On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains

Abstract

Dirichlet problems for second order parabolic operators in space-time domains Ω ⊂ ℝn+1 are of paramount importance in analysis, partial differential equations and applied mathematics. These problems can be approached in many different ways using techniques from partial differential equations, potential theory, stochastic differential equations, stopped diffusions and Monte Carlo methods. The performance of any technique depends on the structural assumptions on the operator, the geometry and smoothness properties of the space-time domain Ω, the smoothness of the Dirichlet data and the smoothness of the coefficients of the operator under consideration. In this paper, which mainly is of numerical nature, we attempt to further understand how Monte Carlo methods based on the numerical integration of stochastic differential equations perform when applied to Dirichlet problems for uniformly elliptic second order parabolic operators and how their performance vary as the smoothness of the boundary, Dirichlet data and coefficients change from smooth to non-smooth. Our analysis is set in the genuinely parabolic setting of time-dependent domains, which in itself adds interesting features previously only modestly discussed in the literature. The methods evaluated and discussed include elaborations on the non-adaptive method proposed by Gobet [ESAIM: Probability and Statistics 5: 261–297, 2001] based on approximation by half spaces and exit probabilities and the adaptive method proposed in [Lecture Notes in Computational Science and Engineering 44: 59–88, 2005] for weak approximation of stochastic differential equations.