具有随机跳和消失扩散项的受控马氏扩散过程出口问题的渐近性
On the asymptotic of exit problems for controlled Markov diffusion processes with random jumps and vanishing diffusion terms
摘要：
本文研究了具有随机跳和消失扩散项的受控Markov扩散过程出口问题的渐近性，其中引入随机跳是为了通过从一种动力学模式切换到另一种动力学模式来改变受控扩散过程的演化.也就是说，取决于状态位置和状态转换信息，受控扩散的动力学在不同的漂移和扩散项之间随机切换。在这里，我们分两步具体研究关于此类受控马尔可夫扩散过程的渐近退出问题：（i）首先，对于每个受控扩散模型，我们寻找一个允许的马尔可夫控制过程，它使相应的无穷小生成元的主特征值最小，且Dirichlet边界条件为零--其中这种容许控制过程还迫使受控扩散过程在给定的有界开放域中保持更长的持续时间。(ii)然后，利用大偏差理论，我们确定了具有随机跳和扩散项消失的受控马氏扩散过程的出口位置和出口时刻的分布类型。此外，在出口时间的渐近结果也使我们能够确定相应的椭圆型偏微分方程系统的狄利克雷问题的极限行为包含一个小的消失参数。
英文摘要：
In this paper, we study the asymptotic of exit problem for controlled Markov diffusion processes with random jumps and vanishing diffusion terms, where the random jumps are introduced in order to modify the evolution of the controlled diffusions by switching from one mode of dynamics to another. That is, depending on the state-position and state-transition information, the dynamics of the controlled diffusions randomly switches between the different drift and diffusion terms. Here, we specifically investigate the asymptotic exit problem concerning such controlled Markov diffusion processes in two steps: (i) First, for each controlled diffusion model, we look for an admissible Markov control process that minimizes the principal eigenvalue for the corresponding infinitesimal generator with zero Dirichlet boundary conditions -- where such an admissible control process also forces the controlled diffusion process to remain in a given bounded open domain for a longer duration. (ii) Then, using large deviations theory, we determine the exit place and the type of distribution at the exit time for the controlled Markov diffusion processes coupled with random jumps and vanishing diffusion terms. Moreover, the asymptotic results at the exit time also allow us to determine the limiting behavior of the Dirichlet problem for the corresponding system of elliptic partial differential equations containing a small vanishing parameter.
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