On Anticipated backward stochastic differential equations with Markov chain noise

TL;DR

This paper extends the theory of anticipated backward stochastic differential equations driven by Markov chain noise by providing solution estimates, exploring duality with delayed equations, and establishing a new comparison theorem.

Contribution

It introduces new estimates for solutions, investigates duality with stochastic delayed equations, and presents a novel comparison theorem based solely on the drivers.

Findings

Provided bounds for solutions of anticipated BSDEs with Markov chain noise.

Established duality between these BSDEs and stochastic delayed equations.

Derived a new comparison theorem depending only on the drivers.

Abstract

In 2013, Lu and Ren \cite {luren} considered anticipated backward stochastic differential equations driven by finite state, continuous time Markov chain noise and established the existence and uniqueness of the solutions of these equations and a scalar comparison theorem. In this paper, we provide an estimate for their solutions and study the duality between these equations and stochastic differential delayed equations with Markov chain noise. Finally we derive another comparison theorem for these solutions depending only on the two drivers.