Almost Sure Convergence of the Numerical Discretization of Stochastic Jump Diffusions

Authors: Li, C.W.1; Liu, X.Q.2

Source: Acta Applicandae Mathematicae, Volume 62, Number 3, July 2000 , pp. 225-244(20)

Publisher: Springer

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Abstract:

Based on the shuffle product expansion of exponential Lie series in terms of a Philip Hall basis for the stochastic differential equations of jump-diffusion type, we can establish Stratonovich–Taylor–Hall (STH) schemes. However, the STHr scheme converges only at order r in the mean-square sense. In order to have the almost sure Stratonovich–Taylor–Hall (ASTH) schemes, we have to include all the terms related to multiple Poissonian integrals as the moments of multiple Poissonian integrals always have lower orders of magnitudes as compared with those of multiple Brownian integrals.

Keywords: Philip Hall basis; Stratonovich–Taylor expansion; almost sure convergence; exponential Lie series; jump diffusion; shuffle product

Document Type: Regular Paper

Affiliations: 1: Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong 2: Institute of Applied Mathematics, Academia Sinica, Beijing 100080, P.R. China

Publication date: July 1, 2000

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