ABSTRACT

We consider a stochastic optimal control problem with state variable dynamics described by a stochastic differential equation of diffusive type modulated by a semi-Markov process with a finite state space. The time horizon is both deterministic and finite. Within such setup, we provide a detailed proof of the dynamic programming principle and use it to characterize the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We illustrate our results with an application to Mathematical Finance: the generalization of Merton's optimal consumption-investment problem to financial markets with semi-Markov switching.

摘要

我们考虑具有状态变量动力学的随机最优控制问题，该问题由具有有限状态空间的半马尔可夫过程调制的扩散型随机微分方程描述。时间范围既是确定性的又是有限的。在这样的设置中，我们提供了动态规划原理的详细证明，并使用它来将值函数表征为相应 Hamilton-Jacobi-Bellman 方程的粘度解。我们用数学金融的应用来说明我们的结果：将默顿的最优消费投资问题推广到具有半马尔可夫转换的金融市场。