Discretization of Stochastic Differential Equations.

Summary This chapter develops discretization schemes for stochastic differential equations and their applications to the probabilistic numerical resolution of deterministic parabolic partial differential equations. It starts with some important properties of Ito’s Brownian stochastic calculus, and the existence and uniqueness theorem for stochastic differential equations with Lipschitz coefficients. Then, using probabilistic techniques only, existence, uniqueness, and smoothness properties are proved for solutions of parabolic partial differential equations. To this end, we show that stochastic differential equations with smooth coefficients define stochastic flows, and we prove some properties of such flows. We are then in a position to prove an optimal convergence rate result for the discretization schemes.