Comparison and converse comparison theorems for backward stochastic differential equations with Markov chain noise

TL;DR

This paper extends comparison and converse comparison theorems to one-dimensional backward stochastic differential equations driven by Markov chain noise, introducing a new nonlinear expectation framework.

Contribution

It generalizes existing theorems for BSDEs with Markov chain noise and establishes a converse comparison theorem using a novel nonlinear expectation called $f$-expectation.

Findings

Extended comparison theorems under simplified hypotheses

Established a converse comparison theorem for BSDEs with Markov chain noise

Introduced and analyzed properties of $f$-expectation

Abstract

Comparison and converse comparison theorems are important parts of the research on backward stochastic differential equations. In this paper, we obtain comparison results for one dimensional backward stochastic differential equations with Markov chain noise, extending and generalizing previous work under natural and simplified hypotheses, and establish a converse comparison theorem for the same type of equation after giving the definition and properties of a type of nonlinear expectation: $f$-expectation.