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An explicit approximation for super-linear stochastic functional differential equations

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  • Li, Xiaoyue
  • Mao, Xuerong
  • Song, Guoting

Abstract

Since it is difficult to implement implicit schemes on the infinite-dimensional space, we aim to develop the explicit numerical method for approximating super-linear stochastic functional differential equations (SFDEs). Precisely, borrowing the truncation idea and linear interpolation we propose an explicit truncated Euler–Maruyama (EM) scheme for SFDEs, and obtain the boundedness and convergence in Lp(p≥2). We also prove the convergence rate with 1/2 order. Different from some previous works (Mao, 2003; Zhang et al., 2018), we release the global Lipschitz restriction on the diffusion coefficient. Furthermore, we reveal that numerical solutions preserve the underlying exponential stability. Moreover, we give several examples to support our theory.

Suggested Citation

  • Li, Xiaoyue & Mao, Xuerong & Song, Guoting, 2024. "An explicit approximation for super-linear stochastic functional differential equations," Stochastic Processes and their Applications, Elsevier, vol. 169(C).
    • Handle: RePEc:eee:spapps:v:169:y:2024:i:c:s0304414923002478
    DOI: 10.1016/j.spa.2023.104275
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    References listed on IDEAS

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    1. Flavia Sancier & Salah Mohammed, 2017. "An Option Pricing Model with Memory," Papers 1709.00468, arXiv.org.
    2. Nguyen, Dang H. & Nguyen, Nhu N. & Yin, George, 2021. "Stochastic functional Kolmogorov equations, I: Persistence," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 319-364.
    3. Bruno Dupire, 2019. "Functional Itô calculus," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 721-729, May.
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