Sequential control variates for functionals of Markov processes

Abstract : Using a sequential control variates algorithm, we compute Monte Carlo approximations of solutions of linear partial differential equations connected to linear Markov processes by the Feynman--Kac formula. It includes diffusion processes with or without absorbing/reflecting boundary and jump processes. We prove that the bias and the variance decrease geometrically with the number of steps of our algorithm. Numerical examples show the efficiency of the method on elliptic and parabolic problems.