Product expansion for stochastic jump diffusions and its application to numerical approximation
type: Item
instance of: scholarly article
title: Product expansion for stochastic jump diffusions and its application to numerical approximation
zbMATH Open document ID: 0934.65006
DOI: 10.1016/S0377-0427(99)00095-3
author: Yanyan Li
published in: Journal of Computational and Applied Mathematics
publication date: 2000-04-25T00:00:00Z
review text: \textit{H. J. Sussmann} [Sel. Pap. 7th Int. Symp. Math. Theory Networks Syst., Stockholm 1985, 323-335 (1986; Zbl 0619.93021)] expressed the Chen series as the infinite product of the exponential Lie series in terms of a Philip Hall basis for the Lie algebra generated by the ordinary differential operator. Based on this result, one can obtain numerical schemes of an appropriate order by truncating up to a certain number of terms of the exponential Lie series. The aim of this paper is to develop a new product expansion of the exponential Lie series for jump diffusions with a single Poisson noise while that with multi-Poisson noises is expressed as shuffle products which implicitly represent the ordinary part of products of multiple stochastic integrals. On the basis of this expansion, the authors can construct the Stratonovich-Taylor-Hall (STH) schemes which are proved to be more efficient than Stratonovich-Taylor schemes, as they involve only the minimum number of multiple stochastic integrals, which can be regarded as systems of stochastic jump-diffusion equations and approximated by a lower-order scheme with an appropriate stepsize to ensure the necessary accuracy. Mean-square convergence of the STH schemes is shown, and numerical results are provided to illustrate the proposed approach.
reviewed by: Neculai Curteanu
Mathematics Subject Classification ID: 65C50
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 41A58
zbMATH DE Number: 1367885
zbMATH Keywords: stochastic jump diffusions
zbMATH Keywords: Stratonovich-Taylor-Hall schemes
zbMATH Keywords: mean-square convergence
zbMATH Keywords: Chen series
zbMATH Keywords: exponential Lie series
zbMATH Keywords: Philip Hall basis
zbMATH Keywords: product expansion
zbMATH Keywords: shuffle products
zbMATH Keywords: multiple stochastic integrals
zbMATH Keywords: numerical results
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1367885