Solving 2 nd order parabolic system by simulations of Markov jump processes ∗

There are known methods of approximating the solution of parabolic 2 order systems by solving stochastic differential equations instead. The main idea is based on the fact that a stochastic differential equation defines a diffusion process, generated by an elliptic differential operator on Rd. We propose a difference scheme for the elliptic operator, which possesses the structure of Markov (jump) process. The existence of such a scheme is proved, the proof relying on the choice of new coordinates in which the elliptic operator is “almost” Laplacian, and has the properties necessary for discretization. Time discretization, which involves difference schemes for parabolic equations with known stability difficulties, can thus be replaced by space discretization and simulation of the associated Markov (jump) process.