Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance. (English) Zbl 1257.35200
Summary: In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretization on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretization is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretization of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives.
MSC:
35R60 | PDEs with randomness, stochastic partial differential equations |
35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
65C05 | Monte Carlo methods |
65N06 | Finite difference methods for boundary value problems involving PDEs |
65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
91G40 | Credit risk |
91G60 | Numerical methods (including Monte Carlo methods) |