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The fields of applications are numerous : insurance and risk, biology, communication networks, dependability, supply management, etc.    Indeed, the PDMPs studied so far are in fact deterministic functions of CSMPs (Completed Semi-Markov Processes), i.e. semi-Markov processes completed to become Markov processes. This remark leads to considerably broaden the definition of PDMPs and allows their properties to be deduced from those of CSMPs, which are easier to grasp. Stability is studied within a very general framework. In the other chapters, the results become more accurate as the assumptions become more precise. 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First we consider completely observable control problems with finite horizons. Using a time discretization we construct a nonlinear semigroup related to the dynamic programming principle (DPP), whose generator provides the Hamilton–Jacobi–Bellman (HJB) equation, and we characterize the value function via the nonlinear semigroup, besides the viscosity solution theory. When we control not only the dynamics of a system but also the terminal time of its evolution, control-stopping problems arise. This problem is treated in the same frameworks, via the nonlinear semigroup. Its results are applicable to the American option price problem.  Zero-sum two-player time-homogeneous stochastic differential games and viscosity solutions of the Isaacs equations arising from such games are studied via a nonlinear semigroup related to DPP (the min-max principle, to be precise). Using semi-discretization arguments, we construct the nonlinear semigroups whose generators provide lower and upper Isaacs equations.  Concerning partially observable control problems, we refer to stochastic parabolic equations driven by colored Wiener noises, in particular, the Zakai equation. The existence and uniqueness of solutions and regularities as well as Itô's formula are stated. A control problem for the Zakai equations has a nonlinear semigroup whose generator provides the HJB equation on a Banach space. The value function turns out to be a unique viscosity solution for the HJB equation under mild conditions.  This edition provides a more generalized treatment of the topic than does the earlier book Lectures on Stochastic Control Theory (ISI Lecture Notes 9), where time-homogeneous cases are dealt with. Here, for finite time-horizon control problems, DPP was formulated as aone-parameter nonlinear semigroup, whose generator provides the HJB equation, by using a time-discretization method. The semigroup corresponds to the value function and is characterized as the envelope of Markovian transition semigroups of responses for constant control processes. 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Reiß, zbMATH    “…the … monograph is a modern classic in probability theory…  …every … expert in one of the various topics covered by this monograph will reconsider his own point of view and gain deeper insight into his subject.” Klaus D. Schmidt, Mathematical Reviews","brand":"WoB","offers":[{"title":"GB \/ NEW \/ GARDNERS","offer_id":49802538156305,"sku":"NGR9783030618735","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/3030618730.jpg?v=1750999092"},{"product_id":"probabilistic-theory-of-mean-field-games-with-applications-ii-book-franois-delarue-9783319564357","title":"Probabilistic Theory of Mean Field Games with Applications II","description":"This two-volume book offers a comprehensive treatment of the probabilistic approach to mean field game models and their applications. The book is self-contained in nature and includes original material and applications with explicit examples throughout, including numerical solutions.    Volume II tackles the analysis of mean field games in which the players are affected by a common source of noise. The first part of the volume introduces and studies the concepts of weak and strong equilibria, and establishes general solvability results. The second part is devoted to the study of the master equation, a partial differential equation satisfied by the value function of the game over the space of probability measures. Existence of viscosity and classical solutions are proven and used to study asymptotics of games with finitely many players.    Together, both Volume I and Volume II will greatly benefit mathematical graduate students and researchers interested in mean field games. 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Topics include the generation and analysis of pseudorandom numbers (which are intended to imitate truly random numbers on a computer), the design and justification of Monte Carlo algorithms, and advanced approaches such as Markov chain Monte Carlo and stochastic optimization. In contrast to deterministic numerical methods, the outcome of a Monte Carlo algorithm is itself random – and one needs the tools of probability and statistics to interpret these results meaningfully. The theoretical foundations, particularly the law of large numbers and central limit theorem, are combined with practical algorithms that reveal both the strengths and subtleties of stochastic simulation.   The book includes numerous exercises, both theoretical and computational. Each chapter features step-by-step algorithms, illustrated examples, and results presented through numerical computations, tables, and a variety of plots and figures. 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