Abstract
This paper is concerned with the solvability of a new kind of backward stochastic differential equations whose generator f is affected by a finite-state Markov chain. We also present the asymptotic property of backward stochastic differential equations involving a singularly perturbed Markov chain with weak and strong interactions and then apply this result to the homogenization of a system of semilinear parabolic partial differential equations.MSC:60H10, 35R60.
Highlights
Pioneered by the works of Pardoux and Peng [ ] and Duffie and Epstein [ ] about the nonlinear backward stochastic differential equations (BSDEs) driven by a Brownian motion, BSDEs have been extensively studied because of their deep connections with mathematical finance, stochastic control, and partial differential equations (PDEs), such as [, ], etc
Many researchers have been devoted to more general cases in the framework of continuous time diffusions or jump diffusions, such as BSDE driven by a Lévy process, BSDE with respect to both a Brownian motion and a Poisson random measure
Combining with the asymptotic property of the singularly perturbed Markov chain αε, we show that the solution sequence {(Ytε, t converges weakly with the limit formed by the solution of a simpler BSDE which involves the limit aggregated Markov chain
Summary
Pioneered by the works of Pardoux and Peng [ ] and Duffie and Epstein [ ] about the nonlinear backward stochastic differential equations (BSDEs) driven by a Brownian motion, BSDEs have been extensively studied because of their deep connections with mathematical finance, stochastic control, and partial differential equations (PDEs), such as [ , ], etc. Consider the following BSDE in the probability space ( , F, P): Yt = ξ + f (s, Ys, Zs, αs) ds – Zs dBs. When studying the solvability for BSDE ( ) with the Markov chain α, since {FtB ∨ Ftα,T } ≤t≤T is not a filtration, the conventional approach [ , ] to use the Itô representation theorem and contraction mapping cannot be applied straightforwardly. When studying the solvability for BSDE ( ) with the Markov chain α, since {FtB ∨ Ftα,T } ≤t≤T is not a filtration, the conventional approach [ , ] to use the Itô representation theorem and contraction mapping cannot be applied straightforwardly To tackle this problem, inspired by the method dealing with backward doubly stochastic differential equations [ ], we construct an enlarged filtration generated by the Brownian motion B and the Markov chain α and propose a corresponding extended Itô representation theorem. Together with the form of BSDE ( ) and the Burkholder-Davis-Gundy inequality, we can conclude that Y is continuous and Y ∈ SF t ( , T; Rk)
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