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Strong approximations of stochastic differential equations with jumps

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Abstract

This paper is a survey of strong discrete time approximations of jump-diffusion processes described by stochastic differential equations (SDEs). It also presents new results on strong discrete time approximations for the specific case of pure jump SDEs. Strong approximations based on jump-adapted time discretizations, which produce no discretization error in the case of pure jump processes, are analyzed. The computational complexity of these approximations is proportional to the jump intensity. By exploiting a stochastic expansion for pure jump processes, higher order discrete time approximations, whose computational complexity is not dependent on the jump intensity, are proposed. For the specific case of pure jump SDEs, the strong order of convergence of strong Taylor schemes is established under weaker conditions than those currently known in the literature.

Suggested Citation

  • NicolaBruti-Liberati & Eckhard Platen, 2007. "Strong approximations of stochastic differential equations with jumps," Published Paper Series 2007-7, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    • Handle: RePEc:uts:ppaper:2007-7
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    Citations

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    Cited by:

    1. Amr Abou-Senna & Boping Tian, 2022. "Almost Sure Exponential Stability of Numerical Solutions for Stochastic Pantograph Differential Equations with Poisson Jumps," Mathematics, MDPI, vol. 10(17), pages 1-18, September.
    2. Yang, Xu & Zhao, Weidong, 2018. "Finite element methods and their error analysis for SPDEs driven by Gaussian and non-Gaussian noises," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 58-75.
    3. Kohatsu-Higa, Arturo & Tankov, Peter, 2010. "Jump-adapted discretization schemes for Lévy-driven SDEs," Stochastic Processes and their Applications, Elsevier, vol. 120(11), pages 2258-2285, November.
    4. Szimayer, Alex & Maller, Ross A., 2007. "Finite approximation schemes for Lévy processes, and their application to optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1422-1447, October.
    5. Fan, Zhencheng, 2017. "Convergence of numerical solutions to stochastic differential equations with Markovian switching," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 176-187.
    6. Arturo Kohatsu-Higa & Salvador Ortiz-Latorre & Peter Tankov, 2012. "Optimal simulation schemes for L\'evy driven stochastic differential equations," Papers 1204.4877, arXiv.org.
    7. Sabbar, Yassine & Kiouach, Driss & Rajasekar, S.P. & El-idrissi, Salim El Azami, 2022. "The influence of quadratic Lévy noise on the dynamic of an SIC contagious illness model: New framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    8. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007.
    9. Huang Xiao, 2013. "Quasi-maximum likelihood estimation of multivariate diffusions," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 17(2), pages 179-197, April.
    10. Todorov, Viktor, 2009. "Estimation of continuous-time stochastic volatility models with jumps using high-frequency data," Journal of Econometrics, Elsevier, vol. 148(2), pages 131-148, February.

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