SIAM Journal on Control and Optimization, Volume 58, Issue 1, Page 393-424, January 2020.
In this paper, we investigate the solvability of matrix valued backward stochastic Riccati equations with jumps (BSREJ), which are associated with a stochastic linear quadratic (SLQ) optimal control problem with random coefficients and driven by both Brownian motion and a Poisson process. By the dynamic programming principle, Doob--Meyer decomposition, and inverse flow technique, the existence and uniqueness of the solution for the BSREJ are established. The difficulties addressed in this issue not only are brought from the high nonlinearity of the generator of the BSREJ like the case driven only by Brownian motion, but also from that (i) the inverse flow of the controlled linear stochastic differential equation driven by Poisson process may not exist without additional technical conditions, and (ii) showing that the inverse matrix term involving a jump process in the generator is well-defined. Utilizing the structure of the optimal control problem, we overcome these difficulties and establish the existence of the solution. In addition, we show the construction of the optimal feedback control with the help of the Riccati equation and the relation between the solution of the Riccati equation and the value function of the SLQ problem, which also implies the uniqueness of BSREJ.

中文翻译：

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带跳的倒向随机Riccati方程与带有跳和随机系数的随机线性二次最优控制

SIAM控制与优化杂志，第58卷，第1期，第393-424页，2020年1月。
在本文中，我们研究具有跳变的矩阵值后向随机Riccati方程（BSREJ）的可解性，该方程与具有随机系数的随机线性二次（SLQ）最优控制问题相关，并且受布朗运动和泊松过程的驱动。根据动态规划原理，Doob-Meyer分解和逆流技术，确定了BSREJ解的存在性和唯一性。这个问题解决的困难不仅是由BSREJ发生器的高非线性引起的，例如仅由布朗运动驱动的情况，而且还来自（i）由泊松过程驱动的受控线性随机微分方程的逆流没有附加的技术条件可能不会存在，（ii）表明涉及发生器中跳跃过程的逆矩阵项是定义明确的。利用最优控制问题的结构，我们克服了这些困难并建立了解决方案的存在。此外，我们借助Riccati方程展示了最优反馈控制的构造，以及Riccati方程的解与SLQ问题的值函数之间的关系，这也暗示了BSREJ的独特性。