## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 21 (2016), paper no. 25, 10 pp.

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

Zhe Yang, Dimbinirina Ramarimbahoaka, and Robert J. Elliott

#### 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.

#### Article information

**Source**

Electron. Commun. Probab., Volume 21 (2016), paper no. 25, 10 pp.

**Dates**

Received: 1 August 2014

Accepted: 12 February 2016

First available in Project Euclid: 10 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1457617916

**Digital Object Identifier**

doi:10.1214/16-ECP4102

**Mathematical Reviews number (MathSciNet)**

MR3485394

**Zentralblatt MATH identifier**

1338.60164

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60]

**Keywords**

BSDEs comparison theorem converse comparison Markov chain

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Yang, Zhe; Ramarimbahoaka, Dimbinirina; Elliott, Robert J. Comparison and converse comparison theorems for backward stochastic differential equations with Markov chain noise. Electron. Commun. Probab. 21 (2016), paper no. 25, 10 pp. doi:10.1214/16-ECP4102. https://projecteuclid.org/euclid.ecp/1457617916