A Discrete Construction for Gaussian Markov Processes

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 $\{X_{k\mathrm{/}{2}^N}\}$ for $0\le k\le 2^N$. Moreover, we prove that the process $X^N$ converges in distribution toward $X$.