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A New Simplified Weak Second-Order Scheme for Solving Stochastic Differential Equations with Jumps

Author

Listed:
  • Yang Li

    (College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China)

  • Yaolei Wang

    (College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China)

  • Taitao Feng

    (College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China)

  • Yifei Xin

    (College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China)

Abstract

In this paper, we propose a new weak second-order numerical scheme for solving stochastic differential equations with jumps. By using trapezoidal rule and the integration-by-parts formula of Malliavin calculus, we theoretically prove that the numerical scheme has second-order convergence rate. To demonstrate the effectiveness and the second-order convergence rate, three numerical experiments are given.

Suggested Citation

  • Yang Li & Yaolei Wang & Taitao Feng & Yifei Xin, 2021. "A New Simplified Weak Second-Order Scheme for Solving Stochastic Differential Equations with Jumps," Mathematics, MDPI, vol. 9(3), pages 1-14, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:224-:d:485920
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    References listed on IDEAS

    as
    1. 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.
    2. Geman, Helyette, 2002. "Pure jump Levy processes for asset price modelling," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1297-1316, July.
    3. Li, Jing & Li, Lingfei & Zhang, Gongqiu, 2017. "Pure jump models for pricing and hedging VIX derivatives," Journal of Economic Dynamics and Control, Elsevier, vol. 74(C), pages 28-55.
    4. Jang, Jiwook, 2007. "Jump diffusion processes and their applications in insurance and finance," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 62-70, July.
    5. Li, Min & Huang, Chengming & Chen, Ziheng, 2021. "Compensated projected Euler-Maruyama method for stochastic differential equations with superlinear jumps," Applied Mathematics and Computation, Elsevier, vol. 393(C).
    6. Mikulevicius, R., 2012. "On the rate of convergence of simple and jump-adapted weak Euler schemes for Lévy driven SDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2730-2757.
    7. Maekawa, Koichi & Lee, Sangyeol & Morimoto, Takayuki & Kawai, Ken-ichi, 2008. "Jump diffusion model with application to the Japanese stock market," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 78(2), pages 223-236.
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