Martingale Decomposition and BSDE on Time Scales

TL;DR

This paper introduces a unified framework for backward stochastic equations on time scales, generalizing differential and difference equations, and proves their existence and uniqueness.

Contribution

It develops a martingale decomposition and establishes backward stochastic dynamic equations on time scales, unifying differential and difference equations.

Findings

Established existence and uniqueness of BS$ abla$E

Unified backward stochastic equations on time scales

Generalized previous differential and difference results

Abstract

In this paper, we present martingale decomposition on time scales. We establish the related backward stochastic dynamic equations on time scales (this paper BS$\nabla$E for short, concerning $\nabla$-integral on time scales) which unify backward stochastic differential equations and backward stochastic difference equations. We prove the existence and uniqueness theorem of BS$\nabla$E. This work can be considered as a unification and a generalization of similar results in backward stochastic difference equations and backward stochastic differential equations.