A Discrete Construction for Gaussian Markov Processes

Abstract

In the Lévy 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 $X_t=g\left(t\right)\cdot {\int }_0^{t}f\left(t\right)d{W}_t$ 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 $\left\{X_{k/2^N}\right\}$ for $0\le k\le 2^N$. Moreover, we prove that the process $X^N$ converges in distribution toward $X$.