论文标题
关于与具有小的随机扰动的分布式系统链的扩散过程的罕见事实模拟
On the rare-event simulations of diffusion processes pertaining to a chain of distributed systems with small random perturbations
论文作者
论文摘要
在本文中，我们考虑了某些稀有事实模拟的重要性抽样问题，该模拟涉及与具有随机扰动的分布式系统链有关的扩散过程的行为。我们还假设由$ n $ -subsystems形成的分布式系统 - 其中一个小的随机扰动进入了第一个子系统，然后随后传输到其他子系统 - 满足适当的Hörmander条件。在这里，我们提供了有效的重要性采样估计器，并具有指数差异率的有效性抽样估计量，涉及涉及这种扩散过程的稀有事件的概率的渐近性，该概率也确保了小噪声限制中的最小相对估计误差。这种分析的框架基本上依赖于大偏差的概率理论与与基础分布式系统相关的随机控制问题家族的概率理论之间的联系，在这种情况下，这种连接提供了基于指数倾斜的偏置偏置分布的计算范式，用于构建有效的质量采样估算器的稀有式估算器，以实现稀有式启动式的模拟。此外，作为副产品，该框架还使我们能够得出一个汉密尔顿 - 雅各比 - 贝尔曼（Hamilton-Jacobi-Bellman）的家族，我们还为相应的最佳控制问题提供了解决条件。
In this paper, we consider an importance sampling problem for a certain rare-event simulations involving the behavior of a diffusion process pertaining to a chain of distributed systems with random perturbations. We also assume that the distributed system formed by $n$-subsystems -- in which a small random perturbation enters in the first subsystem and then subsequently transmitted to the other subsystems -- satisfies an appropriate Hörmander condition. Here we provide an efficient importance sampling estimator, with an exponential variance decay rate, for the asymptotics of the probabilities of the rare events involving such a diffusion process that also ensures a minimum relative estimation error in the small noise limit. The framework for such an analysis basically relies on the connection between the probability theory of large deviations and the values functions for a family of stochastic control problems associated with the underlying distributed system, where such a connection provides a computational paradigm -- based on an exponentially-tilted biasing distribution -- for constructing efficient importance sampling estimators for the rare-event simulation. Moreover, as a by-product, the framework also allows us to derive a family of Hamilton-Jacobi-Bellman for which we also provide a solvability condition for the corresponding optimal control problem.
