Multi-Resolution Schauder Approach to Multidimensional Gauss-Markov Processes

TL;DR

This paper introduces a novel multi-resolution Schauder basis expansion for multidimensional Gauss-Markov processes, offering a natural, convergent, and optimal approximation method that simplifies analysis and applications in stochastic process theory.

Contribution

It proposes an alternative Schauder basis expansion for Gauss-Markov processes, enabling multi-resolution analysis and optimal finite-dimensional approximations.

Findings

Strong almost-sure convergence of the partial sums.

Finite-dimensional processes minimize Dirichlet energy.

Simplifies treatment of complex stochastic processes.

Abstract

The study of 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 L\'{e}vy-Cieselski 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 unitary variance. We thereby offer a natural multi-resolution description of Gauss-Markov processes as limits of the 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 simpler treatment in many applied and theoretical fields and we provide a short overview of applications we are currently developing.