Communications in Mathematical Sciences

Volume 15 (2017)

Number 1

Computation of the local time of reflecting Brownian motion and the probabilistic representation of the Neumann problem

Pages: 237 – 259

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n1.a11

Authors

Yijing Zhou (Department of Mathematics and Statistics, University of North Carolina, Charlotte, N.C., U.S.A.)

Wei Cai (INS, Shanghai Jiao Tong University, Shanghai, China; and Department of Mathematics and Statistics, University of North Carolina, Charlotte, N.C., U.S.A.)

Elton Hsu (Department of Mathematics, Northwestern University, Evanston, Illinois, U.S.A.)

Abstract

In this paper, we propose numerical methods for computing the boundary local time of reflecting Brownian motion (RBM) for a bounded domain in $\mathbb{R}^3$ and the probabilistic solution of the Laplace equation with the Neumann boundary condition. Approximations of RBM based on walk-on-spheres (WOS) and random walk on lattices are discussed and tested for sampling RBM paths and their applicability in finding accurate approximation of the local time and discretization of the probabilistic representation of the Neumann problems using the computed local time. Numerical tests for several domains (a cube, a sphere, an ellipsoid, and a non-convex non-smooth domain made of multiple spheres) have shown the convergence of the numerical methods as the time length of RBM paths and number of paths sampled increase.

Keywords

reflecting Brownian motion, Brownian motion, boundary local time, Skorohod problem, WOS, random walk, Laplace equation

2010 Mathematics Subject Classification

65C05, 65N99, 78M25, 92C45

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Published 10 January 2017