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The present monograph, based mainly on studies of the authors and their - authors, and also on lectures given by the authors in the past few years, has the following particular aims: To present basic results (with proofs) of optimal stopping theory in both discrete and continuous time using both martingale and Mar- vian approaches; To select a seriesof concrete problems ofgeneral interest from the t- ory of probability, mathematical statistics, and mathematical ?nance that can be reformulated as problems of optimal stopping of stochastic processes and solved by reduction to free-boundary problems of real analysis (Stefan problems). The table of contents found below gives a clearer idea of the material included in the monograph. Credits and historical comments are given at the end of each chapter or section. The bibliography contains a material for further reading. Acknowledgements.TheauthorsthankL.E.Dubins,S.E.Graversen,J.L.Ped- sen and L. A. Shepp for useful discussions. The authors are grateful to T. B. To- zovafortheexcellenteditorialworkonthemonograph.Financialsupportandh- pitality from ETH, Zur ¨ ich (Switzerland), MaPhySto (Denmark), MIMS (Man- ester) and Thiele Centre (Aarhus) are gratefully acknowledged. The authors are also grateful to INTAS and RFBR for the support provided under their grants. The grant NSh-1758.2003.1 is gratefully acknowledged. Large portions of the text were presented in the “School and Symposium on Optimal Stopping with App- cations” that was held in Manchester, England from 17th to 27th January 2006.

Optimal stopping (Mathematical statistics) --- Boundary value problems. --- Nonlinear integral equations. --- Economics, Mathematical. --- Economics --- Mathematical economics --- Econometrics --- Mathematics --- Methodology --- Integral equations, Nonlinear --- Integral equations --- Nonlinear theories --- Boundary conditions (Differential equations) --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Stopping, Optimal (Mathematical statistics) --- Sequential analysis --- Optimal stopping (Mathematical statistics). --- Boundary value problems --- Nonlinear integral equations --- Economics, Mathematical --- Arrêt optimal (Statistique mathématique) --- Problèmes aux limites --- Equations intégrales non linéaires --- Mathématiques économiques --- EPUB-LIV-FT SPRINGER-B LIVMATHE --- Distribution (Probability theory. --- Mathematical optimization. --- Differential equations, partial. --- Finance. --- Probability Theory and Stochastic Processes. --- Calculus of Variations and Optimal Control; Optimization. --- Partial Differential Equations. --- Quantitative Finance. --- Funding --- Funds --- Currency question --- Partial differential equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Probabilities. --- Calculus of variations. --- Partial differential equations. --- Economics, Mathematical . --- Isoperimetrical problems --- Variations, Calculus of --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk