In this paper, we prove the moderate deviations principle (MDP) for a general system of slow-fast dynamics. We provide a unified approach, based on weak convergence ideas and stochastic control arguments, that cover both the averaging and the homogenization regimes. We allow the coefficients to be in the whole space and not just the torus and allow the noises driving the slow and fast processes to be correlated arbitrarily. Similar to the large deviation case, the methodology that we follow allows construction of provably efficient Monte Carlo methods for rare events that fall into the moderate deviations regime.For convenience, we refer to the state space of Y ε as Y. The parameter ε ≪ 1 represents the strength of the noise while δ ≪ 1 is the time-scale separation parameter. W t and B t are independent m-dimensional Brownian motions.In (1), X ε is the slow motion and Y ε is the fast motion. Depending on the order in which ε, δ go to zero, we get different behavior, and in particular

No abstract

We consider systems of slow-fast diffusions with small noise in the slow component. We construct provably logarithmic asymptotically optimal importance schemes for the estimation of rare events based on the moderate deviations principle. Using the subsolution approach we construct schemes and identify conditions under which the schemes will be asymptotically optimal. Moderate deviations based importance sampling offers a viable alternative to large deviations importance sampling when the events are not too rare. In particular, in many cases of interest one can indeed construct the required change of measure in closed form, a task which is more complicated using the large deviations based importance sampling, especially when it comes to multiscale dynamically evolving processes. The presence of multiple scales and the fact that we do not make any periodicity assumptions for the coefficients driving the processes, complicates the design and the analysis of efficient importance sampling schemes. Simulation studies illustrate the theory.

No abstract