Large Deviations and Importance Sampling for Systems of Slow-Fast Motion

Abstract

In this paper we develop the large deviations principle and a rigorous mathematical framework for asymptotically efficient importance sampling schemes for general, fully dependent systems of stochastic differential equations of slow and fast motion with small noise in the slow component. We assume periodicity with respect to the fast component. Depending on the interaction of the fast scale with the smallness of the noise, we get different behavior. We examine how one range of interaction differs from the other one both for the large deviations and for the importance sampling. We use the large deviations results to identify asymptotically optimal importance sampling schemes in each case. Standard Monte Carlo schemes perform poorly in the small noise limit. In the presence of multiscale aspects one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. It turns out that one has to consider the so called cell problem from the homogenization theory for Hamilton-Jacobi-Bellman equations in order to guarantee asymptotic optimality. We use stochastic control arguments.

More detailed proofs. Differences from the published version are editorial and typographical

References in corpus (4)

• Small-time asymptotics for fast mean-reverting stochastic volatility models
• On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control
• Importance Sampling for Multiscale Diffusions
• Large Deviations for Multiscale Diffusions via Weak Convergence Methods

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• Jarzynski's equality, fluctuation theorems, and variance reduction: Mathematical analysis and numerical algorithms
• Large deviations of slow-fast systems driven by fractional Brownian motion
• Large deviations for stochastic nonlinear systems of slow-fast diffusions with non-Gaussian Lévy noises