| Abstract: | In this article, we extend a Milstein finite difference scheme introduced in 8 Giles, M. B. and Reisinger, C. 2012. Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance. SIAM Financ. Math., 3(1): 572–592. (doi:10.1137/110841916)Crossref] , Google Scholar] for a certain linear stochastic partial differential equation (SPDE) to semi-implicit and fully implicit time-stepping as introduced by Szpruch 32 Szpruch, L. 2010. Numerical approximations of nonlinear stochastic systems PhD Thesis, University of Strathclyde Google Scholar] for stochastic differential equations (SDEs). We combine standard finite difference Fourier analysis for partial differential equations with the linear stability analysis in 3 Buckwar, E. and Sickenberger, T. 2011. A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods. Math. Comput. Simulation, 81: 1110–1127. (doi:10.1016/j.matcom.2010.09.015)Crossref], Web of Science ®] , Google Scholar] for SDEs to analyse the stability and accuracy. The results show that Crank–Nicolson time-stepping for the principal part of the drift with a partially implicit but negatively weighted double Itô integral gives unconditional stability over all parameter values and converges with the expected order in the mean-square sense. This opens up the possibility of local mesh refinement in the spatial domain, and we show experimentally that this can be beneficial in the presence of reduced regularity at boundaries. |
