Achievement

We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics are 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 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. As a stochastic approach, the proposed method completely avoids the challenge of iteratively solving non-sparse linear systems, arising from the nature of nonlocality. This allows for embarrassingly parallel implementation and also enables adaptive approximation techniques to be incorporated in a straightforward fashion. Moreover, our method recovers the convergence rates of classic deterministic approaches (e.g. finite element or collocation methods), due to the use of high-order temporal and spatial discretization schemes. In addition, our approach can handle a broad class of problems with general inhomogenous forcing terms as long as they are globally Lipchitz continuous. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.

Publication:

• G. Zhang, W. Zhao, M. Gunzburger and C. Webster, Numerical methods for a class of nonlocal diffusion problems with the use of backward SDEs, Computers & Mathematics with Applications, 71 (2016), pp. 2479-2496.

• W. Zhao, W. Zhang and G. Zhang, Second-order numerical schemes for decoupled forward-backward stochastic differential equations with jumps, Journal of Computational Mathematics, 35 (2017), pp. 213-244.