛−1_𝑘=0(−1)^𝑘𝑘=2𝑛−1−𝑖_{2𝑛−1−𝑖𝑘=𝑛}(−1)^𝑘+2𝑛−1−𝑖2𝑛−1−𝑖−𝑘^{𝑛−1𝑘=0}(−1)^𝑘𝑘=2𝑛−1−𝑖^{2𝑛−1−𝑖}_𝑘=0(−1)^𝑘𝑘2𝑛−1−𝑖=𝛿_{𝑖,2𝑛−1}, (3.9) and then ^𝑛−1_𝑘=0𝜑_𝑘(𝑡)𝜓_𝑘^(𝑖)(𝑡)−𝜑_𝑘^(𝑖)(𝑡)𝜓_𝑘(𝑡)=(−1)^𝑛𝛿_{𝑖,2𝑛−1}. (3.10) On the other hand, setting 𝒟=(−1)^𝑛d^2𝑛d𝑡^2𝑛,𝒟^𝑖₀=d^𝑖d𝑡^𝑖,𝒟^𝑖₁=d^𝑖+𝑛d𝑡^𝑖+𝑛, (3.11) we clearly see that, for any 𝑘∈{0,1,…,𝑛−1}, 𝒟𝜑_𝑘=𝒟𝜓_𝑘=0,𝒟^𝑖₀𝜑_𝑘(0)=𝒟^𝑖₁𝜓_𝑘(1)=0for𝑖∈{0,1,…,𝑛−1}. (3.12) Consequently, by Theorem 2.1, we see that the function 𝑣 solves the boundary value problem (3.7). The uniqueness part will follow from a more general argument stated in the proof of Theorem 4.4.

4. Various Bridges of the (𝑛−1)-Fold Integral of Brownian Motion

In this section, we construct various bridges related to (𝑋_𝑛(𝑡))_𝑡∈[0,1]. More precisely, we take (𝑋_𝑛(𝑡))_𝑡∈[0,1] conditioned on the event that certain derivatives vanish at time 1. Let us recall that all the derivatives at time 0 naturally vanish: 𝑋_𝑛(0)=𝑋_𝑛−1(0)=⋯=𝑋₂(0)=𝑋₁(0)=0.

For any 𝑚∈{0,1,…,𝑛}, let 𝐽={𝑗₁,𝑗₂,…,𝑗_𝑚} be a subset of {1,2,…,𝑛} with 1≤𝑗₁<𝑗₂<⋯<𝑗_𝑚≤𝑛 and the convention that for 𝑚=0, 𝐽=∅. We see that, for each 𝑚∈{0,1,…,𝑛}, we can define (^𝑛_𝑚) subsets of indices 𝐽, and the total number of sets 𝐽 is then ∑^𝑛_𝑚=0(^𝑛_𝑚)=2^𝑛. Set for any 𝑡∈[0,1]𝑌(𝑡)=𝑋_𝑛(𝑡)∣𝑋_𝑗(1)=0,𝑗∈𝐽=𝑋_𝑛(𝑡)∣𝑋_𝑗1(1)=𝑋_𝑗2(1)=⋯=𝑋_𝑗𝑚(1)=0. (4.1) In this way, we define 2^𝑛 processes (𝑌(𝑡))_𝑡∈[0,1] that we will call “bridges’’ of (𝑋_𝑛(𝑡))_𝑡∈[0,1]. In particular, (i)for 𝐽=∅, we simply have (𝑌(𝑡))_𝑡∈[0,1]=(𝑋_𝑛(𝑡))_𝑡∈[0,1]; (ii)for 𝐽={1}, the corresponding process is the (𝑛−1)-fold integral of Brownian bridge 𝑋_𝑛(𝑡)∣𝑋₁(1)=0_𝑡∈[0,1]=^𝑡₀(𝑡−𝑠)^𝑛−1()!(𝑠)𝑛−1d𝛽_𝑡∈[0,1]; (4.2)(iii)for 𝐽={𝑛}, the corresponding process is the “single’’ bridge of (𝑛−1)-fold integral of Brownian motion: 𝑋_𝑛(𝑡)∣𝑋_𝑛(1)=0_𝑡∈[0,1]=^𝑡₀(𝑡−𝑠)^𝑛−1()!(𝑠)∣𝑛−1d𝐵¹₀(1−𝑠)^𝑛−1()!(𝑠)𝑛−1d𝐵=0_𝑡∈[0,1]; (4.3)(iv)for 𝐽={1,2,…,𝑛}, the corresponding process is 𝑋_𝑛(𝑡)∣𝑋_𝑛(1)=𝑋_𝑛−1(1)=⋯=𝑋₁(1)=0_𝑡∈[0,1]. (4.4) This is the natural bridge related to the 𝑛-dimensional Markov process (𝑋_𝑛(𝑡),𝑋_𝑛−1(𝑡),…,𝑋₁(𝑡))_𝑡∈[0,1].

In this section, we exhibit several interesting properties of the various processes (𝑌(𝑡))_𝑡∈[0,1]. One of the main goals is to relate these bridges to additional boundary value conditions at 1. For this, we introduce the following subset 𝐼 of {0,1,…,2𝑛−1}: 𝐼=(𝑛−𝐽)∪[{𝑛,𝑛+1,…,2𝑛−1}⧵(𝐽+𝑛−1)] (4.5) with 𝑛−𝐽=𝑛−𝑗,𝑗∈𝐽=𝑛−𝑗₁,…,𝑛−𝑗_𝑚,𝐽+𝑛−1=𝑗+𝑛−1,𝑗∈𝐽=𝑗₁+𝑛−1,…,𝑗_𝑚+𝑛−1. (4.6) The cardinality of 𝐼 is 𝑛. Actually, the set 𝐼 will be used further for enumerating the boundary value problems which can be related to the bridges labeled by 𝐽. Conversely, 𝐼 yields 𝐽 through 𝐽=(𝑛−𝐼)∩{1,2,…,𝑛}. (4.7) In Table 1, we give some examples of sets 𝐼 and 𝐽.

Table 1 

4.1. Polynomial Drift Description

Below, we provide a representation of (𝑌(𝑡))_𝑡∈[0,1] by means of (𝑋_𝑛(𝑡))_𝑡∈[0,1] subject to a random polynomial drift.

Theorem 4.1. One has the distributional identity (𝑌(𝑡))^𝑑_𝑡∈[0,1]=𝑋_𝑛(𝑡)−_𝑗∈𝐽𝑃_𝑗(𝑡)𝑋_𝑗(1)_𝑡∈[0,1], (4.8) where the functions 𝑃_𝑗, 𝑗∈𝐽, are Hermite interpolation polynomials on [0,1] characterized by 𝑃_𝑗^(2𝑛)𝑃=0,_𝑗(𝑖)(0){},𝑃=0𝑓𝑜𝑟𝑖∈0,1,…,𝑛−1_𝑗(𝑖)(1)=𝛿_{𝑗,𝑛−𝑖}𝑓𝑜𝑟𝑖∈𝐼. (4.9)

Remark 4.2. In the case where 𝑛=2, we retrieve a result of [16]. Moreover, conditions (4.9) characterize the polynomials 𝑃_𝑗, 𝑗∈𝐽. We prove this fact in Lemma A.1 in the Appendix.

Proof. By invoking classical arguments of Gaussian processes theory, we have the distributional identity (𝑌(𝑡))^𝑑_𝑡∈[0,1]=𝑋_𝑛(𝑡)−_𝑗∈𝐽𝑃_𝑗(𝑡)𝑋_𝑗(1)_𝑡∈[0,1], (4.10) where the functions 𝑃_𝑗, 𝑗∈𝐽, are such that 𝔼[𝑌(𝑡)𝑋_𝑘(1)]=0 for all 𝑘∈𝐽. We get the linear system _𝑗∈𝐽𝔼𝑋_𝑗(1)𝑋_𝑘(1)𝑃_𝑗(𝑡)=𝔼𝑋_𝑛(𝑡)𝑋_𝑘(1),𝑘∈𝐽. (4.11) We plainly have 𝔼𝑋_𝑗(𝑠)𝑋_𝑘(𝑡)=₀^𝑠∧𝑡(𝑠−𝑢)^𝑗−1!()𝑗−1𝑡−𝑢^𝑘−1(𝑘−1)!d𝑢. (4.12) Then, the system (4.11) writes _𝑗∈𝐽1𝑃𝑗+𝑘−1^𝑗(𝑡)!=𝑗−1^𝑡₀(𝑡−𝑢)^𝑛−1()!()𝑛−11−𝑢^𝑘−1d𝑢. (4.13) The matrix of this system (1/(𝑗+𝑘−1))_{𝑗,𝑘∈𝐽} is regular as it can be seen by introducing the related quadratic form which is definite positive. Indeed, for any real numbers 𝑥_𝑗, 𝑗∈𝐽, we have _{𝑗,𝑘∈𝐽}𝑥_𝑗𝑥_𝑘=𝑗+𝑘−1¹_{0𝑗,𝑘∈𝐽}𝑥_𝑗𝑥_𝑘𝑢^{𝑗+𝑘−2}d𝑢=¹_0𝑗∈𝐽𝑥_𝑗𝑢^𝑗−12d𝑢≥0,_{𝑗,𝑘∈𝐽}𝑥_𝑗𝑥_𝑘𝑗+𝑘−1=0iff∀𝑗∈𝐽,𝑥_𝑗=0. (4.14) Thus, the system (4.11) has a unique solution. As a result, the 𝑃_𝑗 are linear combinations of the functions 𝑡↦∫^𝑡₀(𝑡−𝑢)^𝑛−1(1−𝑢)^𝑘−1d𝑢 which are polynomials of degree less than 𝑛+𝑘. Hence, the  𝑃_𝑗 are polynomials of degree at most 2𝑛−1.
We now compute the derivatives of 𝑃_𝑗 at 0 in 1. We have 𝑃_𝑗^(𝑖)(0)=0 for 𝑖∈{0,1,…,𝑛−1} since the functions 𝑡↦∫^𝑡₀(𝑡−𝑢)^𝑛−1(1−𝑢)^𝑘−1d𝑢 plainly enjoy this property. For checking that 𝑃_𝑗^(𝑖)(1)=𝛿_{𝑗,𝑛−𝑖} for 𝑖∈𝐼, we successively compute (i)for 𝑖∈{0,1,…,𝑛−1}, d^𝑖d𝑡^𝑖𝔼𝑋_𝑛(𝑡)𝑋_𝑘(1)=𝔼𝑋_𝑛−𝑖(𝑡)𝑋_𝑘(1); (4.15)(ii)for 𝑖=𝑛−1, d^𝑖d𝑡^𝑖𝔼𝑋_𝑛(𝑡)𝑋_𝑘(1)=𝔼𝐵(𝑡)𝑋_𝑘(1)=^𝑡₀(1−𝑢)^𝑘−1(𝑘−1)!d𝑢; (4.16)(iii)for 𝑖=𝑛, d^𝑖d𝑡^𝑖𝔼𝑋_𝑛(𝑡)𝑋_𝑘(1=())1−𝑡^𝑘−1()!;𝑘−1 (4.17)(iv)for 𝑖∈{𝑛,𝑛+1,…,𝑛+𝑘−1}, d^𝑖d𝑡^𝑖𝔼𝑋_𝑛(𝑡)𝑋_𝑘(1)=(−1)^𝑖+𝑛(1−𝑡)^{𝑘+𝑛−𝑖−1}()!;𝑘+𝑛−𝑖−1 (4.18)(v)for 𝑖=𝑘+𝑛−1, d^𝑖d𝑡^𝑖𝔼𝑋_𝑛(𝑡)𝑋_𝑘(1)=(−1)^𝑘−1; (4.19)(vi)for 𝑖≥𝑘+𝑛, d^𝑖d𝑡^𝑖𝔼𝑋_𝑛(𝑡)𝑋_𝑘(1)=0. (4.20) Consequently, at time 𝑡=1, for 𝑖≥𝑛, d^𝑖d𝑡^𝑖𝔼𝑋_𝑛(𝑡)𝑋_𝑘(1)_𝑡=1=(−1)^𝑘−1𝛿_{𝑖,𝑘+𝑛−1}. (4.21) Now, by differentiating (4.11), we get for 𝑖∈{0,1,…,𝑛−1}, _𝑗∈𝐽𝔼𝑋_𝑗(1)𝑋_𝑘(1)𝑃_𝑗^(𝑖)(1)=𝔼𝑋_𝑛−𝑖(1)𝑋_𝑘(1),𝑘∈𝐽. (4.22) In particular, if 𝑖∈(𝑛−𝐽), this can be rewritten as _𝑗∈𝐽𝔼𝑋_𝑗(1)𝑋_𝑘(1)𝑃_𝑗^(𝑖)(1)=_𝑗∈𝐽𝛿_{𝑗,𝑛−𝑖}𝔼𝑋_𝑗(1)𝑋_𝑘(1),𝑘∈𝐽, (4.23) which by identification yields 𝑃_𝑗^(𝑖)(1)=𝛿_{𝑗,𝑛−𝑖}. Similarly, for 𝑖∈{𝑛,𝑛+1,…,2𝑛−1}, in view of (4.21), we have _𝑗∈𝐽𝔼𝑋_𝑗(1)𝑋_𝑘(1)𝑃_𝑗^(𝑖)(1)=(−1)^𝑘−1𝛿_{𝑖,𝑘+𝑛−1},𝑘∈𝐽. (4.24) In particular, if 𝑖∈{𝑛,𝑛+1,…,2𝑛−1}⧵(𝐽+𝑛−1), we have 𝛿_{𝑖,𝑘+𝑛−1}=0 for 𝑘∈𝐽, and then _𝑗∈𝐽𝔼𝑋_𝑗(1)𝑋_𝑘(1)𝑃_𝑗^(𝑖)(1)=0,𝑘∈𝐽, (4.25) which by identification yields 𝑃_𝑗^(𝑖)(1)=0=𝛿_{𝑗,𝑛−𝑖}. The proof of Theorem 4.1 is finished.

4.2. Covariance Function

Let 𝑐_𝑌 be the covariance function of (𝑌(𝑡))_𝑡∈[0,1]: 𝑐_𝑌(𝑠,𝑡)=𝔼[𝑌(𝑠)𝑌(𝑡)]. In the next theorem, we supply a representation of 𝑐_𝑌 of the form (2.1).

Theorem 4.3. The covariance function of (𝑌(𝑡))_𝑡∈[0,1] admits the following representation: for 𝑠,𝑡∈[0,1], 𝑐_𝑌(𝑠,𝑡)=^𝑛−1_𝑘=0𝜑_k(𝑠∧𝑡)𝜓_𝑘(𝑠∨𝑡) (4.26) with, for any 𝑘∈{0,1,…,𝑛−1}, 𝜓_𝑘(𝑡)=𝜓_𝑘(𝑡)−_𝑗∈𝐽𝜓_𝑘^{(𝑛−𝑗)}(1)𝑃_𝑗(𝑡). (4.27) Moreover, the functions 𝜓_𝑘, 𝑘∈{0,1,…,𝑛−1}, are Hermite interpolation polynomials such that 𝜓_𝑘^(2𝑛)𝜓=0,_𝑘(𝑖)(0)=𝛿_i,𝑘{},𝜓𝑓𝑜𝑟𝑖∈0,1,…,𝑛−1_𝑘(𝑖)(1)=0𝑓𝑜𝑟𝑖∈𝐼. (4.28)

Proof. We decompose 𝑌(𝑡) into the difference 𝑌(𝑡)=𝑋_𝑛(𝑡)−𝑍(𝑡)with𝑍(𝑡)=_𝑗∈𝐽𝑃_𝑗(𝑡)𝑋_𝑗(1). (4.29) We have 𝑐_𝑌(𝑠,𝑡)=𝔼𝑋_𝑛(𝑠)𝑋_𝑛(𝑡)+𝔼[𝑍(𝑠)𝑍(𝑡)]−𝔼𝑋_𝑛(𝑠)𝑍(𝑡)−𝔼𝑍(𝑠)𝑋_𝑛(𝑡)=𝑐_𝑋𝑛(𝑠,𝑡)+_{𝑗,𝑘∈𝐽}𝔼𝑋_𝑗(1)𝑋_𝑘(1)𝑃_𝑗(𝑠)𝑃_𝑘(𝑡)−_𝑗∈𝐽𝔼𝑋_𝑛(𝑡)𝑋_𝑗(1)𝑃_𝑗(𝑠)−_𝑗∈𝐽𝔼𝑋_𝑛(𝑠)𝑋_𝑗(1)𝑃_𝑗(𝑡). (4.30) By definition (4.11) of the 𝑃_𝑗’s, we observe that _{𝑗,𝑘∈𝐽}𝔼𝑋_𝑗(1)𝑋_𝑘(1)𝑃_𝑗(𝑠)𝑃_𝑘(𝑡)−_𝑘∈𝐽𝔼𝑋_𝑛(𝑠)𝑋_𝑘(1)𝑃_𝑘(𝑡)=_{𝑘∈𝐽𝑗∈𝐽}𝔼𝑋_𝑗(1)𝑋_𝑘(1)𝑃_𝑗(𝑠)−𝔼𝑋_𝑛(𝑠)𝑋_𝑘(1)𝑃_𝑘(𝑡)=0. (4.31) Then, we can simplify 𝑐_𝑌(𝑠,𝑡) into 𝑐_𝑌(𝑠,𝑡)=𝑐_𝑋𝑛(𝑠,𝑡)−_𝑗∈𝐽𝔼𝑋_𝑛(𝑡)𝑋_𝑗(1)𝑃_𝑗(𝑠). (4.32) Since the covariance functions 𝑐_𝑌 and 𝑐_𝑋𝑛 are symmetric, we also have 𝑐_𝑌(𝑠,𝑡)=𝑐_𝑋𝑛(𝑠,𝑡)−_𝑗∈𝐽𝔼𝑋_𝑛(𝑠)𝑋_𝑗(1)𝑃_𝑗(𝑡). (4.33) Let us introduce the symmetric polynomial 𝑄(𝑠,𝑡)=_𝑗∈𝐽𝔼𝑋_𝑛(𝑠)𝑋_𝑗(1)𝑃_𝑗(𝑡)=_𝑗∈𝐽𝔼𝑋_𝑛(𝑡)𝑋_𝑗(1)𝑃_𝑗(𝑠). (4.34) It can be expressed by means of the functions 𝜑_𝑘,𝜓_𝑘’s as follows: 𝑄(𝑠,𝑡)=_𝑗∈𝐽^𝑛−1_𝑘=0𝜑_𝑘(𝑠∧𝑡)𝜓_𝑘^{(𝑛−𝑗)}(1)𝑃_𝑗(𝑠∨𝑡). (4.35) We can rewrite 𝑐_𝑌(𝑠,𝑡) as 𝑐_𝑌(𝑠,𝑡)=𝑐_𝑋𝑛(𝑠,𝑡)−𝑄(𝑠,𝑡)=^𝑛−1_𝑘=0𝜑_𝑘(𝑠∧𝑡)𝜓_𝑘(𝑠∨𝑡)−_𝑗∈𝐽𝜓_𝑘^{(𝑛−𝑗)}(1)𝑃_𝑗(𝑠∨𝑡), (4.36) and then, for 𝑠,𝑡∈[0,1], 𝑐_𝑌(𝑠,𝑡)=^𝑛−1_𝑘=0𝜑_𝑘(𝑠∧𝑡)𝜓_𝑘(𝑠∨𝑡)with𝜓_𝑘(𝑡)=𝜓_𝑘(𝑡)−_𝑗∈𝐽𝜓_𝑘^{(𝑛−𝑗)}(1)𝑃_𝑗(𝑡). (4.37) We immediately see that 𝜓_𝑘 is a polynomial of degree less than 2𝑛 such that 𝜓_𝑘^(𝑖)(0)=𝛿_𝑖,𝑘 for 𝑖∈{0,1,…,𝑛−1} and, since 𝑃_𝑗^(𝑖)(1)=𝛿_{𝑗,𝑛−𝑖}, 𝜓_𝑘^(𝑖)(1)=𝜓_𝑘^(𝑖)(1)−_𝑗∈𝐽𝜓_𝑘^{(𝑛−𝑗)}(1)𝑃_𝑗^(𝑖)(1)=1−𝟙_{{𝑖∈𝑛−𝐽}}𝜓_𝑘^(𝑖)(1). (4.38) We deduce that 𝜓_𝑘^(𝑖)(1)(),𝜓=0if𝑖∈𝑛−𝐽_𝑘(𝑖)(1)=𝜓_𝑘^(𝑖)(1)=0if𝑖∈{𝑛,𝑛+1,…,2𝑛−1}⧵(𝐽+𝑛−1). (4.39) Then 𝜓_𝑘^(𝑖)(1)=0 for any 𝑖∈𝐼. This ends up the proof of Theorem 4.3.

4.3. Boundary Value Problem

In this section, we write out the natural boundary value problem which is associated with the process (𝑌(𝑡))_𝑡∈[0,1]. The following statement is the main connection between the different boundary value conditions associated with the operator d^2𝑛/d𝑡^2𝑛 and the different bridges of the process (𝑋_𝑛)_𝑡∈[0,1] introduced in this work.

Theorem 4.4. Let 𝑢 be a fixed continuous function on [0,1]. The function 𝑣 defined on [0,1] by 𝑣(𝑡)=¹₀𝑐_𝑌(𝑠,𝑡)𝑢(𝑠)d𝑠 (4.40) is the solution of the boundary value problem 𝑣^(2𝑛)=(−1)^𝑛[],𝑣𝑢𝑜𝑛0,1_(𝑖)(0){},𝑣=0𝑓𝑜𝑟𝑖∈0,1,…,𝑛−1_(𝑖)(1)=0𝑓𝑜𝑟𝑖∈𝐼. (4.41)

Proof. Step 1. Recall that 𝑐_𝑌(𝑠,𝑡)=𝑐_𝑋𝑛(𝑠,𝑡)−𝑄(𝑠,𝑡). We decompose the function 𝑣 into the difference 𝑤−𝑧, where, for 𝑡∈[0,1], 𝑤(𝑡)=¹₀𝑐_𝑋𝑛(𝑠,𝑡)𝑢(𝑠)d𝑠,𝑧(𝑡)=¹₀𝑄(𝑠,𝑡)𝑢(𝑠)d𝑠. (4.42) We know from Theorem 3.1 that 𝑤^(2𝑛)=(−1)^𝑛𝑢, 𝑤^(𝑖)(0)=(𝜕^𝑖𝑄/𝜕𝑡^𝑖)(𝑠,0)=0 for 𝑖∈{0,1,…,𝑛−1} and 𝑤^(𝑖)(1)=0 for 𝑖∈{𝑛,𝑛+1,…,2𝑛−1}. Moreover, with the function 𝑡↦𝑄(𝑠,𝑡) being a polynomial of degree less than 2𝑛, the function 𝑧 is also a polynomial of degree less than 2𝑛. Then 𝑧^(2𝑛)=0, 𝑧^(𝑖)(0)=0 for 𝑖∈{0,1,…,𝑛−1}, and 𝑣^(2𝑛)=𝑤^(2𝑛)−𝑧^(2𝑛)=(−1)^𝑛𝑢,𝑣^(𝑖)(0)=𝑤^(𝑖)(0)−𝑧^(𝑖)(0)=0for𝑖∈{0,1,…,𝑛−1}. (4.43) On the other hand, we learn from (2.13) that, for 𝑖∈{0,1,…,2𝑛−1}, 𝑤^(𝑖)(1)=¹₀𝜕^𝑖𝑐_𝑋𝑛𝜕𝑡^𝑖(𝑠,1)𝑢(𝑠)d𝑠, (4.44) and then, for 𝑖∈{0,1,…,𝑛−1}, 𝑤^(𝑖)(1)=¹₀𝔼𝑋_𝑛(𝑠)𝑋_𝑛−𝑖(1)𝑢(𝑠)d𝑠. (4.45) We also have, for any 𝑖∈{0,1,…,2𝑛−1}, 𝜕^𝑖𝑄𝜕𝑡^𝑖(𝑠,1)=_𝑗∈𝐽𝔼𝑋_𝑛(𝑠)𝑋_𝑗(1)𝑃_𝑗^(𝑖)(1)=𝟙_{{𝑖∈(𝑛−𝐽)}}𝔼𝑋_𝑛(𝑠)𝑋_𝑛−𝑖(1). (4.46) As a result, we see that 𝑧^(𝑖)(1)=¹₀𝜕^𝑖𝑄𝜕𝑡^𝑖(𝑠,1)𝑢(𝑠)d𝑠=𝟙_{{𝑖∈(𝑛−𝐽)}}𝑤^(𝑖)(1). (4.47) This implies that, for 𝑖∈(𝑛−𝐽), 𝑧^(𝑖)(1)=𝑤^(𝑖)(1) and for 𝑖∈{𝑛,𝑛+1,…,2𝑛−1}⧵(𝐽+𝑛−1), 𝑧^(𝑖)(1)=0=𝑤^(𝑖)(1). Then 𝑧^(𝑖)(1)=𝑤^(𝑖)(1) for any 𝑖∈𝐼, that is, 𝑣^(𝑖)(1)=0. The function 𝑣 is a solution of (4.41).Step 2. We now check the uniqueness of the solution of (4.41). Let 𝑣₁ and 𝑣₂ be two solutions of 𝒟𝑣=𝑢. Then the function 𝑤=𝑣₁−𝑣₂ satisfies 𝒟𝑤=0, 𝑤^(𝑖)(0)=0 for 𝑖∈{0,1,…,𝑛−1} and 𝑤^(𝑖)(1)=0 for 𝑖∈𝐼. We compute the following “energy” integral: ¹₀𝑤^(𝑛)(𝑡)²d𝑡=(−1)^{𝑛+1𝑛−1}_𝑖=0(−1)^𝑖𝑤^(𝑖)(𝑡)𝑤^{(2𝑛−1−𝑖)}(𝑡)¹₀+(−1)^𝑛1₀𝑤(𝑡)𝑤^(2𝑛)(𝑡)=()d𝑡−1^{𝑛+1𝑛−1}_𝑖=0(−1)^𝑖𝑤^(𝑖)(1)𝑤^{(2𝑛−1−𝑖)}(1). (4.48) We have constructed the set 𝐼 in order to have 𝑤^(𝑖)(1)𝑤^{(2𝑛−1−𝑖)}(1)=0 for any 𝑖∈{0,1,…,𝑛−1}: when we pick an index 𝑖∈{0,1,…,𝑛−1}, either 𝑖∈𝐼 or 2𝑛−1−𝑖∈𝐼. Indeed, (i)if 𝑖∈𝐼∩{0,1,…,𝑛−1}, 𝑤^(𝑖)=0;(ii)if 𝑖∈{0,1,…,𝑛−1}⧵𝐼, by observing that {0,1,…,𝑛−1}⧵𝐼={0,1,…,𝑛−1}⧵(𝑛−𝐽), we have 2𝑛−1−𝑖∈{𝑛,𝑛+1,…,2𝑛−1}⧵(𝐽+𝑛−1). Since {𝑛,𝑛+1,…,2𝑛−1}⧵(𝐽+𝑛−1)⊂𝐼, we see that 2𝑛−1−𝑖∈𝐼 and then 𝑤^{(2𝑛−1−𝑖)}=0. Next, ∫¹₀𝑤^(𝑛)(𝑡)²d𝑡=0 which entails 𝑤^(𝑛)=0, that is, 𝑤 is a polynomial of degree less than 𝑛. Moreover, with the boundary value conditions at 0, we obtain 𝑤=0 or 𝑣₁=𝑣₂.
The proof of Theorem 4.4 is finished.

Remark 4.5. We have seen in the previous proof that uniqueness is assured as soon as the boundary value conditions at 1 satisfy 𝑤^(𝑖)(1)𝑤^{(2𝑛−1−𝑖)}(1)=0 for any 𝑖∈{0,1,…,𝑛−1}. These conditions are fulfilled when the set 𝐼={𝑖₁,𝑖₂,…,𝑖_𝑛}⊂{0,1,…,2𝑛−1} is such that 𝑖₁∈{0,2𝑛−1}, 𝑖₂∈{1,2𝑛−2}, 𝑖₃∈{2,2𝑛−3},…,𝑖_𝑛∈{𝑛−1,𝑛}. This is equivalent to say that 𝐼 and (2𝑛−1−𝐼) make up a partition of {0,1,…,2𝑛−1}, or 2𝑛−1−𝐼={0,1,…,2𝑛−1}⧵𝐼. (4.49) In this manner, we get 2^𝑛 different boundary value problems which correspond to the 2^𝑛 different bridges we have constructed. We will see in Section 6 that the above identity concerning the differentiating set 𝐼 characterizes the possibility for the Green function of the boundary value problem (4.41) to be the covariance of a Gaussian process.

4.4. Prediction

Now, we tackle the problem of the prediction for the process (𝑌(𝑡))_𝑡∈[0,1].

Theorem 4.6. Fix 𝑡₀∈[0,1]. The shifted process (𝑌(𝑡+𝑡₀))_{𝑡∈[0,1−𝑡0]} admits the following representation: 𝑌𝑡+𝑡_{0𝑡∈[0,1−𝑡0]}=𝑌_𝑡0(𝑡)+^𝑛−1_𝑖=0𝑄_𝑖,𝑡0(𝑡)𝑌^(𝑖)𝑡_{0𝑡∈[0,1−𝑡0]}, (4.50) where 𝑌_𝑡0(𝑡)=𝑋_𝑛(𝑡)−_𝑗∈𝐽𝑃_𝑗,𝑡0(𝑡)𝑋_𝑗1−𝑡₀. (4.51) The process (𝑋_𝑛(𝑡))_{𝑡∈[0,1−𝑡0]} is a copy of (𝑋_𝑛(𝑡))_{𝑡∈[0,1−𝑡0]} which is independent of (𝑋_𝑛(𝑡))_{𝑡∈[0,𝑡0]}.
The functions 𝑃_𝑗,𝑡0, 𝑗∈𝐽, and 𝑄_𝑖,𝑡0, 𝑖∈{0,1,…,𝑛−1}, are Hermite interpolation polynomials on [0,1−t₀] characterized by 𝑃^(2𝑛)_𝑗,𝑡0𝑃=0,_{(𝜄)𝑗,𝑡0}(0){},𝑃=0𝑓𝑜𝑟𝜄∈0,1,…,𝑛−1_{(𝜄)𝑗,𝑡0}1−𝑡₀=𝛿_{𝜄,𝑛−𝑗}𝑄𝑓𝑜𝑟𝜄∈𝐼,_{(2𝑛)𝑖,𝑡0}𝑄=0,_{(𝜄)𝑖,𝑡0}(0)=𝛿_𝜄,𝑖{},𝑄𝑓𝑜𝑟𝜄∈0,1,…,𝑛−1_{(𝜄)𝑖,𝑡0}1−𝑡₀=0𝑓𝑜𝑟𝜄∈𝐼. (4.52) Actually, these functions can be expressed by means of the functions 𝑃_𝑗, 𝑗∈𝐽, and 𝜓_𝑖, 𝑖∈{0,1,…,𝑛−1}, as follows: 𝑃_𝑗,𝑡0(𝑡)=1−𝑡₀^𝑛−𝑗𝑃_𝑗𝑡1−𝑡₀,𝑄_𝑖,𝑡0=1−𝑡₀^𝑖𝜓_𝑖𝑡1−𝑡₀. (4.53) In other words, the process (𝑌_𝑡0(𝑡))_{𝑡∈[0,1−𝑡0]} is a bridge of length (1−𝑡₀) which is independent of (𝑌(𝑡))_{𝑡∈[0,𝑡0]}, that is, 𝑌_𝑡0(𝑡)=_{𝑡∈[0,1−𝑡0]}^𝑑𝑋_𝑛(𝑡)∣𝑋_𝑗1−𝑡₀=0,𝑗∈𝐽_{𝑡∈[0,1−𝑡0]}. (4.54)

Proof. Step 1. Fix 𝑡₀∈[0,1]. We have the well-known decomposition, based on the classical prediction property of Brownian motion stipulating that (𝐵(𝑡+𝑡₀))_{𝑡∈[0,1−𝑡0]}=(𝐵_𝑡0(𝑡)+𝐵(𝑡₀))_{𝑡∈[0,1−𝑡0]}, where (𝐵_𝑡0(𝑡))_{𝑡∈[0,1−𝑡0]} is a Brownian motion independent of (𝐵(𝑡))_{𝑡∈[0,𝑡0]}, 𝑋_𝑛𝑡+𝑡_{0𝑡∈[0,1−𝑡0]}=𝑋_𝑛(𝑡)+^𝑛−1_𝑘=0𝑡^𝑘𝑋𝑘!^𝑛−𝑘𝑡_{0𝑡∈[0,1−𝑡0]} (4.55) with 𝑋_𝑛(𝑡)=∫¹₀((𝑡−𝑠)^𝑛−1/(𝑛−1)!)d𝐵(𝑠). Differentiating this equality (𝑛−𝑗) times, 𝑗∈{1,2,…,𝑛}, we obtain 𝑋_𝑗𝑡+𝑡₀=𝑋_𝑗(𝑡)+^𝑛−1_{𝑘=𝑛−𝑗}𝑡^{𝑗+𝑘−𝑛}!𝑋𝑗+𝑘−𝑛^𝑛−𝑘𝑡₀. (4.56) Therefore, 𝑌𝑡+𝑡₀=𝑋_𝑛𝑡+𝑡₀−_𝑗∈𝐽𝑃_𝑗𝑡+𝑡₀𝑋_𝑗(1)=𝑋_𝑛(𝑡)+^𝑛−1_𝑘=0𝑡^𝑘𝑋𝑘!^𝑛−𝑘𝑡₀−_𝑗∈𝐽𝑃_𝑗𝑡+𝑡₀𝑋_𝑗1−𝑡₀+^𝑛−1_{𝑘=𝑛−𝑗}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑋𝑗+𝑘−𝑛^𝑛−𝑘𝑡₀=𝑋_𝑛(𝑡)−_𝑗∈𝐽𝑃_𝑗𝑡+𝑡₀𝑋_𝑗1−𝑡₀+^𝑛−1_𝑘=0𝑡^𝑘𝑘!−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗𝑡+𝑡₀𝑋_𝑛−𝑘𝑡₀. (4.57) We are going to express the 𝑋_𝑛−𝑘(𝑡₀), 𝑘∈{1,…,𝑛}, by means of the 𝑋_𝑗(1−𝑡₀), 𝑗∈𝐽. We have, by differentiating (4.57) 𝑖 times, for 𝑖∈{0,1,…,𝑛−1}, 𝑌^(𝑖)𝑡+𝑡₀=𝑋_𝑛−𝑖(𝑡)−_𝑗∈𝐽𝑃_𝑗^(𝑖)𝑡+𝑡₀𝑋_𝑗1−𝑡₀+^𝑛−1_𝑘=0𝑡^𝑘−𝑖()!𝟙𝑘−𝑖^{{𝑘≥𝑖}}−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗(𝑖)𝑡+𝑡₀𝑋_𝑛−𝑘𝑡₀. (4.58) For 𝑡=0, this yields 𝑌^(𝑖)𝑡₀=−_𝑗∈𝐽𝑃_𝑗^(𝑖)𝑡₀𝑋_𝑗1−𝑡₀+^𝑛−1_𝑘=0𝛿_𝑖,𝑘−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗(𝑖)𝑡₀𝑋_𝑛−𝑘𝑡₀. (4.59) Set for 𝑖,𝑘∈{0,1,…,𝑛−1}𝑎_𝑖𝑘=𝛿_𝑖,𝑘−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗(𝑖)𝑡₀, (4.60) and let us introduce the matrix 𝐴=𝑎_{𝑖𝑘0≤𝑖,𝑘≤𝑛−1} (4.61) together with its inverse matrix 𝐵=𝐴⁻¹=𝑏_{𝑖𝑘0≤𝑖,𝑘≤𝑛−1}. (4.62) Equalities (4.59) for 𝑖∈{0,1,…,𝑛−1} read as a linear system of (𝑛−1) equations and (𝑛−1) unknowns: ^𝑛−1_𝑘=0𝑎_𝑖𝑘𝑋_𝑛−𝑘𝑡₀=𝑌^(𝑖)𝑡₀+_𝑗∈𝐽𝑃_𝑗^(𝑖)𝑡₀𝑋_𝑗1−𝑡₀,𝑖∈{0,1,…,𝑛−1}, (4.63) which can be rewritten into a matrix form as 𝐴𝑋_𝑛𝑡₀𝑋_𝑛−1𝑡₀⋮𝑋₂𝑡₀𝑋₁𝑡₀=𝑌𝑡₀𝑌^′𝑡₀⋮𝑌^(𝑛−2)𝑡₀𝑌^(𝑛−1)𝑡₀+_𝑗∈𝐽𝑋_𝑗1−𝑡₀𝑃_𝑗𝑡₀𝑃^′_𝑗𝑡₀⋮𝑃_𝑗^(𝑛−2)𝑡₀𝑃_𝑗^(𝑛−1)𝑡₀. (4.64) The solution is given by 𝑋_𝑛𝑡₀𝑋_𝑛−1𝑡₀⋮𝑋₂𝑡₀𝑋₁𝑡₀𝑌𝑡=𝐵⁰𝑌^′𝑡₀⋮𝑌^(𝑛−2)𝑡₀𝑌^(𝑛−1)𝑡₀+_𝑗∈𝐽𝑋_𝑗1−𝑡₀𝐵𝑃_𝑗𝑡₀𝑃^′_𝑗𝑡₀⋮𝑃_𝑗^(𝑛−2)𝑡₀𝑃_𝑗^(𝑛−1)𝑡₀, (4.65) and we see that 𝑋_𝑛−𝑘(𝑡₀), 𝑘∈{0,1,…,𝑛−1}, is of the form 𝑋_𝑛−𝑘𝑡₀=^𝑛−1_𝑖=0𝑏_𝑘𝑖𝑌^(𝑖)𝑡₀+_𝑗∈𝐽^𝑛−1_𝑖=0𝑏_𝑘𝑖𝑃_𝑗^(𝑖)𝑡₀𝑋_𝑗1−𝑡₀. (4.66) Therefore, by plugging (4.66) into (4.57), we obtain 𝑌𝑡+𝑡₀=𝑋_𝑛(𝑡)−_𝑗∈𝐽𝑃_𝑗𝑡+𝑡₀𝑋_𝑗1−𝑡₀+^𝑛−1_𝑘=0𝑡^𝑘𝑘!−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗𝑡+𝑡₀×^𝑛−1_𝑖=0𝑏_𝑘𝑖𝑌^(𝑖)𝑡₀+_𝑗∈𝐽^𝑛−1_𝑖=0𝑏_𝑘𝑖𝑃_𝑗^(𝑖)𝑡₀𝑋_𝑗1−𝑡₀=𝑋_𝑛(𝑡)−_𝑗∈𝐽𝑃_𝑗𝑡+𝑡₀−_{0≤𝑖,𝑘≤𝑛−1}𝑏_𝑘𝑖𝑃_𝑗^(𝑖)𝑡₀𝑡^𝑘𝑘!−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗𝑡+𝑡₀𝑋_𝑗1−𝑡₀+_{0≤𝑖,𝑘≤𝑛−1}𝑏_𝑘𝑖𝑡^𝑘−𝑘!_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗𝑡+𝑡₀𝑌^(𝑖)𝑡₀. (4.67) Finally, 𝑌(𝑡+𝑡₀) can be written as 𝑌𝑡+𝑡₀=𝑌_𝑡0(𝑡)+^𝑛−1_𝑖=0𝑄_𝑖,𝑡0(𝑡)𝑌^(𝑖)𝑡₀, (4.68) where 𝑄_𝑖,𝑡0(𝑡)=^𝑛−1_𝑘=0𝑏_𝑘𝑖𝑡^𝑘𝑘!−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗𝑡+𝑡₀,𝑌_𝑡0(𝑡)=𝑋_𝑛(𝑡)−_𝑗∈𝐽𝑃_𝑗,𝑡0(𝑡)𝑋_𝑗1−𝑡₀ (4.69) with 𝑃_𝑗,𝑡0(𝑡)=𝑃_𝑗𝑡+𝑡₀−_{0≤𝑖,𝑘≤𝑛−1}𝑏_𝑘𝑖𝑃_𝑗^(𝑖)𝑡₀𝑡^𝑘𝑘!−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗𝑡+𝑡₀=𝑃_𝑗𝑡+𝑡₀−^𝑛−1_𝑖=0𝑃_𝑗^(𝑖)𝑡₀𝑄_𝑖,𝑡0(𝑡). (4.70)Step 2. We easily see that the functions 𝑃_𝑗,𝑡0 and 𝑄_𝑖,𝑡0 are polynomials of degree less than 2𝑛. Let us compute now their derivatives at 0 and 𝑡₀. First, concerning 𝑄_𝑖,𝑡0 we have 𝑄^(𝜄)_𝑖,𝑡0(𝑡)=^𝑛−1_𝑘=0𝑏_𝑘𝑖𝑡^𝑘−𝜄()!𝟙𝑘−𝜄^{{𝑘≥𝜄}}−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗(𝜄)𝑡+𝑡₀. (4.71) Choosing 𝑡=0 and recalling the definition of 𝑎_𝜄𝑘 and the fact that the matrices (𝑎_𝑖𝑘)_{0≤𝑖,𝑘≤𝑛} and (𝑏_𝑖𝑘)_{0≤𝑖,𝑘≤𝑛} are inverse, this gives, for 𝜄∈{0,1,…,𝑛−1}, 𝑄^(𝜄)_𝑖,𝑡0(0)=^𝑛−1_𝑘=0𝑏_𝑘𝑖𝛿_𝜄,𝑘−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗(𝜄)𝑡₀=^𝑛−1_𝑘=0𝑎_𝜄𝑘𝑏_𝑘𝑖=𝛿_𝜄,𝑖. (4.72) Choosing 𝑡=1−𝑡₀, we have for 𝜄∈𝐼𝑄^(𝜄)_𝑖,𝑡01−𝑡₀=^𝑛−1_𝑘=0𝑏_𝑘𝑖1−𝑡₀^𝑘−𝜄()!𝟙𝑘−𝜄^{{𝑘≥𝜄}}−_{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗(𝜄)(1). (4.73) By Theorem 4.1, we know that 𝑃_𝑗^(𝜄)(1)=𝛿_{𝑗,𝑛−𝜄} for 𝜄∈𝐼; then _{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗(𝜄)(1)=1−𝑡₀^𝑘−𝜄()!𝟙𝑘−𝜄^{{𝑘≥𝜄}}𝟙_{{𝜄∈(𝑛−𝐽)}}. (4.74) Observing that, if 𝜄≤𝑘≤𝑛−1, the conditions 𝜄∈(𝑛−𝐽) and 𝜄∈𝐼 are equivalent, we simply have _{𝑗∈𝐽∶𝑗≥𝑛−𝑘}1−𝑡₀^{𝑗+𝑘−𝑛}!𝑃𝑗+𝑘−𝑛^𝑗(𝜄)(1)=1−𝑡₀^𝑘−𝜄()!𝟙𝑘−𝜄^{{𝑘≥𝜄}} (4.75) which immediately entails, by (4.73), 𝑄^(𝜄)_𝑖,𝑡01−𝑡₀=0for𝜄∈𝐼. (4.76) Next, concerning 𝑃^(𝜄)_𝑗,𝑡0, we have 𝑃^(𝜄)_𝑗,𝑡0(𝑡)=𝑃_𝑗^(𝜄)𝑡+𝑡₀−^𝑛−1_𝑖=0𝑃_𝑗^(𝑖)𝑡₀𝑄^(𝜄)_𝑖,𝑡0(𝑡). (4.77) Choosing 𝑡=0, this gives, for 𝜄∈{0,1,…,𝑛−1}, since 𝑄^(𝜄)_𝑖,𝑡0(0)=𝛿_𝜄,𝑖, 𝑃^(𝜄)_𝑗,𝑡0(0)=𝑃_𝑗^(𝜄)𝑡₀−^𝑛−1_𝑖=0𝑃_𝑗^(𝑖)𝑡₀𝑄^(𝜄)_𝑖,𝑡0(0)=0. (4.78) Choosing 𝑡=1−𝑡₀, we have, for 𝜄∈𝐼, since 𝑃_𝑗^(𝜄)(1)=𝛿_{𝜄,𝑛−𝑗} and 𝑄^(𝜄)_𝑖,𝑡0(1−𝑡₀)=0, 𝑃^(𝜄)_𝑗,𝑡01−𝑡₀=𝑃_𝑗^(𝜄)(1)−^𝑛−1_𝑖=0𝑃_𝑗^(𝑖)𝑡₀𝑄^(𝜄)_𝑖,𝑡01−𝑡₀=𝛿_{𝜄,𝑛−𝑗}. (4.79) The polynomials 𝑃_𝑗,𝑡0 (resp., 𝑄_𝑖,𝑡0) enjoy the same properties than the 𝑃_𝑗’s (resp., the 𝜓_𝑖’s), regarding the successive derivatives, they can be deduced from these latter by a rescaling according as 𝑃_𝑗,𝑡0(𝑡)=1−𝑡₀^𝑛−𝑗𝑃_𝑗𝑡1−𝑡₀resp.𝑄_𝑖,𝑡0=1−𝑡₀^𝑖𝜓_𝑖𝑡1−𝑡₀. (4.80) It is then easy to extract the identity in distribution below, by using the property of Gaussian conditioning: 𝑋_𝑛(𝑡)−_𝑗∈𝐽𝑃_𝑗,𝑡0(𝑡)𝑋_𝑗1−𝑡=_{0𝑡∈[0,1−𝑡0]}^𝑑𝑋_𝑛(𝑡)∣𝑋_𝑗1−𝑡₀=0,𝑗∈𝐽_{𝑡∈[0,1−𝑡0]}. (4.81) Theorem 4.6 is established.

5. Example: Bridges of Integrated Brownian Motion (𝑛=2)

Here, we have a look on the particular case where 𝑛=2 for which the corresponding process (𝑋_𝑛(𝑡))_𝑡∈[0,1] is nothing but integrated Brownian motion (the so-called Langevin process): 𝑋₂(𝑡)=^𝑡₀𝐵(𝑠)d𝑠. (5.1) The underlying Markov process is the so-called Kolmogorov diffusion (𝑋₂(𝑡),𝑋₁(𝑡))_𝑡∈[0,1]. All the associated conditioned processes that will be constructed are related to the equation 𝑣⁽⁴⁾(𝑡)=𝑢(𝑡) with boundary value conditions at time 0: 𝑣(0)=𝑣^′(0)=0. There are four such processes: (i)(𝑋₂(𝑡))_𝑡∈[0,1] (integrated Brownian motion); (ii)(𝑋₂(𝑡)∣𝑋₁(1)=0)_𝑡∈[0,1] (integrated Brownian bridge); (iii)(𝑋₂(𝑡)∣𝑋₂(1)=0)_𝑡∈[0,1] (bridge of integrated Brownian motion); (iv)(𝑋₂(𝑡)∣𝑋₁(1)=𝑋₂(1)=0)_𝑡∈[0,1] (another bridge of integrated Brownian motion).

On the other hand, when adding two boundary value conditions at time 1 to the foregoing equation, we find six boundary value problems: 𝑣(1)=𝑣^′(1)=0, 𝑣(1)=𝑣^′′(1)=0, 𝑣(1)=𝑣^′′′(1)=0, 𝑣^′(1)=𝑣^′′(1)=0, 𝑣^′(1)=𝑣^′′′(1)=0, 𝑣^′′(1)=𝑣^′′′(1)=0. Actually, only four of them can be related to some Gaussian processes—the previously listed processes—in the sense of our work whereas two others cannot be.

For each process, we provide the covariance function, the representation by means of integrated Brownian motion subject to a random polynomial drift, the related boundary value conditions at 1, and the decomposition related to the prediction problem. Since the computations are straightforward, we will omit them and we only report here the results.

For an account on integrated Brownian motion in relation with the present work, we refer the reader to, for example, [16] and references therein.

5.1. Integrated Brownian Motion

The process corresponding to the set 𝐼=∅ is nothing but integrated Brownian motion: 𝑋₂(𝑡)_𝑡∈[0,1]=^𝑡₀𝐵(𝑠)d𝑠_𝑡∈[0,1]. (5.2) The covariance function is explicitly given by 𝑐()=1𝑠,𝑡6[𝑠∧𝑡]²[3(𝑠∨𝑡)−𝑠∧𝑡]. (5.3) This process is related to the boundary value conditions at 1 (𝐽={2,3}): 𝑣^′′(1)=𝑣^′′′(1)=0. The prediction property can be stated as follows: 𝑋₂𝑡+𝑡_{0𝑡∈[0,1−𝑡0]}=𝑋₂(𝑡)+𝑋₂𝑡₀+𝑡𝑋₁𝑡_{0𝑡∈[0,1−𝑡0]}. (5.4)

5.2. Integrated Brownian Bridge

The process corresponding to the set 𝐼={1} is integrated Brownian bridge: (𝑌(𝑡))_𝑡∈[0,1]=^𝑡₀𝐵(𝑠)d𝑠∣𝐵(1)=0_𝑡∈[0,1]=^𝑡₀𝛽(𝑠)d𝑠_𝑡∈[0,1]=𝑋₂(𝑡)∣𝑋₁(1)=0_𝑡∈[0,1]. (5.5) This process can be represented as (𝑌(𝑡))_𝑡∈[0,1]=𝑋₂(𝑡)−12𝑡²𝑋₁(1)_𝑡∈[0,1]. (5.6) The covariance function is explicitly given by 𝑐()=1𝑠,𝑡6[𝑠∧𝑡]²[3()]−1𝑠∨𝑡−𝑠∧𝑡4𝑠²𝑡². (5.7) The process (𝑌(𝑡))_𝑡∈[0,1] is related to the boundary value conditions at 1 (𝐽={1,3}): 𝑣^′(1)=𝑣^′′′(1)=0. The prediction property says that 𝑌𝑡+𝑡_{0𝑡∈[0,1−𝑡0]}=𝑌_𝑡0(𝑡)+𝑌𝑡₀+𝑡𝑡−²21−𝑡₀𝑌^′𝑡_{0𝑡∈[0,1−𝑡0]}. (5.8)

5.3. Bridge of Integrated Brownian Motion

The process corresponding to the set 𝐼={2} is the bridge of integrated Brownian motion: (𝑌(𝑡))_𝑡∈[0,1]=^𝑡₀𝐵(𝑠)d𝑠∣¹₀𝐵(𝑠)d𝑠=0_𝑡∈[0,1]=𝑋₂(𝑡)∣𝑋₂(1)=0_𝑡∈[0,1]. (5.9) The bridge is understood as the process is pinned at its extremities: 𝑌(0)=𝑌(1)=0. This process can be represented as (𝑌(𝑡))_𝑡∈[0,1]=𝑋₂(𝑡)−12𝑡²(3−𝑡)𝑋₂(1)_𝑡∈[0,1]. (5.10) The covariance function is explicitly given by 𝑐()=1𝑠,𝑡6[𝑠∧𝑡]²[3()]−1𝑠∨𝑡−𝑠∧𝑡𝑠12²𝑡²(3−𝑠)(3−𝑡). (5.11) The process (𝑌(𝑡))_𝑡∈[0,1] is related to the boundary value conditions at 1 (𝐽={0,2}): 𝑣(1)=𝑣^′′(1)=0. The prediction property says that 𝑌𝑡+𝑡_{0𝑡∈[0,1−𝑡0]}=𝑌_𝑡0(𝑡)+𝑡³−31−𝑡₀𝑡²+21−𝑡₀³21−𝑡₀³𝑌𝑡₀+𝑡³−31−𝑡₀𝑡²+21−𝑡₀²𝑡21−𝑡₀²𝑌^′𝑡_{0𝑡∈[0,1−𝑡0]}. (5.12)

5.4. Other Bridge of Integrated Brownian Motion

The process corresponding to the set 𝐼={1,2} is another bridge of integrated Brownian motion (actually of the two-dimensional Kolmogorov diffusion): (𝑌(𝑡))_𝑡∈[0,1]=^𝑡₀𝐵(𝑠)d𝑠∣¹₀𝐵(𝑠)d𝑠=𝐵(1)=0_𝑡∈[0,1]=𝑋₂(𝑡)∣𝑋₂(1)=𝑋₁(1)=0_𝑡∈[0,1]. (5.13) The bridge here is understood as the process is pinned at its extremities together with its derivatives: 𝑌(0)=𝑌^′(0)=𝑌(1)=𝑌^′(1)=0. This process can be represented as (𝑌(𝑡))_𝑡∈[0,1]=𝑋₂(𝑡)−𝑡²(𝑡−1)𝑋₁(1)−𝑡²(3−2𝑡)𝑋₂(1)_𝑡∈[0,1]. (5.14) The covariance function is explicitly given by 𝑐()=1𝑠,𝑡6[𝑠∧𝑡]²[1−𝑠∨𝑡]²[3(𝑠∨𝑡)−𝑠∧𝑡−2𝑠𝑡]. (5.15) The process (𝑌(𝑡))_𝑡∈[0,1] is related to the boundary value conditions at 1 (𝐽={0,1}): 𝑣(1)=𝑣^′(1)=0. The prediction property says that 𝑌𝑡+𝑡_{0𝑡∈[0,1−𝑡0]}=𝑌_𝑡0(𝑡)+𝑡²𝑡+𝑡₀−11−𝑡₀³𝑌𝑡₀+𝑡²3−3𝑡₀−2𝑡1−𝑡₀²𝑌^′𝑡_{0𝑡∈[0,1−𝑡0]}. (5.16)

5.5. Two Counterexamples

(i) The solution of the problem associated with the boundary value conditions 𝑣(1)=𝑣^′′′(1)=0 (which corresponds to the set 𝐼₁={0,3}) is given by 𝑣(𝑡)=¹₀𝐺₁(𝑠,𝑡)𝑢(𝑠)d𝑠,𝑡∈[0,1], (5.17) where 𝐺₁()=1𝑠,𝑡6[𝑠∧𝑡]²[3()]+1𝑠∨𝑡−𝑠∧𝑡6𝑠²𝑡²(𝑠−3). (5.18)

(ii) The solution of the problem associated with the boundary value conditions 𝑣^′(1)=𝑣^′′(1)=0 (which corresponds to the set 𝐼₂={1,2}) is given by 𝑣(𝑡)=¹₀𝐺₂(𝑠,𝑡)𝑢(𝑠)d𝑠,𝑡∈[0,1], (5.19) where 𝐺₂()=1𝑠,𝑡6[𝑠∧𝑡]²[3()]+1𝑠∨𝑡−𝑠∧𝑡6𝑠²𝑡²(𝑡−3). (5.20) We can observe the relationships 𝐺₁(𝑠,𝑡)=𝐺₂(𝑡,𝑠) and 𝐼₂={0,1,2,3}⧵(3−𝐼₁). The Green functions 𝐺₁ and 𝐺₂ are not symmetric, so they cannot be viewed as the covariance functions of any Gaussian process. In the next section, we give an explanation of these observations.

6. General Boundary Value Conditions

In this last part, we address the problem of relating the general boundary value problem𝑣^(2𝑛)=(−1)^𝑛[],𝑣(0)𝑢on0,1=𝑣^′(0)=⋯=𝑣^(𝑛−1)(0)𝑣=0,_(𝑖1)(1)=𝑣^(𝑖2)(1)=⋯=𝑣^(𝑖𝑛)(1)=0, (6.1) for any indices 𝑖₁,𝑖₂,…,𝑖_𝑛 such that 0≤𝑖₁<𝑖₂<⋯<𝑖_𝑛≤2𝑛−1, to some possible Gaussian process. Set 𝐼={𝑖₁,𝑖₂,…,𝑖_𝑛}. We have noticed in Theorem 4.4 and Remark 4.5 that, when 𝐼 satisfies the relationship 2𝑛−1−𝐼={0,1,…,2𝑛−1}⧵𝐼, the system (6.1) admits a unique solution. We proved this fact by computing an energy integral. Actually, this fact holds for any set of indices 𝐼; see Lemma A.1.

Our aim is to characterize the set of indices 𝐼 for which the Green function of (6.1) can be viewed as the covariance function of a Gaussian process. A necessary condition for a function of two variables to be the covariance function of a Gaussian process is that it must be symmetric. So, we characterize the set of indices 𝐼 for which the Green function of (6.1) is symmetric and we will see that in this case this function is a covariance function. For this, we make use of general results concerning the relationships between boundary value problems and symmetric linear operators in Hilbert spaces. Since our results below are direct applications of Theorem 2 in Section 18.2 of [17, Chapter V], which describes the general form of boundary conditions corresponding to the self-adjoint extensions of standard symmetric operator, we only state them without any proof. We refer the reader to a draft [18] for more details and self-contained proofs.

6.1. Representation of the Solution

We first write out a representation for the Green function of (6.1).

Theorem 6.1. The boundary value problem (6.1) has a unique solution. The corresponding Green function admits the following representation, for 𝑠,𝑡∈[0,1]: 𝐺_𝐼(𝑠,𝑡)=(−1)^𝑛𝟙_{{𝑠≤𝑡}}(𝑡−𝑠)^2𝑛−1()!−()2𝑛−1−1^𝑛_𝜄∈𝐼(1−𝑠)^{2𝑛−1−𝜄}()!𝑅2𝑛−1−𝜄^𝐼,𝜄(𝑡), (6.2) where the 𝑅_𝐼,𝜄, 𝜄∈𝐼, are Hermite interpolation polynomials satisfying 𝑅^(2𝑛)_𝐼,𝜄𝑅=0,_{(𝑖)𝐼,𝜄}(0){},𝑅=0𝑓𝑜𝑟𝑖∈0,1,…,𝑛−1_{(𝑖)𝐼,𝜄}(1)=𝛿_𝜄,𝑖𝑓𝑜𝑟𝑖∈𝐼. (6.3)

Remark 6.2. Conditions (6.3) completely characterize the polynomials 𝑅_𝐼,𝜄, 𝜄∈𝐼 (see Lemma A.1 in the appendix).

A necessary condition for 𝐺_𝐼 to be the covariance function of a Gaussian process is that it must be symmetric: 𝐺_𝐼(𝑠,𝑡)=𝐺_𝐼(𝑡,𝑠) for any 𝑠,𝑡∈[0,1]. The theorem below asserts that if the set of indices 𝐼 is not of the form displayed in the preamble of Section 4, that is, 𝐼≠{0,1,…,2𝑛−1}⧵(2𝑛−1−𝐼), the Green function 𝐺_𝐼 is not symmetric and consequently this function cannot be viewed as a covariance function, that is, we cannot relate the boundary value problem (6.1) to any Gaussian process.

Theorem 6.3. The Green function 𝐺_𝐼 is symmetric (and it corresponds to a covariance function) if and only if the set of indices 𝐼 satisfies 2𝑛−1−𝐼={0,1,…,2𝑛−1}⧵𝐼.

6.2. Example: Bridges of Twice Integrated Brownian Motion (𝑛=3)

Here, we have a look on the particular case where 𝑛=3 for which the corresponding process (𝑋_𝑛(𝑡))_𝑡∈[0,1] is the twice integrated Brownian motion: 𝑋₃(𝑡)=^𝑡₀(𝑡−𝑠)𝐵(𝑠)d𝑠=^𝑡₀^𝑠2₀𝐵𝑠₁d𝑠₁d𝑠₂. (6.4) All the associated conditioned processes that can be constructed are related to the equation 𝑣⁽⁶⁾(𝑡)=−𝑢(𝑡) with boundary value conditions at time 0: 𝑣(0)=𝑣^′(0)=𝑣^′′(0)=0. There are 2³=8 such processes. Since the computations are tedious and the explicit results are cumbersome, we only report the correspondence between bridges and boundary value conditions at time 1 through the sets of indices 𝐼 and 𝐽. These are written in Table 2.

Table 2 

The Green functions related to the other sets cannot be related to some Gaussian processes. The sets are written in Table 3 with the correspondence 𝐼₂={0,1,2,3,4,5}⧵(5−𝐼₁).

Table 3 

6.3. Example: Bridges of Thrice Integrated Brownian Motion (𝑛=4)

For 𝑛=4, only the 2⁴=16 following differentiating sets can be related to bridges: {},{},{},{},{},{},{},{},{},{},{},{},{},{},{},{}.0,1,2,30,1,2,40,1,3,50,1,4,50,2,3,60,2,4,60,3,5,60,4,5,61,2,3,71,2,4,71,3,5,71,4,5,72,3,6,72,4,6,73,5,6,74,5,6,7 (6.5)

Appendix

Hermite Interpolation Polynomials

Lemma A.1. Let 𝑎_𝑖, 𝑖∈{0,1,…,𝑛−1}, and 𝑏_𝑖, 𝑖∈𝐼, be real numbers. There exists a unique polynomial 𝑃 such that 𝑃^(2𝑛)𝑃=0,_(𝑖)(0)=𝑎_𝑖{},𝑃for𝑖∈0,1,…,𝑛−1_(𝑖)(1)=𝑏_𝑖for𝑖∈𝐼. (A.1)

Remark A.2. Conditions (A.1) characterize the Hermite interpolation polynomial at points 0 and 1 with given values of the successive derivatives at 0 up to order 𝑛−1 and given values of the derivatives at 1 with selected orders in 𝐼. When 𝐼≠{0,1,…,𝑛−1}, these polynomials differ from the usual Hermite interpolation polynomials which involve the successive derivatives at certain points progressively from zero order up to certain orders.

Proof. We look for polynomials 𝑃 in the form 𝑃(𝑡)=∑^2𝑛−1_𝑗=0𝑐_𝑗𝑡^𝑗/𝑗!. We have 𝑃^(𝑖)(𝑡)=^2𝑛−1_𝑗=0𝑐_𝑗𝑡^𝑗−𝑖!.𝑗−𝑖 (A.2) We will adopt the convention 1/[(𝑗−𝑖)!]=0 for 𝑖>𝑗. Conditions (A.1) yield the linear system (with the convention that 𝑖 and 𝑗 denote, respectively the raw and column indices) 𝑐_𝑖=𝑎_𝑖if𝑖∈{0,1,…,𝑛−1_{2𝑛−1𝑗=0}},𝑐_𝑗𝑗−𝑖!=𝑏_𝑖if𝑖∈𝐼. (A.3) The (2𝑛)×(2𝑛) matrix of this system writes 𝛿𝐀=1^{𝑖,𝑗𝑖∈{0,1,…,𝑛−1}𝑗∈{0,1,…,2𝑛−1}}𝑗−𝑖!^𝑖∈𝐼_{𝑗∈{0,1,…,2𝑛−1}}=𝛿_{𝑖,𝑗𝑖∈{0,1,…,𝑛−1}𝑗∈{0,1,…,𝑛−1}}(0)_{𝑖∈{0,1,…,𝑛−1}𝑗∈{𝑛,𝑛+1,…,2𝑛−1}}1𝑗−𝑖!^𝑖∈𝐼_{𝑗∈{0,1,…,𝑛−1}}1𝑗−𝑖!^𝑖∈𝐼_{𝑗∈{𝑛,𝑛+1,…,2𝑛−1}}. (A.4) Proving the statement of Lemma A.1 is equivalent to proving that the matrix 𝐀 is regular. In view of the form of 𝐀 as a bloc-matrix, we see, since the north-west bloc is the unit matrix and the north-east bloc is the null matrix, that this is equivalent to proving that the south-east bloc of 𝐀 is regular. Let us call this latter 𝐀₀ and label its columns 𝐂_𝑗⁽⁰⁾, 𝑗∈{0,1,…,𝑛−1}: 𝐀₀=1𝑛+𝑗−𝑖!^𝑖∈𝐼_{𝑗∈{0,1,…,𝑛−1}}=𝐂₀⁽⁰⁾𝐂₁⁽⁰⁾⋯𝐂⁽⁰⁾_𝑛−1. (A.5) For proving that 𝐀₀ is regular, we factorize 𝐀₀ into the product of two regular triangular matrices. The method consists in performing several transformations on the columns of 𝐀₀ which do not affect its rank. We provide in this way an algorithm leading to an 𝐋𝐔-factorization of 𝐀₀, where 𝐋 is a lower triangular matrix and 𝐔 is an upper triangular matrix with no vanishing diagonal term.
We begin by performing the transformation 𝐂_𝑗⁽¹⁾=𝐂_𝑗⁽⁰⁾𝐂if𝑗=0,_𝑗(0)−𝐂⁽⁰⁾_𝑗−1𝑛+𝑗−𝑖₁if𝑗∈{1,2,…,𝑛−1}. (A.6) The generic term of the column 𝐂_𝑗⁽¹⁾, for 𝑗∈{1,2,…,𝑛−1}, is 1!−1𝑛+𝑗−𝑖𝑛+𝑗−𝑖₁1!=𝑛+𝑗−𝑖−1𝑖−𝑖₁𝑛+𝑗−𝑖₁1!.𝑛+𝑗−𝑖 (A.7) This transformation supplies a matrix 𝐀₁ with columns 𝐂_𝑗⁽¹⁾,  𝑗∈{0,1,…,𝑛−1}, which writes 𝐀₁=𝐂₀⁽¹⁾𝐂₁⁽¹⁾⋯𝐂⁽¹⁾_𝑛−1=1(𝑛−𝑖)!_𝑖∈𝐼𝑖−𝑖₁𝑛+𝑗−𝑖₁1𝑛+𝑗−𝑖!^𝑖∈𝐼_{𝑗∈{1,2,…,𝑛−1}}. (A.8) We have written 𝐀₁=𝐀₀𝐔₁, (A.9) where 𝐔₁ is the triangular matrix with a diagonal made of 1 below: 𝐔₁=𝛿_𝑖,𝑗−𝛿_{𝑖,𝑗−1}𝟙_{𝑗≥1}𝑛+𝑗−𝑖_{10≤𝑖,𝑗≤𝑛−1}. (A.10) We now perform the transformation 𝐂_𝑗⁽²⁾=𝐂_𝑗⁽¹⁾{},𝐂if𝑗∈0,1_𝑗(1)−𝑛+𝑗−𝑖₁−1𝑛+𝑗−𝑖₁𝐂⁽¹⁾_𝑗−1𝑛+𝑗−𝑖₂if𝑗∈{2,3,…,𝑛−1}. (A.11) The generic term of the column 𝐂_𝑗⁽²⁾, for 𝑗∈{2,3,…,𝑛−1}, is 𝑖−𝑖₁𝑛+𝑗−𝑖₁1!−𝑛+𝑗−𝑖𝑖−𝑖₁𝑛+𝑗−𝑖₁𝑛+𝑗−𝑖₂1!=𝑛+𝑗−𝑖−1𝑖−𝑖₁𝑖−𝑖₂𝑛+𝑗−𝑖₁𝑛+𝑗−𝑖₂1!.𝑛+𝑗−𝑖 (A.12) This transformation supplies a matrix 𝐀₂ with columns 𝐂_𝑗⁽²⁾,  𝑗∈{0,1,…,𝑛−1}, which writes 𝐀₂=𝐂₀⁽²⁾𝐂₁⁽²⁾⋯𝐂⁽²⁾_𝑛−1=1(𝑛−𝑖)!_𝑖∈𝐼𝑖−𝑖₁𝑛+1−𝑖₁1(𝑛+1−𝑖)!_𝑖∈𝐼𝑖−𝑖₁𝑖−𝑖₂𝑛+𝑗−𝑖₁𝑛+𝑗−𝑖₂1𝑛+𝑗−𝑖!^𝑖∈𝐼_{𝑗∈{2,3,…,𝑛−1}}. (A.13) We have written 𝐀₂=𝐀₁𝐔₂=𝐀₀𝐔₁𝐔₂, (A.14) where 𝐔₂ is the triangular matrix with a diagonal made of 1: 𝐔₂=𝛿_𝑖,𝑗−𝑛+𝑗−𝑖₁−1𝑛+𝑗−𝑖₁𝑛+𝑗−𝑖₂𝛿_{𝑖,𝑗−1}𝟙_{{𝑗≥2}0≤𝑖,𝑗≤𝑛−1}. (A.15) In a recursive manner, we easily see that we can construct a sequence of matrices 𝐀_𝑘,𝐔_𝑘, 𝑘∈{1,2,…,𝑛−1}, such that 𝐀_𝑘=𝐀_𝑘−1𝐔_𝑘, where 𝐔_𝑘 is the triangular matrix with a diagonal made of 1: 𝐔_𝑘=𝛿_𝑖,𝑗−𝑛+𝑗−𝑖₁−1⋯𝑛+𝑗−𝑖_𝑘−1−1𝑛+𝑗−𝑖₁⋯𝑛+𝑗−𝑖_𝑘𝛿_{𝑖,𝑗−1}𝟙_{{𝑗≥𝑘}0≤𝑖,𝑗≤𝑛−1},𝐀_𝑘=1(𝑛−𝑖)!_𝑖∈𝐼𝑖−𝑖₁𝑛+1−𝑖₁1(𝑛+1−𝑖)!_𝑖∈𝐼⋯(𝑖−𝑖₁)⋯(𝑖−𝑖_𝑘−1)(𝑛+𝑘−1−𝑖₁)⋯(𝑛+𝑘−1−𝑖_𝑘−1)1(𝑛+𝑘−1−𝑖)!_𝑖∈𝐼(𝑖−𝑖₁)⋯(𝑖−𝑖_𝑘)(𝑛+𝑗−𝑖₁)⋯(𝑛+𝑗−𝑖_𝑘)1(𝑛+𝑗−𝑖)!^𝑖∈𝐼_{𝑗∈{𝑘,…,𝑛−1}}. (A.16) We finally obtain, since all the 𝐔_𝑘, 𝑘∈{1,2,…,𝑛−1}, are regular, that 𝐀₀=𝐀_𝑛−1𝐔⁻¹_𝑛−1⋯𝐔₁⁻¹=𝐋𝐔 (A.17) with 𝐔=𝐔⁻¹_𝑛−1⋯𝐔₁⁻¹ and 𝐋=𝐀_𝑛−1=1(𝑛−𝑖)!_𝑖∈𝐼𝑖−𝑖₁𝑛+1−𝑖₁1(𝑛+1−𝑖)!_𝑖∈𝐼⋯𝑖−𝑖₁⋯𝑖−𝑖_𝑛−12𝑛−1−𝑖₁⋯2𝑛−1−𝑖_𝑛−11(2𝑛−1−𝑖)!_𝑖∈𝐼. (A.18) It is clear that the matrices 𝐔 and 𝐋 are triangular and regular, and then 𝐀₀ (and 𝐀) is also regular. Moreover, the inverse of 𝐀₀ can be computed as 𝐀₀⁻¹=𝐔⁻¹𝐋⁻¹=𝐔₁⋯𝐔_𝑛−1𝐋⁻¹. (A.19) The proof of Lemma A.1 is finished.