Moderate Deviations for Recursive Stochastic Algorithms

Abstract

We prove a moderate deviation principle for the continuous time interpolation of discrete time recursive stochastic processes. The methods of proof are somewhat different from the corresponding large deviation result, and in particular the proof of the upper bound is more complicated. The results can be applied to the design of accelerated Monte Carlo algorithms for certain problems, where schemes based on moderate deviations are easier to construct and in certain situations provide performance comparable to those based on large deviations.

Importance sampling for stochastic reaction–diffusion equations in the moderate deviation regime

“…The interested reader is referred to [25] for the original development of subsolutionbased importance sampling. As we will show below (Theorem 3.1 and Remark 11), when x = x * , the MDP action functional takes the form (20) with…”

“…Moderate Deviation Principles (MDPs) have been studied in many different contexts such as multiscale and interacting particle systems, Markov processes with jumps, small-noise stochastic dynamics, statistical estimation, option pricing and stochastic recursive algorithms see e.g. [30,55] for SRDEs as well as [7,11,20,29,33,34,36,38,44].…”

Large Deviation Methods for the Analysis and Design of Monte Carlo Schemes in Physics and Chemistry. Final Technical Report

“…The solution of something known as a "cell problem," which was essential in analyzing the large deviation properties, not surprisingly is a needed ingredient in the design of nearly optimal importance sampling schemes. The paper [1] focuses on use of "moderate deviations" as a tool for the development of algorithms when events are rare but not too rare, and [4] is the …rst paper to consider the particularly di¢ cult situation where a rest point is within the domain of the simulation.…”