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Volume 35, Issue 2
Second-Order Numerical Schemes for Decoupled Forward-Backward Stochastic Differential Equations with Jumps

Weidong Zhao, Wei Zhang & Guannan Zhang

DOI: 10.4208/jcm.1612-m2015-0245

J. Comp. Math., 35 (2017), pp. 213-244.

Published online: 2017-04

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  • Abstract

We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics is driven by a $d$-dimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [20, 28] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.

  • Keywords

Decoupled FBSDEs with Lévy jumps, Backward Kolmogorov equation, Nonlinear Feynman-Kac formula, Second-order convergence, Error estimates.

  • AMS Subject Headings

60H35, 60H10, 65C20, 65C30.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wdzhao@sdu.edu.cn (Weidong Zhao)

weizhang0313@bjut.edu.cn (Wei Zhang)

zhangg@ornl.gov (Guannan Zhang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-213, author = {Zhao , WeidongZhang , Wei and Zhang , Guannan}, title = {Second-Order Numerical Schemes for Decoupled Forward-Backward Stochastic Differential Equations with Jumps}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {2}, pages = {213--244}, abstract = {

We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics is driven by a $d$-dimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [20, 28] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1612-m2015-0245}, url = {http://global-sci.org/intro/article_detail/jcm/9770.html} }
TY - JOUR T1 - Second-Order Numerical Schemes for Decoupled Forward-Backward Stochastic Differential Equations with Jumps AU - Zhao , Weidong AU - Zhang , Wei AU - Zhang , Guannan JO - Journal of Computational Mathematics VL - 2 SP - 213 EP - 244 PY - 2017 DA - 2017/04 SN - 35 DO - http://doi.org/10.4208/jcm.1612-m2015-0245 UR - https://global-sci.org/intro/article_detail/jcm/9770.html KW - Decoupled FBSDEs with Lévy jumps, Backward Kolmogorov equation, Nonlinear Feynman-Kac formula, Second-order convergence, Error estimates. AB -

We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics is driven by a $d$-dimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [20, 28] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.

Zhao , WeidongZhang , Wei and Zhang , Guannan. (2017). Second-Order Numerical Schemes for Decoupled Forward-Backward Stochastic Differential Equations with Jumps. Journal of Computational Mathematics. 35 (2). 213-244. doi:10.4208/jcm.1612-m2015-0245
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