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

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...

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Bibliographic Details
Published inBIT Vol. 61; no. 4; pp. 1223 - 1269
Main Authors Deligiannidis, G., Maurer, S., Tretyakov, M. V.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.12.2021
Springer Nature B.V
Subjects
Algorithms
Computational Mathematics and Numerical Analysis
Cost analysis
Differential equations
Dirichlet problem
Error analysis
Mathematics
Mathematics and Statistics
Numeric Computing
Numerical methods
Random walk
Weak approximation of stochastic differential equations
35R09
60J75
60H10
65C30
Jump processes
Integro-differential equations
SDEs driven by Lévy processes
Feynman–Kac formula
60H35
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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.
ISSN:0006-3835
1572-9125
DOI:10.1007/s10543-021-00863-2