A Finite Horizon Optimal Stochastic Impulse Control Problem with A Decision Lag
A Finite Horizon Optimal Stochastic Impulse Control Problem with A Decision Lag
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发表时间：
2020-05
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通讯作者：
Chang Li;J. Yong
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作者：
Chang Li;J. Yong
本文研究了有限时间内具有决策滞后的最优随机脉冲控制问题，即在一个脉冲产生后，在下一个脉冲产生之前必须经过一定的时间单位.证明了值函数的连续性。建立了一个合适的动态规划原理，它考虑了状态过程对时间的依赖性。导出了相应的Hamilton-Jacobi-Bellman（HJB）方程，该方程体现了问题的一些特殊性质。该最优脉冲控制问题的值函数被刻画为相应HJB方程的唯一粘性解。在给定值函数的条件下，构造了最优脉冲控制。此外，极限情况下的等待时间接近0 $进行了讨论。
This paper studies an optimal stochastic impulse control problem in a finite horizon with a decision lag, by which we mean that after an impulse is made, a fixed number units of time has to be elapsed before the next impulse is allowed to be made. The continuity of the value function is proved. A suitable version of dynamic programming principle is established, which takes into account the dependence of state process on the elapsed time. The corresponding Hamilton-Jacobi-Bellman (HJB) equation is derived, which exhibit some special feature of the problem. The value function of this optimal impulse control problem is characterized as the unique viscosity solution to the corresponding HJB equation. An optimal impulse control is constructed provided the value function is given. Moreover, a limiting case with the waiting time approaching $0$ is discussed.