| 1 |
Front Matter |
|
|
Abstract
|
| 2 |
|
|
|
Abstract
|
| 3 |
General Theory |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
Throughout this chapter . is a sequence of random variables defined on a probability space . and taking values in a complete separable metric space .. As is usual, we will refer to such a space as a .. The metric of . is denoted by .(., .), and expectation with respect to . by .. The theory of large deviations focuses on random variables . for which the probabilities . converge to 0 exponentially fast for a class of Borel sets ..
|
| 4 |
Relative Entropy and Tightness of Measures |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
In this chapter we will collect results on relative entropy and tightness of probability measures that will be used many times in this book.
|
| 5 |
Examples of Representations and Their Application |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
Our approach to the study of large deviations is based on convenient variational representations for expected values of nonnegative functionals. In this chapter we give three examples of such representations and show how they allow easy proofs of some classical results.
|
| 6 |
|
|
|
Abstract
|
| 7 |
Recursive Markov Systems with Small Noise |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
In Chap. 3 we presented several examples of representations and how they could be used for large deviation analysis.
|
| 8 |
Moderate Deviations for Recursive Markov Systems |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
In this chapter we consider .-valued discrete time processes of the same form as in Chap. ., but instead of analyzing the large deviation behavior, we consider deviations closer to the LLN limit.
|
| 9 |
Empirical Measure of a Markov Chain |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
In this chapter we develop the large deviation theory for the empirical measure of a Markov chain, thus generalizing Sanov’s theorem from Chap. .. The ideas developed here are useful in other contexts, such as proving sample path large deviation properties of processes with multiple time scales as described in Sect. ..
|
| 10 |
Models with Special Features |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
Chapters . through . considered small noise large deviations of stochastic recursive equations, small noise moderate deviations for processes of the same type, and large deviations for the empirical measure of a Markov chain. These chapters thus consider models that are both standard and fairly general for each setting. In this chapter we consider discrete time models that are somewhat less standard, with the aim being to show how the weak convergence methodology can be adapted.
|
| 11 |
|
|
|
Abstract
|
| 12 |
Representations for Continuous Time Processes |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
In previous chapters we developed and applied representations for the large deviation analysis of discrete time processes. The derivation of useful representations in this setting follows from a straightforward application of the chain rule. The only significant issue is to decide on the ordering used for the underlying “driving noises” when the chain rule is applied, since controls are allowed to depend on the “past,” which is determined by this ordering.
|
| 13 |
Abstract Sufficient Conditions for Large and Moderate Deviations in the Small Noise Limit |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
In this chapter we use the representations derived in Chap. . to study large and moderate deviations for stochastic systems driven by Brownian and/or Poisson noise, and consider a “small noise” limit, as in Sects. . and .. We will prove general abstract large deviation principles, and in later chapters apply these to models in which the noise enters the system in an additive and independent manner (In our terminology, this includes systems with multiplicative noise, namely settings in which the noise term is multiplied by a state-dependent coefficient). For these systems, one can view the mapping that takes the noise into the state of the system as “nearly” continuous, and it is this property that allows a unified and relatively straightforward treatment.
|
| 14 |
Large and Moderate Deviations for Finite Dimensional Systems |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
In this chapter we use the abstract sufficient conditions from Chap. . to prove large and moderate deviation principles for small noise finite dimensional jump-diffusions. We will consider only Laplace principles rather than uniform Laplace principles, since, as was noted in Chap. ., the extension from the nonuniform to the uniform case is straightforward. The first general results on large deviation principles for jump-diffusions of the form considered in this chapter are due to Wentzell [245–248] and Freidlin and Wentzell [140]. The conditions for an LDP identified in the current chapter relax some of the assumptions made in these works. Results on moderate deviation principles in this chapter are based on the recent work [41]. We do not aim for maximal generality, and from the proofs it is clear that many other models (e.g., time inhomogeneous jump diffusions, SDEs with delay) can be treated in an analogous fashion.
|
| 15 |
Systems Driven by an Infinite Dimensional Brownian Noise |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
In Chap. . we gave a representation for positive functionals of a Hilbert space valued Brownian motion. This chapter will apply the representation to study the large deviation properties of infinite dimensional small noise stochastic dynamical systems. In the application, the driving noise is given by a Brownian sheet, and so in this chapter we will present a sufficient condition analogous to Condition . (but there will be no Poisson noise in this chapter) that covers the setting of such noise processes (see Condition .). Another formulation of an infinite dimensional Brownian motion that will be needed in Chap. . is as a sequence of independent Brownian motions regarded as a .-valued random variable. We also present the analogous sufficient condition (Condition .) for an LDP to hold for this type of driving noise.
|
| 16 |
Stochastic Flows of Diffeomorphisms and Image Matching |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
The previous chapter considered in detail an example driven by a Brownian sheet, namely a stochastic reaction–diffusion equation. In this chapter we consider an application of one of the other formulations of infinite dimensional Brownian motion, which is the infinite sequence of independent one-dimensional Brownian motions. Such a collection will be used to define a general class of Brownian flows of diffeomorphisms [178], which are a special case of the stochastic flows of diffeomorphisms studied in [16, 25, 124, 178]. We will consider small noise asymptotics, prove the corresponding LDP, and then use it to give a Bayesian interpretation of an estimator used for image matching.
|
| 17 |
Models with Special Features |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
Chapters 8 through 12 considered representations in continuous time and their application to large and moderate deviation analyses of finite and infinite dimensional systems described by stochastic differential equations. In this chapter we complete our study of continuous time processes by considering additional problems with features that benefit from a somewhat different use of the representations and/or weak convergence arguments.
|
| 18 |
|
|
|
Abstract
|
| 19 |
Rare Event Monte Carlo and Importance Sampling |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
Suppose that in the analysis of some system, the value of a probability or expected value that is largely determined by one or a few events is important. Examples include the data loss in a communication network; depletion of capital reserves in a model for insurance; motion between metastable states in a chemical reaction network; and exceedance of a regulatory threshold in a model for pollution in a waterway. In previous chapters we have described how large deviation theory gives approximations for such quantities. The approximations take the form of logarithmic asymptotics, i.e., exponential decay rates.
|
| 20 |
Performance of an IS Scheme Based on a Subsolution |
Amarjit Budhiraja,Paul Dupuis |
|
Abstract
In Chap. . we considered the problem of rare event simulation associated with small noise discrete time Markov processes of the form analyzed in Chap. .. Two types of events were emphasized: those that are described by process behavior on a bounded time interval (finite-time problems) and those that concern properties of the process over unbounded time horizons (e.g., exit probability problems).
|
发表于 2025-3-21 19:54:29
