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An implicit numerical scheme for a class of backward doubly stochastic differential equations

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  • Hu, Yaozhong
  • Nualart, David
  • Song, Xiaoming

Abstract

In this paper, we consider a class of backward doubly stochastic differential equations (BDSDEs for short) with general terminal value and general random generator. Those BDSDEs do not involve any forward diffusion processes. By using the techniques of Malliavin calculus, we are able to establish the Lp-Hölder continuity of the solution pair. Then, an implicit numerical scheme for the BDSDE is proposed and the rate of convergence is obtained in the Lp-sense. As a by-product, we obtain an explicit representation of the process Y in the solution pair to a linear BDSDE with random coefficients.

Suggested Citation

  • Hu, Yaozhong & Nualart, David & Song, Xiaoming, 2020. "An implicit numerical scheme for a class of backward doubly stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3295-3324.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:6:p:3295-3324
    DOI: 10.1016/j.spa.2019.09.014
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    References listed on IDEAS

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    1. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    2. N. El Karoui & S. Peng & M. C. Quenez, 1997. "Backward Stochastic Differential Equations in Finance," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 1-71, January.
    3. Bender, Christian & Denk, Robert, 2007. "A forward scheme for backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1793-1812, December.
    4. Imkeller, Peter & Dos Reis, Gonçalo, 2010. "Path regularity and explicit convergence rate for BSDE with truncated quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 348-379, March.
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