STOCHASTIC DIFFERENTIAL EQUATIONS AND STOCHASTIC LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM WITH LEVY PROCESSES

在这篇论文，作者首先作为噪音来源与 L é vy 过程学习二种随机的微分方程(SDE ) 。基于这些 SDE 和 Lévy 进程驾驶的多维的向后的随机的微分方程( BSDE )的解决方案的存在和唯一，作者继续学习一随机线性二次( LQ )有 Lévy 进程的最佳的控制问题，在状态和控制的费用 weighting 矩阵被允许不定的地方。包含平等和不平等限制的一个种新随机的 Riccati 方程从方形的结束的想法被导出，它的解决之可能性被证明为 well-posedness 和能具有州的反馈或 LQ 问题的开环的形式的最佳的控制的存在足够。而且，作者获得一些特殊情况的 Riccati 方程的答案的存在和唯一。最后，二个例子被举说明这些理论结果。

In this paper, tile authors first study two kinds of stochastic differential equations （SDEs） with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations （BSDEs） driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic （LQ） optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.