SOLVING 2 nd ORDER PARABOLIC SYSTEM

There are known methods of approximating the solution of parabolic 2 nd order systems by solving stochastic diierential equations instead. The main idea is based on the fact that stochastic diierential equation deenes a diiusion process, generated by an elliptic diierential operator on R d. We propose a diierence scheme for the ellip-tic 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 diierence schemes for parabolic equations with known stability diiculties, can thus be replaced by space discretization and simulation of the associated Markov (jump) process.