Stochastic regression in terms of Brownian motion

摘要

Given a continuous stochastic process (X_t)_(t∈[0,T]), this article provides, in the first part, a stochastic process that is the best mean square approximation of the form (X_t){top}∧=b(t)+a(t)W_t, with W_t Brownian motion. The function coefficients a(t) and b(t) depend on the process X_t and are calculated in the case of several classical examples. In the second part, we extend the method for mean square approximations of the form (X_t){top}∧=b(t)+a(t)W_t+c(t)(W_t)~2. We also present simulations for each example, and show that replacing (W_t)~2 by the martingale (W_t)~2 - t is a more natural framework for the problem.