A Monte-Carlo method with inherent parallelism for numerical solving partial differential equations with boundary conditions

Abstract

One can construct the representation of solutions to various problems for parabolic differential equation with boundary conditions in the framework of classical stochastic analysis. This fact yields Monte Carlo methods for solving PDE’s numerically. Instead of solving a PDE by common numeric techniques, one can simulate a stochastic system which allows for a simple parallelism with linear speed up. A number of numerical schemes exists for solving stochastic differential equations, i.e. the Euler scheme. If reflection is concerned, most methods have some shortcomings. In this article an efficient algorithm for simulating a reflected stochastic differential equation is developed. Results of numerical experiments are referenced revealing significant reduction in computational time gained by parallelization.