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

TL;DR

This paper investigates backward stochastic differential equations influenced by Markov chains with large state spaces, analyzing their asymptotic behavior and applying findings to homogenize PDE systems with singularly perturbed Markov chains.

Contribution

It introduces a hierarchical approach to analyze BSDEs with Markov chains in large state spaces and demonstrates their application in PDE homogenization.

Findings

Asymptotic properties of BSDEs with singularly perturbed Markov chains are established.

Theoretical results enable homogenization of PDE systems influenced by Markov chains.

A hierarchical reduction method effectively manages complexity in large state space Markov chains.

Abstract

Stemmed from the derivation of the optimal control to a stochastic linear-quadratic control problem with Markov jumps, we study one kind of backward stochastic differential equations (BSDEs) that the generator f is affected by a Markovian switching. Then, the case that the Markov chain is involved in a large state space is considered. Following the classical approach, a hierarchical approach is adopted to reduce the complexity and a singularly perturbed Markov chain is involved. We will study the asymptotic property of BSDE with the singularly perturbed Markov chain. At last, as an application of our theoretical result, we show the homogenization of one system of partial differential equations (PDEs) with a singularly perturbed Markov chain.