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 analysis 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.

我们研究了具有控制过程的二次惩罚项的性能指标的一般扩散的随机最优控制的反问题。在系统动力学、成本函数和最优控制过程的温和条件下，我们证明我们的反问题使用随机最大值原理是适定的。然后，利用适定性，我们将逆问题简化为具有唯一解的值函数所涉及的随机变量期望的一些求根问题。基于这一结果，我们通过用观察到的最优控制过程和相应状态过程的算术平均值代替上述期望，提出了一种反问题的数值方法。Hamilton-Jacobi-Bellman 方程的数值分析的最新进展使所提出的方法可用于多维情况。特别是，在 Hamilton-Jacobi-Bellman 方程的基于核的配置方法的帮助下，即使在值函数的显式形式不可用时，我们的反问题方法仍然可以很好地工作。多项数值实验表明，该数值方法可以高精度地恢复未知惩罚参数。