Backward stochastic differential equations with Markov switching driven by Brownian motion and Poisson random measure

Abstract

This paper is concerned with backward stochastic differential equations with Markov switching driven by Brownian motion and Poisson random measure. The motivation is a constrained stochastic Riccati equation derived from a stochastic linear quadratic optimal control problem with both Poisson and Markovian jumps. The existence and uniqueness of an adapted solution under global Lipschitz condition on the generator is obtained. The continuous dependence of the solution on parameters is proved. Two comparison theorems are also derived by a generalized Girsanov transformation theorem.

Keywords::

• backward stochastic differential equation
• Poisson random measure
• Markov switching
• comparison theorem

AMS Subject Classification::

• 60H10
• 60J28
• 93E20

Acknowledgements

The authors would like to thank the anonymous referees for many constructive comments that led to improved versions of this paper. And many thanks are devoted to Dr Huaibin Tang for her helpful suggestion during the revising process.

Additional information

Funding

Research of Jingtao Shi is supported by National Natural Sciences Foundations of China [grant number 11301011], [grant number 11201264]; Shandong Province [grant number ZR2011AQ012]. Research of Zhen Wu is supported by National Natural Sciences Foundations of China [grant number 11921101], [grant number 61174092]; National Natural Sciences Fund for Distinguished Young Scholars of China [grant number 11125102].