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Estimació d'un paràmetre --- Equacions diferencials estocàstiques --- Anàlisi numèrica --- Equació de Fokker-Planck --- Estimació estadística --- Estimació (Estadística) --- Estimació de paràmetres --- Estimació paramètrica --- Paràmetres d'estimació --- Sistemes estocàstics --- Teoria de l'estimació --- Parameter estimation. --- Stochastic differential equations. --- Differential equations --- Fokker-Planck equation --- Estimation theory --- Stochastic systems

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Birth and death processes (Stochastic processes) --- Gaussian distribution. --- Normal distribution --- Distribution (Probability theory) --- Branching processes --- Markov processes --- Processos estocàstics --- Distribució de Gauss --- Distribució gaussiana --- Distribució normal --- Distribució (Teoria de la probabilitat) --- Càlcul estocàstic --- Funcions aleatòries --- Processos aleatoris --- Probabilitats --- Anàlisi estocàstica --- Aproximació estocàstica --- Camps aleatoris --- Filtre de Kalman --- Fluctuacions (Física) --- Martingales (Matemàtica) --- Mètode de Montecarlo --- Processos de Markov --- Processos de ramificació --- Processos gaussians --- Processos puntuals --- Rutes aleatòries (Matemàtica) --- Semimartingales (Matemàtica) --- Sistemes estocàstics --- Teoremes de límit (Teoria de probabilitats) --- Teoria de cues --- Teoria de l'estimació --- Teoria de la predicció

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Stochastic games are have an element of chance: the state of the next round is determined probabilistically depending upon players' actions and the current state. Successful players need to balance the need for short-term payoffs while ensuring future opportunities remain high. The various techniques needed to analyze these often highly non-trivial games are a showcase of attractive mathematics, including methods from probability, differential equations, algebra, and combinatorics. This book presents a course on the theory of stochastic games going from the basics through to topics of modern research, focusing on conceptual clarity over complete generality. Each of its chapters introduces a new mathematical tool - including contracting mappings, semi-algebraic sets, infinite orbits, and Ramsey's theorem, among others - before discussing the game-theoretic results they can be used to obtain. The author assumes no more than a basic undergraduate curriculum and illustrates the theory with numerous examples and exercises, with solutions available online.

Game theory. --- Mathematical models --- Mathematics --- Games, Theory of --- Theory of games --- Stochastic processes. --- Random processes --- Probabilities --- Teoria de jocs --- Processos estocàstics --- Càlcul estocàstic --- Funcions aleatòries --- Processos aleatoris --- Probabilitats --- Anàlisi estocàstica --- Aproximació estocàstica --- Camps aleatoris --- Filtre de Kalman --- Fluctuacions (Física) --- Martingales (Matemàtica) --- Mètode de Montecarlo --- Processos de Markov --- Processos de ramificació --- Processos gaussians --- Processos puntuals --- Rutes aleatòries (Matemàtica) --- Semimartingales (Matemàtica) --- Sistemes estocàstics --- Teoremes de límit (Teoria de probabilitats) --- Teoria de cues --- Teoria de l'estimació --- Teoria de la predicció --- Presa de decisions (Estadística) --- Jocs cooperatius (Matemàtica) --- Jocs d'atzar (Matemàtica) --- Jocs d'estratègia (Matemàtica) --- Jocs diferencials --- Jocs no cooperatius (Matemàtica) --- Presa de decisions

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This textbook explores two distinct stochastic processes that evolve at random: weakly stationary processes and discrete parameter Markov processes. Building from simple examples, the authors focus on developing context and intuition before formalizing the theory of each topic. This inviting approach illuminates the key ideas and computations in the proofs, forming an ideal basis for further study. After recapping the essentials from Fourier analysis, the book begins with an introduction to the spectral representation of a stationary process. Topics in ergodic theory follow, including Birkhoff’s Ergodic Theorem and an introduction to dynamical systems. From here, the Markov property is assumed and the theory of discrete parameter Markov processes is explored on a general state space. Chapters cover a variety of topics, including birth–death chains, hitting probabilities and absorption, the representation of Markov processes as iterates of random maps, and large deviation theory for Markov processes. A chapter on geometric rates of convergence to equilibrium includes a splitting condition that captures the recurrence structure of certain iterated maps in a novel way. A selection of special topics concludes the book, including applications of large deviation theory, the FKG inequalities, coupling methods, and the Kalman filter. Featuring many short chapters and a modular design, this textbook offers an in-depth study of stationary and discrete-time Markov processes. Students and instructors alike will appreciate the accessible, example-driven approach and engaging exercises throughout. A single, graduate-level course in probability is assumed.

Stochastic processes. --- Markov processes. --- Distribution (Probability theory). --- Probabilities. --- Stochastic Processes. --- Markov Process. --- Distribution Theory. --- Probability Theory. --- Processos estocàstics --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Stochastic processes --- Random processes --- Càlcul estocàstic --- Funcions aleatòries --- Processos aleatoris --- Probabilitats --- Anàlisi estocàstica --- Aproximació estocàstica --- Camps aleatoris --- Filtre de Kalman --- Fluctuacions (Física) --- Martingales (Matemàtica) --- Mètode de Montecarlo --- Processos de Markov --- Processos de ramificació --- Processos gaussians --- Processos puntuals --- Rutes aleatòries (Matemàtica) --- Semimartingales (Matemàtica) --- Sistemes estocàstics --- Teoremes de límit (Teoria de probabilitats) --- Teoria de cues --- Teoria de l'estimació --- Teoria de la predicció

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In this book, the optimal transportation problem (OT) is described as a variational problem for absolutely continuous stochastic processes with fixed initial and terminal distributions. Also described is Schrödinger’s problem, which is originally a variational problem for one-step random walks with fixed initial and terminal distributions. The stochastic optimal transportation problem (SOT) is then introduced as a generalization of the OT, i.e., as a variational problem for semimartingales with fixed initial and terminal distributions. An interpretation of the SOT is also stated as a generalization of Schrödinger’s problem. After the brief introduction above, the fundamental results on the SOT are described: duality theorem, a sufficient condition for the problem to be finite, forward–backward stochastic differential equations (SDE) for the minimizer, and so on. The recent development of the superposition principle plays a crucial role in the SOT. A systematic method is introduced to consider two problems: one with fixed initial and terminal distributions and one with fixed marginal distributions for all times. By the zero-noise limit of the SOT, the probabilistic proofs to Monge’s problem with a quadratic cost and the duality theorem for the OT are described. Also described are the Lipschitz continuity and the semiconcavity of Schrödinger’s problem in marginal distributions and random variables with given marginals, respectively. As well, there is an explanation of the regularity result for the solution to Schrödinger’s functional equation when the space of Borel probability measures is endowed with a strong or a weak topology, and it is shown that Schrödinger’s problem can be considered a class of mean field games. The construction of stochastic processes with given marginals, called the marginal problem for stochastic processes, is discussed as an application of the SOT and the OT.

Stochastic processes. --- Random processes --- Probabilities --- Processos estocàstics --- Càlcul estocàstic --- Funcions aleatòries --- Processos aleatoris --- Probabilitats --- Anàlisi estocàstica --- Aproximació estocàstica --- Camps aleatoris --- Filtre de Kalman --- Fluctuacions (Física) --- Martingales (Matemàtica) --- Mètode de Montecarlo --- Processos de Markov --- Processos de ramificació --- Processos gaussians --- Processos puntuals --- Rutes aleatòries (Matemàtica) --- Semimartingales (Matemàtica) --- Sistemes estocàstics --- Teoremes de límit (Teoria de probabilitats) --- Teoria de cues --- Teoria de l'estimació --- Teoria de la predicció --- Probabilities. --- Geometry, Differential. --- Differential equations. --- Functional analysis. --- Measure theory. --- Probability Theory. --- Differential Geometry. --- Differential Equations. --- Functional Analysis. --- Measure and Integration. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- 517.91 Differential equations --- Differential equations --- Differential geometry --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk