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Stemmed from the derivation of the optimal control to a stochasticlinear-quadratic control problem with Markov jumps, we study one kind ofbackward stochastic differential equations (BSDEs) that the generator f isaffected by a Markovian switching. Then, the case that the Markov chain isinvolved in a large state space is considered. Following the classicalapproach, a hierarchical approach is adopted to reduce the complexity and asingularly perturbed Markov chain is involved. We will study the asymptoticproperty of BSDE with the singularly perturbed Markov chain. At last, as anapplication of our theoretical result, we show the homogenization of one systemof partial differential equations (PDEs) with a singularly perturbed Markovchain.

Backward Stochastic Differential Equations with Markov Chains and The Application: Homogenization of PDEs System