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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.

Intuitively, multidimensional continuous stochastic processes are easily conceived as solutions to randomly perturbed differential equations of the form

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

The Lévy-Ciesielski construction of the

It is also important for our purpose to realize that the Schauder elements

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

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 [

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

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.

Let

Note that the processes considered in this paper are defined on the time interval

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.

Let

Note that the functions

Let

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 [

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

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

More precisely, starting from

For the sake of compactness of notations, we define

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.

For

The innovation process

Because of the Markovian property of the process

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

For all

The two processes

Using the expressions obtained in Proposition

Let us now define the function

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

The Gauss-Markov process

We have

We therefore identified a collection of functions

The above analysis motivates the introduction of a set of functions

To complete this program, we need to introduce some quantities that will play a crucial role in expressing the family

For all

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

Given two invertible matrices

Directly stems from Lemma

We have

Let us define

For every

The definition implies that

Before studying the property of the functions

We first introduce a family of functions

For every

Remark that the definitions make apparent the fact that these two families of functions are linked for all

Let

The space

First of all, since the functions

The proof now amounts showing the density of the family of functions we consider. Before showing this density property, we introduce for all

In fact, we show that the span of functions

We prove that

The fact that the column functions of

If

Indeed it easy to verify that, for all

From now on, abusing language, we will say that the family of

The basis

In order to exhibit a dual family of functions to the basis

Here, we make the assumptions that

We are now in a position to use the orthonormality of

As a consequence, defining the functions

The expression of the basis

For

Notice that the basis

We can now formulate the following.

Given the dual pairing in

We have to demonstrate that, for all

Otherwise, if

This proposition directly implies the main result of the section.

The collection of functions

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

The Schauder basis of functions with compact supports constructed allows to define functions by considering the coefficients on this basis, which constitute sequences of real numbers in the space

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

One denotes by

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

For every

This proposition is proved in Appendix

It is important to realize that, in the multidimensional case, the space

Yet, if we additionally relax the hypothesis that

However, the situation is much simpler in the one-dimensional case: because the uniform convergence of the sample paths is preserved as long as

In the case of the

This remark does not hold that the space

We equip the space

The function

We therefore conclude that we dispose of a continuous bijection mapping the coefficients onto the sample paths,

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

One calls

Should a function

The function

The proof of this theorem is provided in Appendix

Up to this point, we have rigorously defined the dual spaces of sample paths

Considering the infinite-dimensional subspace

The results of the preceding sections straightforwardly extend on the equivalence classes, and in particular we see that the functions

Similarly, for the matrix representation of

Denoting

The Cholesky decomposition of the finite-dimensional covariance block matrix

For every

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

The Cholesky decomposition of the finite-dimensional inverse covariance matrix

The result stems for the equalities

We now show that, asymptotically, the bases

The sequence of processes

For all fixed

Now that these preliminary remarks have been made, we can evaluate, for any

For any

As

We stress the fact that the relation

The fact that this covariance is equal to the covariance of the process

The process

Our multiresolution representation of the Gauss-Markov processes appears to be the direct consequence of the fact that, because of the Markov property, the Cholesky decomposition of the finite-dimensional covariance admits a simple inductive continuous limit. More generally, triangularization of the kernel operators has been studied in depth [

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

In the following, we draw from the theory of interpolating splines to further characterize the nature of our proposed basis for the construction of the Gauss-Markov processes. Essentially adapting the results from the previous works [

In order to define the finite-dimensional sample paths as a nested sequence of RKHSs, let us first define the infinite-dimensional operator

Two straightforward remarks are worth making. First, the space

We know motivate the introduction of the Hilbert space

The Hilbert space

Consider the problem of finding all elements

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

In the sequel, we will use the space

We refer to such spaces as finite-dimensional approximation spaces since we remark that

The Hilbert spaces

The proof this proposition follows the exact same argument as that in the case of

The framework set in the previous section offers a new interpretation of our construction. Indeed, for all

For any

The only hurdle to prove Proposition

Writing the Gauss-Markov process

In the previous section, we have noticed that the kernel

We can now proceed to justify the main result of Proposition

The finite-dimensional processes

Proposition

The central point of this section reads as follows.

Given a function

The space

Let us now show that

When

The Dirichlet energy simply appears as the squared norm induced on

The characterization of the basis as the minimizer of such a Dirichlet energy (

Let us assume that

By Proposition

It is a simple matter of calculus to check that the expression of

Under the hypotheses of Proposition

Compute the

Apply the differential operator

Orthonormalize the column functions

Apply the integral operator

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

In the one-dimensional case, the construction of the Gauss-Markov process is considerably simplified since we do not have to consider the potential degeneracy of matrix-valued functions. Indeed, in this situation, the centered Gauss-Markov process

This reads on

In the multidimensional case, the explicit expressions for the basis functions

We consider in this section that

Recognizing the

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

The simplest example of this kind is the stochastic sine and cosine process corresponding to

Construction of the stochastic sine and cos Ornstein-Uhelenbeck processes for the parameters given in (

(a) Basis for the construction of the Integrated Wiener process (

In applications, it often occurs that people use smooth stochastic processes to model the integration of noisy signals. This is for instance the case of a particular subject of a Brownian forcing or of the synaptic integration of noisy inputs [

Let

Furthermore, because of the simplicity and the sparsity of the matrices involved, we can identify in a compact form all the variables used in the computation of the construction basis for these processes. In particular, the flow

Observe that, for all

We study in more detail the case of the integrated and doubly-integrated Wiener process (

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

The closed-form expressions of the basis of functions

The integral operator

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

A very useful theorem in the stochastic processes theory is the Itô formula. We show here that this formula is consistent with the Hilbert framework introduced. Most of the proofs can be found in Appendix

Let

The proof of this proposition is quite technical and is provided in Appendix

Let

This theorem is proved in Appendix

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

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

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

In the framework we developed, transforming a process

The general problem consists therefore in studying the relationship between two real Gauss-Markov processes

We have noticed that the spaces

The previous remarks allow us to restrict without loss of generality our study to the processes defined for same function

Let us now consider that

Depending on the space we are considering (either coefficients or trajectories), we define the two following operators mapping the process

The

The

The operators

They are linear measurable bijections

For every

The determinants of

The proof of these properties elementary stems from the analysis done on the functions

From the properties proved on the lift operators, we are in position to further analyze the relationship between the probability distributions of

Given the finite-dimensional measures

In the finite-dimensional case, for all