Random walk algorithm for the Dirichlet problem for parabolic integro-differential equation. (English) Zbl 1493.65019
Summary: We consider stochastic differential equations driven by a general Lévy processes (SDEs) with infinite activity and the related, via the Feynman-Kac formula, Dirichlet problem for parabolic integro-differential equation (PIDE). We approximate the solution of PIDE using a numerical method for the SDEs. The method is based on three ingredients: (1) we approximate small jumps by a diffusion; (2) we use restricted jump-adaptive time-stepping; and (3) between the jumps we exploit a weak Euler approximation. We prove weak convergence of the considered algorithm and present an in-depth analysis of how its error and computational cost depend on the jump activity level. Results of some numerical experiments, including pricing of barrier basket currency options, are presented.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 35R09 | Integro-partial differential equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 60G50 | Sums of independent random variables; random walks |
Keywords:
SDEs driven by Lévy processes; jump processes; integro-differential equations; Feynman-Kac formula; weak approximation of stochastic differential equationsReferences:
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