An interpolated Stochastic Algorithm for Quasi-Linear PDEs

Stephane Menozzi Universite Paris VII

Abstract: We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled Forward-Backward SDEs, which provides an efficient probabilistic representation of this type of equations. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length.

The corner stone of the algorithm is a nonlinear dynamic programming principle. Then, the convergence analysis takes advantage of the standing regularity properties of the true solution through an interpolation procedure and also exploits the optimality of the square Gaussian quantization used to approximate the conditional expectations involved.

In particular, our work provides an alternative to the method described in Douglas, Ma and Protter (Ann. Appl. Prob. 96) and weakens the regularity assumptions required in this reference as well as in the schemes introduced by Milstein and Tretyakov (see Stochastic Numerics for Math. Physics, Springer).