The Filtering Equations of Forward-Backward Stochastic Systems with Random Jumps and Applications to Partial Information Stochastic Optimal Control

Abstract

In this article, we consider a filtering problem for forward-backward stochastic systems that are driven by Brownian motions and Poisson processes. This kind of filtering problem arises from the study of partially observable stochastic linear-quadratic control problems. Combining forward-backward stochastic differential equation theory with certain classical filtering techniques, the desired filtering equation is established. To illustrate the filtering theory, the theoretical result is applied to solve a partially observable linear-quadratic control problem, where an explicit observable optimal control is determined by the optimal filtering estimation.

Keywords:

• Forward-backward stochastic differential equation
• Optimal control
• Random jump
• Stochastic filtering

Mathematics Subject Classification:

• 60H10
• 93E20
• 93E11

This project is supported by the National Nature Science Foundation of China (10926098, 11001156, 11071144), the Nature Science Foundation of Shandong Province (ZR2009AQ017), and Independent Innovation Foundation of Shandong University (IIFSDU), China.

Both authors thank the anonymous referee for the careful reading and helpful comments, as well as the editor for the efficient processing of this article. Also, the authors thank Professor Z. Wu for his kind suggestions.