论文标题
随机系数和马尔可夫制度切换系统随机线性二次最佳控制问题
Stochastic Linear Quadratic Optimal Control Problems with Random Coefficients and Markovian Regime Switching System
论文作者
论文摘要
本文彻底研究了马尔可夫政权切换系统的随机线性季度最佳控制问题，在该系统中，状态方程的系数和成本函数的加权矩阵是随机的。我们在均匀的凸条件下证明了随机riccati方程的溶解度，并使用相应的随机Riccati方程的唯一可溶解性获得了开环最佳控制的闭环表示。此外，通过将ITô的公式应用于跳跃，我们可以在希尔伯特空间上获得成本功能的代表，这些成本功能为某些前向后的随机微分方程的适应解决方案。我们表明，开环最佳控制的必要条件是成本功能的凸度，开环最佳控制的足够条件是成本功能的均匀凸度。此外，我们研究了随机线性界面最佳控制问题的随机值流的性能。最后，作为一个应用程序，我们提出了连续的均值变化投资组合选择问题，并证明了其独特的解决性。
This paper thoroughly investigates stochastic linear-quadratic optimal control problems with the Markovian regime switching system, where the coefficients of the state equation and the weighting matrices of the cost functional are random. We prove the solvability of the stochastic Riccati equation under the uniform convexity condition and obtain the closed-loop representation of the open-loop optimal control using the unique solvability of the corresponding stochastic Riccati equation. Moreover, by applying Itô's formula with jumps, we get a representation of the cost functional on a Hilbert space, characterized as the adapted solutions of some forward-backward stochastic differential equations. We show that the necessary condition of the open-loop optimal control is the convexity of the cost functional, and the sufficient condition of the open-loop optimal control is the uniform convexity of the cost functional. In addition, we study the properties of the stochastic value flow of the stochastic linear-quadratic optimal control problem. Finally, as an application, we present a continuous-time mean-variance portfolio selection problem and prove its unique solvability.
