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

Zhe Yang,Robert J Elliott,Dimbinirina Ramarimbahoaka

doi:10.1214/16-ecp4102

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, adapting previous results under simplified hypotheses. We introduce a type of nonlinear expectation, the $f$-expectation, which is an interpretation of the solution to a BSDE, and use it to establish a converse comparison theorem for the same type of equations as those in the comparison results.

Highlights

IntroductionIn 1990 Pardoux and Peng [19] considered general backward stochastic differential equations (BSDEs for short) of the following form: T
In 1990 Pardoux and Peng [19] considered general backward stochastic differential equations (BSDEs for short) of the following form: TYt = ξ + g(s, Ys, Zs)ds − ZsdBs, t t t ∈ [0, T ].Here B is a Brownian Motion and g is the driver, or drift, of the above BSDE
We introduce a type of nonlinear expectation, the f -expectation, which is an interpretation of the solution to a BSDE, and use it to establish a converse comparison theorem for the same type of equations as those in the comparison results

Summary

Introduction

In 1990 Pardoux and Peng [19] considered general backward stochastic differential equations (BSDEs for short) of the following form: T. In 2012, van der Hoek and Elliott [25] introduced a market model where uncertainties are modeled by a finite state Markov chain, instead of Brownian motion or related jump diffusions, which are often used when pricing financial derivatives. Cohen and Elliott [5] and [6] gave some comparison results for multidimensional BSDEs in the Markov Chain model under conditions involving the two drivers and the two solutions. If we consider two one-dimensional BSDEs driven by the Markov chain, we extend the comparison result to a situation involving conditions only on the two drivers. Our result in the Markov chain framework needs less conditions on the drivers compared to those in Crepey and Matoussi [8] which are suitable for more general dynamics.

The model and some preliminary results

A comparison theorem for one-dimensional BSDEs with Markov chain noise

Conclusion