论文标题
逆随机最佳对照
Inverse stochastic optimal controls
论文作者
论文摘要
我们研究了对一般扩散的随机最佳控制的反问题，其性能指数具有控制过程的二次惩罚项。在系统动力学，成本函数和最佳控制过程的轻度条件下，我们表明我们的逆问题使用随机最大原理进行了良好的解决方案。然后，凭借适当的性质，我们将逆问题减少到某个根发现的问题问题，即对价值函数涉及的随机变量的期望，该变量具有独特的解决方案。基于此结果，我们通过用观察到的最佳控制过程和相应状态过程的算术平均值替换上述期望，为我们的反问题提出了一种数值方法。汉密尔顿 - 雅各比 - 贝尔曼方程的数值分析的最新进展使得为多维案例提供了建议的方法。特别是，借助于汉密尔顿 - 雅各比 - 贝尔曼方程的基于内核的搭配方法，即使不可用的值函数的明确形式，我们的反问题方法仍然可以很好地工作。几个数值实验表明，数值方法以高精度恢复了未知的惩罚参数。
We study an inverse problem of the stochastic optimal control of general diffusions with performance index having the quadratic penalty term of the control process. Under mild conditions on the system dynamics, the cost functions, and the optimal control process, we show that our inverse problem is well-posed using a stochastic maximum principle. Then, with the well-posedness, we reduce the inverse problem to some root finding problem of the expectation of a random variable involved with the value function, which has a unique solution. Based on this result, we propose a numerical method for our inverse problem by replacing the expectation above with arithmetic mean of observed optimal control processes and the corresponding state processes. The recent progress of numerical analyses of Hamilton-Jacobi-Bellman equations enables the proposed method to be implementable for multi-dimensional cases. In particular, with the help of the kernel-based collocation method for Hamilton-Jacobi-Bellman equations, our method for the inverse problems still works well even when an explicit form of the value function is unavailable. Several numerical experiments show that the numerical method recovers the unknown penalty parameter with high accuracy.
