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International Journal of Stochastic Analysis
Volume 2011 (2011), Article ID 762486, 32 pages
A Class of Bridges of Iterated Integrals of Brownian Motion Related to Various Boundary Value Problems Involving the One-Dimensional Polyharmonic Operator
1Institut Camille Jordan, CNRS UMR5208, Université de Lyon, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France
2Institut 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.
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.
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
We refer the reader to  for a pioneering work dealing with the connections between general Gaussian processes and Green functions; see also . 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 ) 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
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  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 .
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
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.
Now, we tackle the problem of the prediction for the process .
Theorem 4.6. Fix . The shifted process admits the following representation:
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: