Numerical methods for computing the solutions of Markov backward stochastic differential equations (BSDEs) driven by continuous-time Markov chains (CTMCs) are explored. The main contributions of this paper are as follows: (1) we observe that Euler-Maruyama temporal discretization methods for solving Markov BSDEs driven by CTMCs are equivalent to exponential integrators for solving the associated systems of ordinary differential equations (ODEs); (2) we introduce multi-stage Euler-Maruyama methods for effectively solving "stiff" Markov BSDEs driven by CTMCs; these BSDEs typically arise from the spatial discretization of Markov BSDEs driven by Brownian motion; (3) we propose a multilevel spatial discretization method on sparse grids that efficiently approximates high-dimensional Markov BSDEs driven by Brownian motion with a combination of multiple Markov BSDEs driven by CTMCs on grids with different resolutions. We also illustrate the effectiveness of the presented methods with a number of numerical experiments in which we treat nonlinear BSDEs arising from option pricing problems in finance.

翻译：本文探讨了由连续时间马尔可夫链(CTMC)驱动的马尔可夫倒向随机微分方程(BSDE)的数值求解方法。主要贡献如下:(1)发现求解CTMC驱动马尔可夫BSDE的欧拉-丸山时间离散化方法等价于求解常微分方程(ODE)系统的指数积分器;(2)提出用于有效求解由CTMC驱动的"刚性"马尔可夫BSDE的多阶段欧拉-丸山方法,此类BSDE通常源于布朗运动驱动马尔可夫BSDE的空间离散化;(3)提出基于稀疏网格的多层空间离散化方法,该方法通过组合多个不同网格分辨率下CTMC驱动的马尔可夫BSDE,有效逼近高维布朗运动驱动的马尔可夫BSDE。我们通过一系列数值实验验证了所提方法的有效性,这些实验涉及金融期权定价问题中出现的非线性BSDE。