J. Comp. Math., 28 (2010), pp. 579-605. |
WEAK APPROXIMATION OF OBLIQUELY REFLECTED DIFFUSIONS IN TIME-DEPENDENT DOMAINS Kaj Nystrom ^{1}, Thomas Onskog ^{1} 1 Department of Mathematics and Mathematical Statistics, Ume\aa\ University, SE-901 87 Ume\aa, SwedenReceived 2008-11-5 Accepted 2009-10-20 Available online 2010-5-1 doi:10.4208/jcm.1003-m2957 Abstract In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in $\mathbb{R}^{d}$, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to $1/2$ and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains. Key words: Stochastic differential equations, Oblique reflection, Robin boundary conditions, Skorohod problem, Time-dependent domain, Weak approximation, Monte Carlo method, Parabolic partial differential equations, Projected Euler scheme AMS subject classifications: 65MXX, 35K20, 65CXX, 60J50, 60J60 Email: kaj.nystrom@math.umu.se, thomas.onskog@math.umu.se |