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The Euler Scheme for a Stochastic Differential Equation Driven by Pure Jump Semimartingales

Part of: Limit theorems Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  30 January 2018

Hanchao Wang
Hanchao Wang*
Affiliation:
Zhejiang University
*
∗ Postal address: Department of Mathematics, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, China. Email address: [email protected]
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Abstract

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In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

Keywords

Euler schemeweak convergencepure jump Itô semimartingalestochastic differential equation

MSC classification

Primary: 60J75: Jump processes 65C30: Stochastic differential and integral equations
Secondary: 60F17: Functional limit theorems; invariance principles
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 149 - 166
Copyright
© Applied Probability Trust 

References

Diop, A., Jacod, J. and Todorov, V. (2013). Central limit theorems for approximate quadratic variations of pure Jump Itô semimartingales. Stoch. Process. Appl. 123, 839–886.CrossRefGoogle Scholar
Jacod, J. (2004). The Euler scheme for Lévy driven stochastic differential equations: limit theorems. Ann. Prob. 32, 1830–1872.Google Scholar
Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler methods for stochastic differential equations. Ann. Prob. 26, 267–307.Google Scholar
Jacod, J. and Protter, P. (2012). Discretization of Processes. Springer, Heidelberg.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
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