General linear quadratic optimal stochastic control problem driven by a Brownian motion and a Poisson random martingale measure with random coefficients

Qingxin Meng*

*Corresponding author for this work

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

In this article, we consider a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices. By the duality technique, the dual characterization of the optimal control is derived by the optimality system (so-called stochastic Hamilton system), which turns out to be a linear fully coupled forward-backward stochastic differential equation with jumps. Using a decoupling technique, the connection between the stochastic Hamilton system and the associated Riccati equation is established. As a result, the state feedback representation is obtained for the optimal control. As the coefficients for the LQ problem are random, here, the associated Riccati equation is a highly nonlinear backward stochastic differential equation (BSDE) with jumps, where the generator depends on the unknown variables K, L, and H in a quadratic way (see (5.9) herein). For the case where the generator is bounded and is linearly dependent on the unknown martingale terms L and H, the existence and uniqueness of the solution for the associated Riccati equation are established by Bellman's principle of quasi-linearization.

Original languageEnglish
Pages (from-to)88-109
Number of pages22
JournalStochastic Analysis and Applications
Volume32
Issue number1
DOIs
Publication statusPublished - 2014
Externally publishedYes

Keywords

  • Backward stochastic Riccati equation
  • Dynamic programming
  • Itô-Ventzell formula
  • Linear quadratic optimal stochastic control
  • Poisson random martingale measure
  • Stochastic Hamilton system

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