Random walk algorithm for the Dirichlet problem for parabolic integro-differential equation

Deligiannidis, G.; Maurer, S.; Tretyakov, M. V.

Abstract

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.

Citation

Deligiannidis, G., Maurer, S., & Tretyakov, M. V. (2021). Random walk algorithm for the Dirichlet problem for parabolic integro-differential equation. BIT Numerical Mathematics, 61(4), 1223-1269. https://doi.org/10.1007/s10543-021-00863-2

• Journal Article Type: Article
• Acceptance Date: Mar 26, 2021
• Online Publication Date: Apr 26, 2021
• Publication Date: Dec 1, 2021
• Deposit Date: Mar 16, 2021
• Publicly Available Date: Apr 27, 2022
• Journal: BIT Numerical Mathematics
• Print ISSN: 0006-3835
• Electronic ISSN: 1572-9125
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 61
• Issue: 4
• Pages: 1223-1269
• DOI: https://doi.org/10.1007/s10543-021-00863-2
• Keywords: Computer Networks and Communications; Software; Applied Mathematics; Computational Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/5397449
• Publisher URL: https://link.springer.com/article/10.1007%2Fs10543-021-00863-2