International Journal of Stochastic AnalysisVolume 2011 (2011), Article ID 762486, 32 pagesdoi:10.1155/2011/762486

Research Article

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

Aimé Lachal1,2

¹Institut Camille Jordan, CNRS UMR5208, Université de Lyon, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France²Institut National des Sciences Appliquées de Lyon, Bâtiment Léonard de Vinci, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France

Received 8 August 2011; Accepted 4 October 2011

Academic Editor: R. Liptser

Copyright © 2011 Aimé Lachal. 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.

Abstract.

Let be the linear Brownian motion and the -fold integral of Brownian motion, with being a positive integer: for any In this paper we construct several bridges between times and of the process involving conditions on the successive derivatives of at times and . 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 , its th derivative by or .

Let be the linear Brownian motion started at 0 and be the linear Brownian bridge within the time interval : . These processes are Gaussian processes with covariance functions For a given continuous function , the functions and , respectively, defined on by are the solutions of the respective boundary value problems on : 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 , where is a positive integer. This latter is associated with the -fold integral of Brownian motion : (Notice that all of the derivatives at time 0 naturally vanish: .) Indeed, the following facts, for instance, are known (see, e.g., [1, 2]).(i)The covariance function of the process coincides with the Green function of the boundary value problem (ii)The covariance fonction of the bridge coincides with the Green function of the boundary value problem (iii)The covariance fonction of the bridge coincides with the Green function of the boundary value problem

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 involving different events at time 1: for a certain number of events, , and certain indices such that , and to make the connection with the boundary value problems: for certain indices such that . 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 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 -Markov Gaussian process evolving on the real line . By “-Markov,” it is understood that the trajectory is times differentiable and the -dimensional process is a Markov process. Let us introduce the covariance function of : for , . It is known (see [6]) that the function admits the following representation: where , , are certain functions.

Let be linear differential operators of order less than , and let be a linear differential operator of order defined by where are continuous functions on . More precisely, we have for any times differentiable function and any ,

Theorem 2.1. Assume that the functions , , are times differentiable and satisfy the following conditions, for a certain constant : Then, for any continuous function on , the function defined on by solves the boundary value problem

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

Proof. In view of (2.1), the function can be written as The derivative of is given by and its second-order derivative, since , by More generally, because of the assumptions (2.4), we easily see that, for , and the th order derivative of is given by Actually, we have proved that, for , Finally, due to (2.5), Concerning the boundary value conditions, referring to (2.5), we similarly have The proof of Theorem 2.1 is finished.

In the two next sections, we construct processes connected to the equation subject to the boundary value conditions at 0: for and others at 1 that will be discussed subsequently, where and are the differential operators ( being of order ) defined by

3. The -Fold Integral of Brownian Motion

Let be the linear Brownian motion limited to the time interval and started at 0. We introduce the -fold integral of Brownian motion: for any , In particular, . The trajectories of are times differentiable and we have for . Moreover, we have at time 0 the equalities . The process is an -Markov Gaussian process since the -dimensional process is Markovian. The covariance function of the Gaussian process is given by In order to apply Theorem 2.1, we decompose into the form (2.1). We have for, e.g., , We then obtain the following representation: with, for any , We state below a result of [2] that we revisit here by using Theorem 2.1.

Theorem 3.1. Let be a fixed continuous function on . The function defined on by is the solution of the boundary value problem

Proof. Let us check that conditions (2.4) and (2.5) of Theorem 2.1 are fulfilled. First, we have Performing the transformation in the first sum lying within the last equality, we get and then On the other hand, setting we clearly see that, for any , Consequently, by Theorem 2.1, we see that the function 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 -Fold Integral of Brownian Motion

In this section, we construct various bridges related to . More precisely, we take conditioned on the event that certain derivatives vanish at time 1. Let us recall that all the derivatives at time 0 naturally vanish: .

For any , let be a subset of with and the convention that for , . We see that, for each , we can define subsets of indices , and the total number of sets is then . Set for any In this way, we define processes that we will call “bridges’’ of . In particular, (i)for , we simply have ; (ii)for , the corresponding process is the -fold integral of Brownian bridge (iii)for , the corresponding process is the “single’’ bridge of -fold integral of Brownian motion: (iv)for , the corresponding process is This is the natural bridge related to the -dimensional Markov process .

In this section, we exhibit several interesting properties of the various processes . One of the main goals is to relate these bridges to additional boundary value conditions at 1. For this, we introduce the following subset of : with The cardinality of is . Actually, the set will be used further for enumerating the boundary value problems which can be related to the bridges labeled by . Conversely, yields through In Table 1, we give some examples of sets and .

Table 1

4.1. Polynomial Drift Description

Below, we provide a representation of by means of subject to a random polynomial drift.

Theorem 4.1. One has the distributional identity where the functions , , are Hermite interpolation polynomials on characterized by

Remark 4.2. In the case where , we retrieve a result of [16]. Moreover, conditions (4.9) characterize the polynomials , . 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 where the functions , , are such that for all . We get the linear system We plainly have Then, the system (4.11) writes The matrix of this system is regular as it can be seen by introducing the related quadratic form which is definite positive. Indeed, for any real numbers , , we have Thus, the system (4.11) has a unique solution. As a result, the are linear combinations of the functions which are polynomials of degree less than . Hence, the   are polynomials of degree at most .We now compute the derivatives of at 0 in 1. We have for since the functions plainly enjoy this property. For checking that for , we successively compute (i)for , (ii)for , (iii)for , (iv)for , (v)for , (vi)for , Consequently, at time , for , Now, by differentiating (4.11), we get for , In particular, if , this can be rewritten as which by identification yields . Similarly, for , in view of (4.21), we have In particular, if , we have for , and then which by identification yields . The proof of Theorem 4.1 is finished.

4.2. Covariance Function

Let be the covariance function of : . In the next theorem, we supply a representation of of the form (2.1).

Theorem 4.3. The covariance function of admits the following representation: for , with, for any , Moreover, the functions , , are Hermite interpolation polynomials such that

Proof. We decompose into the difference We have By definition (4.11) of the ’s, we observe that Then, we can simplify into Since the covariance functions and are symmetric, we also have Let us introduce the symmetric polynomial It can be expressed by means of the functions ’s as follows: We can rewrite as and then, for , We immediately see that is a polynomial of degree less than such that for and, since , We deduce that Then for any . 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 . The following statement is the main connection between the different boundary value conditions associated with the operator and the different bridges of the process introduced in this work.

Theorem 4.4. Let be a fixed continuous function on . The function defined on by is the solution of the boundary value problem

Proof. Step 1. Recall that . We decompose the function into the difference , where, for , We know from Theorem 3.1 that , for and for . Moreover, with the function being a polynomial of degree less than , the function is also a polynomial of degree less than . Then , for , and On the other hand, we learn from (2.13) that, for , and then, for , We also have, for any , As a result, we see that This implies that, for , and for , . Then for any , that is, . The function is a solution of (4.41).Step 2. We now check the uniqueness of the solution of (4.41). Let and be two solutions of . Then the function satisfies , for and for . We compute the following “energy” integral: We have constructed the set in order to have for any : when we pick an index , either or . Indeed, (i)if , ;(ii)if , by observing that , we have . Since , we see that and then . Next, which entails , that is, is a polynomial of degree less than . Moreover, with the boundary value conditions at 0, we obtain or .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 for any . These conditions are fulfilled when the set is such that , , . This is equivalent to say that and make up a partition of , or In this manner, we get different boundary value problems which correspond to the different bridges we have constructed. We will see in Section 6 that the above identity concerning the differentiating set 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 .

Theorem 4.6. Fix . The shifted process admits the following representation: where The process is a copy of which is independent of . The functions , , and , , are Hermite interpolation polynomials on characterized by Actually, these functions can be expressed by means of the functions , , and , , as follows: In other words, the process is a bridge of length which is independent of , that is,

Proof. Step 1. Fix . We have the well-known decomposition, based on the classical prediction property of Brownian motion stipulating that , where is a Brownian motion independent of , with . Differentiating this equality times, , we obtain Therefore, We are going to express the , , by means of the , . We have, by differentiating (4.57) times, for , For , this yields Set for and let us introduce the matrix together with its inverse matrix Equalities (4.59) for read as a linear system of equations and unknowns: which can be rewritten into a matrix form as The solution is given by and we see that , , is of the form Therefore, by plugging (4.66) into (4.57), we obtain Finally, can be written as where with Step 2. We easily see that the functions and are polynomials of degree less than . Let us compute now their derivatives at 0 and . First, concerning we have Choosing and recalling the definition of and the fact that the matrices and are inverse, this gives, for , Choosing , we have for By Theorem 4.1, we know that for ; then Observing that, if , the conditions and are equivalent, we simply have which immediately entails, by (4.73), Next, concerning , we have Choosing , this gives, for , since , Choosing , we have, for , since and , The polynomials (resp., ) enjoy the same properties than the ’s (resp., the ’s), regarding the successive derivatives, they can be deduced from these latter by a rescaling according as It is then easy to extract the identity in distribution below, by using the property of Gaussian conditioning: Theorem 4.6 is established.

5. Example: Bridges of Integrated Brownian Motion ()

Here, we have a look on the particular case where for which the corresponding process is nothing but integrated Brownian motion (the so-called Langevin process): The underlying Markov process is the so-called Kolmogorov diffusion . All the associated conditioned processes that will be constructed are related to the equation with boundary value conditions at time 0: . There are four such processes: (i) (integrated Brownian motion); (ii) (integrated Brownian bridge); (iii) (bridge of integrated Brownian motion); (iv) (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: , , , , , . 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 is nothing but integrated Brownian motion: The covariance function is explicitly given by This process is related to the boundary value conditions at 1 (): . The prediction property can be stated as follows:

5.2. Integrated Brownian Bridge

The process corresponding to the set is integrated Brownian bridge: This process can be represented as The covariance function is explicitly given by The process is related to the boundary value conditions at 1 (): . The prediction property says that

5.3. Bridge of Integrated Brownian Motion

The process corresponding to the set is the bridge of integrated Brownian motion: The bridge is understood as the process is pinned at its extremities: . This process can be represented as The covariance function is explicitly given by The process is related to the boundary value conditions at 1 (): . The prediction property says that

5.4. Other Bridge of Integrated Brownian Motion

The process corresponding to the set is another bridge of integrated Brownian motion (actually of the two-dimensional Kolmogorov diffusion): The bridge here is understood as the process is pinned at its extremities together with its derivatives: . This process can be represented as The covariance function is explicitly given by The process is related to the boundary value conditions at 1 (): . The prediction property says that

5.5. Two Counterexamples

(i) The solution of the problem associated with the boundary value conditions (which corresponds to the set ) is given by where

(ii) The solution of the problem associated with the boundary value conditions (which corresponds to the set ) is given by where We can observe the relationships and . The Green functions and 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 problem for any indices such that , to some possible Gaussian process. Set . We have noticed in Theorem 4.4 and Remark 4.5 that, when satisfies the relationship , 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 ; see Lemma A.1.

Our aim is to characterize the set of indices 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 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 : where the , , are Hermite interpolation polynomials satisfying

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

A necessary condition for to be the covariance function of a Gaussian process is that it must be symmetric: for any . The theorem below asserts that if the set of indices is not of the form displayed in the preamble of Section 4, that is, , the Green function 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 is symmetric (and it corresponds to a covariance function) if and only if the set of indices satisfies .

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

Here, we have a look on the particular case where for which the corresponding process is the twice integrated Brownian motion: All the associated conditioned processes that can be constructed are related to the equation with boundary value conditions at time 0: . There are 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 and . These are written in Table 2.

Table 2

Conditioning set

Differentiating set

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 .

Table 3

Differentiating set

Differentiating set

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

For , only the following differentiating sets can be related to bridges:

Appendix

Hermite Interpolation Polynomials

Lemma A.1. Let , , and , , be real numbers. There exists a unique polynomial such that

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 and given values of the derivatives at 1 with selected orders in . When , 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 in the form . We have We will adopt the convention for . Conditions (A.1) yield the linear system (with the convention that and denote, respectively the raw and column indices) The matrix of this system writes Proving the statement of Lemma A.1 is equivalent to proving that the matrix is regular. In view of the form of 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 is regular. Let us call this latter and label its columns , : For proving that is regular, we factorize into the product of two regular triangular matrices. The method consists in performing several transformations on the columns of which do not affect its rank. We provide in this way an algorithm leading to an -factorization of , where is a lower triangular matrix and is an upper triangular matrix with no vanishing diagonal term.We begin by performing the transformation The generic term of the column , for , is This transformation supplies a matrix with columns ,  , which writes We have written where is the triangular matrix with a diagonal made of 1 below: We now perform the transformation The generic term of the column , for , is This transformation supplies a matrix with columns ,  , which writes We have written where is the triangular matrix with a diagonal made of 1: In a recursive manner, we easily see that we can construct a sequence of matrices , , such that , where is the triangular matrix with a diagonal made of 1: We finally obtain, since all the , , are regular, that with and It is clear that the matrices and are triangular and regular, and then (and ) is also regular. Moreover, the inverse of can be computed as The proof of Lemma A.1 is finished.

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A. Lachal, “A class of bridges of iterated integrals of Brownian motion related to various boundary value problems involving the one-dimensional polyharmonic operator,” http://arxiv.org/abs/1108.2132