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Numerical methods for computing the solutions of Markov backward stochasticdifferential equations (BSDEs) driven by continuous-time Markov chains (CTMCs)are explored. The main contributions of this paper are as follows: (1) weobserve that Euler-Maruyama temporal discretization methods for solving MarkovBSDEs driven by CTMCs are equivalent to exponential integrators for solving theassociated systems of ordinary differential equations (ODEs); (2) we introducemulti-stage Euler-Maruyama methods for effectively solving "stiff" Markov BSDEsdriven by CTMCs; these BSDEs typically arise from the spatial discretization ofMarkov BSDEs driven by Brownian motion; (3) we propose a multilevel spatialdiscretization method on sparse grids that efficiently approximateshigh-dimensional Markov BSDEs driven by Brownian motion with a combination ofmultiple Markov BSDEs driven by CTMCs on grids with different resolutions. Wealso illustrate the effectiveness of the presented methods with a number ofnumerical experiments in which we treat nonlinear BSDEs arising from optionpricing problems in finance.

Multi-stage Euler-Maruyama methods for backward stochastic differential equations driven by continuous-time Markov chains