Stable positive Monte Carlo finite difference techniques for random parabolic partial differential equations

Abstract

[EN] Integral Transform technique is a powerful method for solving random partial differencial equations (RPDEs) in unbounded domains but an alternative is needed in the case of bounded domains. In this case finite difference methods have the drawback coming from the complexity of the computation of the statistical moments arising from the operational random calculus throughout the iterative levels of the discretization steps and the necessity to store the information of all the previous levels of the iteration process. In this work we study a RPDE of parabolic type often encountered in heat and mass transfer theory in heterogeneus media. We present a stable numerical random finite difference scheme preserving positivity of the solution stochastic process together with Monte Carlo technique that provides a useful tool to obtain accurate values of the expectation and the variance of the approximating process even for large values of the time variable.