Abstract
We study numerical algorithms for reflected anticipated backward stochastic differential equations (RABSDEs) driven by a Brownian motion and a mutually independent martingale in a defaultable setting. The generator of a RABSDE includes the present and future values of the solution. We introduce two main algorithms, a discrete penalization scheme and a discrete reflected scheme basing on a random walk approximation of the Brownian motion as well as a discrete approximation of the default martingale, and we study these two methods in both the implicit and explicit versions respectively. We give the convergence results of the algorithms, provide a numerical example and an application in American game options in order to illustrate the performance of the algorithms.
Highlights
The backward stochastic differential equation (BSDE) theory plays a significant role in financial modeling
We focus on the study of reflected anticipated BSDE with two obstacles and default risk
(Y, Z, U, K +, K − ) := (Yt, Zt, Ut, Kt+, Kt− )0≤t≤T +δ is a solution for reflected anticipated backward stochastic differential equations (RABSDEs) with the generator f, the terminal value ξ T, the anticipated processes ξ, the anticipated time δ (δ > 0 is a constant), and the obstacles L and V, such that
Summary
The backward stochastic differential equation (BSDE) theory plays a significant role in financial modeling. Pardoux and Peng (1990) studied the general nonlinear BSDEs under a smooth square integrability assumptions on the coefficient and the terminal value, and a Lipschitz condition for the generator f . Cvitanic and Karatzas (1996) first studied reflected BSDEs with continuous lower obstacle and continuous upper obstacle under the smooth square integrability assumption and Lipschitz condition. We focus on the study of reflected anticipated BSDE with two obstacles and default risk. Later Dumitrescu and Labart (2016) extended to RBSDE with two obstacles driven by Brownian motion and an independent compensated Poisson process.
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