A Discrete Construction for Gaussian Markov Processes

TL;DR

This paper extends the L\'evy construction of Brownian motion to a broader class of Gaussian Markov processes, providing a discrete approximation that converges in distribution to the original process.

Contribution

It introduces a novel discrete construction for Gaussian Markov processes that exactly represents conditional expectations and converges to the process in distribution.

Findings

Constructs finite-dimensional processes matching conditional expectations.

Proves convergence of the discrete process to the Gaussian Markov process.

Generalizes the Haar-based basis approach to a wider class of processes.

Abstract

In the L\'evy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process $W^{N}$ and to define the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends to infinity. We generalize such a construction to the class of centered Gaussian Markov processes $X$ which can be written $X_{t} = g(t) \cdot \int_{0}^{t} f(t) dW_{t}$ with $f$ and $g$ being continuous functions. We build the finite-dimensional process $X^{N}$ so that it gives an exact representation of the conditional expectation of $X$ with respect to the filtration generated by ${\lbrace X_{k/2^{N}}\rbrace}$ for $0 \leq k \leq 2^{N}$. Moreover, we prove that the process $X^{N}$ converges in distribution toward $X$.