Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions

TL;DR

This paper develops a theory of nonlinear expectations based on backward stochastic differential equations (BSDEs) on finite state Markov chains, extending the framework beyond continuous diffusions and exploring applications to risk measures and no-arbitrage conditions.

Contribution

It introduces a new framework for nonlinear expectations on Markov chains using BSDEs, including comparison theorems and analysis of arbitrage and risk measures.

Findings

Established properties of nonlinear expectations on Markov chains.

Proved comparison theorems for BSDE solutions.

Analyzed no-arbitrage conditions in scalar cases.

Abstract

Most previous contributions to BSDEs, and the related theories of nonlinear expectation and dynamic risk measures, have been in the framework of continuous time diffusions or jump diffusions. Using solutions of BSDEs on spaces related to finite state, continuous time Markov chains, we develop a theory of nonlinear expectations in the spirit of [Dynamically consistent nonlinear evaluations and expectations (2005) Shandong Univ.]. We prove basic properties of these expectations and show their applications to dynamic risk measures on such spaces. In particular, we prove comparison theorems for scalar and vector valued solutions to BSDEs, and discuss arbitrage and risk measures in the scalar case.