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Large Deviations for Stochastic Processes
Jin Feng, University of Kansas, Lawrence, KS, and Thomas G. Kurtz, University of Wisconsin at Madison, WI
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Mathematical Surveys and Monographs
2006; 410 pp; hardcover
Volume: 131
ISBN-10: 0-8218-4145-9
ISBN-13: 978-0-8218-4145-7
List Price: US$102 Member Price: US$81.60
Order Code: SURV/131

The book is devoted to the results on large deviations for a class of stochastic processes. Following an introduction and overview, the material is presented in three parts. Part 1 gives necessary and sufficient conditions for exponential tightness that are analogous to conditions for tightness in the theory of weak convergence. Part 2 focuses on Markov processes in metric spaces. For a sequence of such processes, convergence of Fleming's logarithmically transformed nonlinear semigroups is shown to imply the large deviation principle in a manner analogous to the use of convergence of linear semigroups in weak convergence. Viscosity solution methods provide applicable conditions for the necessary convergence. Part 3 discusses methods for verifying the comparison principle for viscosity solutions and applies the general theory to obtain a variety of new and known results on large deviations for Markov processes. In examples concerning infinite dimensional state spaces, new comparison principles are derived for a class of Hamilton-Jacobi equations in Hilbert spaces and in spaces of probability measures.