A Discrete Construction for Gaussian Markov Processes
In the Lévy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process and to define the Wiener process as the almost sure path-wise limit of when tends to infinity. We generalize such a construction to the class of centered Gaussian Markov processes which can be written with and being continuous functions. We build the finite-dimensional process so that it gives an exact representation of the conditional expectation of with respect to the filtration generated by for . Moreover, we prove that the process converges in distribution toward .