Dynamic importance sampling for uniformly recurrent markov chains

Importance sampling is a variance reduction technique for efficient estimation of rare-event probabilities by Monte Carlo. In standard importance sampling schemes, the system is simulated using an a priori fixed change of measure suggested by a large deviation lower bound analysis. Recent work, however, has suggested that such schemes do not work well in many situations. In this paper we consider dynamic importance sampling in the setting of uniformly recurrent Markov chains. By ``dynamic'' we mean that in the course of a single simulation, the change of measure can depend on the outcome of the simulation up till that time. Based on a control-theoretic approach to large deviations, the existence of asymptotically optimal dynamic schemes is demonstrated in great generality. The implementation of the dynamic schemes is carried out with the help of a limiting Bellman equation. Numerical examples are presented to contrast the dynamic and standard schemes.Comment: Published at http://dx.doi.org/10.1214/105051604000001016 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Correction. SDEs with oblique reflections on nonsmooth domains

Correction to The Annals of Probability 21 (1993) 554--580 [http://projecteuclid.org/euclid.aop/1176989415]Comment: Published in at http://dx.doi.org/10.1214/07-AOP374 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Large deviations and queueing networks: methods for rate function identification

This paper considers the problem of rate function identification for multidimensional queueing models with feedback. A set of techniques are introduced which allow this identification when the model possesses certain structural properties. The main tools used are representation formulas for exponential integrals, weak convergence methods, and the regularity properties of associated Skorokhod Problems. Two examples are treated as special cases of the general theory: the classical Jackson network and a model for processor sharing

Moderate deviations for recursive stochastic algorithms

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.Comment: Submitted to Stochastic System

Large Deviations for Multiscale Diffusions via Weak Convergence Methods

We study the large deviations principle for locally periodic stochastic differential equations with small noise and fast oscillating coefficients. There are three possible regimes depending on how fast the intensity of the noise goes to zero relative to the homogenization parameter. We use weak convergence methods which provide convenient representations for the action functional for all three regimes. Along the way we study weak limits of related controlled SDEs with fast oscillating coefficients and derive, in some cases, a control that nearly achieves the large deviations lower bound at the prelimit level. This control is useful for designing efficient importance sampling schemes for multiscale diffusions driven by small noise

Splitting for Rare Event Simulation: A Large Deviation Approach to Design and Analysis

Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set $B$ before another set $A$, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented.Comment: Submitted to Stochastic Processes and their Application

Uniform large deviation principles for Banach space valued stochastic differential equations

We prove a large deviation principle (LDP) for a general class of Banach space valued stochastic differential equations (SDE) that is uniform with respect to initial conditions in bounded subsets of the Banach space. A key step in the proof is showing that a uniform large deviation principle over compact sets is implied by a uniform over compact sets Laplace principle. Because bounded subsets of infinite dimensional Banach spaces are in general not relatively compact in the norm topology, we embed the Banach space into its double dual and utilize the weak-$\star$ compactness of closed bounded sets in the double dual space. We prove that a modified version of our stochastic differential equation satisfies a uniform Laplace principle over weak-$\star$ compact sets and consequently a uniform over bounded sets large deviation principle. We then transfer this result back to the original equation using a contraction principle. The main motivation for this uniform LDP is to generalize results of Freidlin and Wentzell concerning the behavior of finite dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of exit times from bounded sets of Banach space valued small noise SDE, including reaction diffusion equations with multiplicative noise and $2$-dimensional stochastic Navier-Stokes equations with multiplicative noise