论文标题
随机线性界面控制，跳跃和策略在随机地平线上开关
Stochastic linear-quadratic control with a jump and regime switching on a random horizon
论文作者
论文摘要
在本文中，我们研究了一个随机系数的随机线性界面控制问题，并且制度在地平线$ [0，t \wedgeτ] $上开关，其中$τ$是给定的基础状态过程的随机跳跃时间，$ t $是常数。我们通过求解随机Riccati方程（SRE）系统的显式最佳状态反馈控制和明确的最佳成本值，并以$ [0，t \ wedgeτ] $上的跳跃。通过过滤扩大理论所产生的分解方法，我们以另一种仅涉及Brownian过滤在确定性的范围$ [0，T] $的SRES方面来表达SRES系统的解决方案。解决后一个系统是本文的关键理论贡献，我们为三种不同的情况确定了这一点，其中一种在文献中似乎是新的。然后将这些结果应用于研究均匀参数的平均变异对冲问题，这些参数均取决于布朗运动和马尔可夫链。最佳投资组合和最佳值以封闭形式呈现，借助线性向后的随机微分方程系统，除了带有跳跃的SRE之外，具有跳跃和无界系数。
In this paper, we study a stochastic linear-quadratic control problem with random coefficients and regime switching on a horizon $[0,T\wedgeτ]$, where $τ$ is a given random jump time for the underlying state process and $T$ is a constant. We obtain an explicit optimal state feedback control and explicit optimal cost value by solving a system of stochastic Riccati equations (SREs) with jumps on $[0,T\wedgeτ]$. By the decomposition approach stemming from filtration enlargement theory, we express the solution of the system of SREs with jumps in terms of another system of SREs involving only Brownian filtration on the deterministic horizon $[0,T]$. Solving the latter system is the key theoretical contribution of this paper and we establish this for three different cases, one of which seems to be new in the literature. These results are then applied to study a mean-variance hedging problem with random parameters that depend on both Brownian motion and Markov chain. The optimal portfolio and optimal value are presented in closed forms with the aid of a system of linear backward stochastic differential equations with jumps and unbounded coefficients in addition to the SREs with jumps.
