A Discrete Construction for Gaussian Markov Processes

2008/05/01 by Taillefumier, Thibaud
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0805.0048

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(t) ⋅ ∫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 ≤ k ≤ 2^N. Moreover, we prove that the process X^N converges in distribution toward X.

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