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驱动的马尔可夫BSDE的欧拉-丸山时间离散方法等价于求解相关常微分方程组（ODEs）的指数积分器；（2）引入多阶段欧拉-丸山方法，有效求解由CTMC驱动的“刚性”马尔可夫BSDE，这类BSDE通常源于布朗运动驱动的马尔可夫BSDE的空间离散化；（3）提出一种基于稀疏网格的多层空间离散化方法，通过结合不同分辨率网格上的多个CTMC驱动马尔可夫BSDE，高效逼近高维布朗运动驱动的马尔可夫BSDE。同时，通过一系列数值实验（包括金融期权定价问题中出现的非线性BSDE）验证了所提方法的有效性。