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In this paper, we consider an importance sampling problem for a certainrare-event simulations involving the behavior of a diffusion process pertainingto a chain of distributed systems with random perturbations. We also assumethat the distributed system formed by $n$-subsystems -- in which a small randomperturbation enters in the first subsystem and then subsequently transmitted tothe other subsystems -- satisfies an appropriate H\"{o}rmander condition. Herewe provide an efficient importance sampling estimator, with an exponentialvariance decay rate, for the asymptotics of the probabilities of the rareevents involving such a diffusion process that also ensures a minimum relativeestimation error in the small noise limit. The framework for such an analysisbasically relies on the connection between the probability theory of largedeviations and the values functions for a family of stochastic control problemsassociated with the underlying distributed system, where such a connectionprovides a computational paradigm -- based on an exponentially-tilted biasingdistribution -- for constructing efficient importance sampling estimators forthe rare-event simulation. Moreover, as a by-product, the framework also allowsus to derive a family of Hamilton-Jacobi-Bellman for which we also provide asolvability condition for the corresponding optimal control problem.

On the rare-event simulations of diffusion processes pertaining to a chain of distributed systems with small random perturbations