Topic
Stochastic process
About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.
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Book•
01 Jan 1997
TL;DR: In this article, the authors discuss the relationship between Markov Processes and Ergodic properties of Markov processes and their relation with PDEs and potential theory. But their main focus is on the convergence of random processes, measures, and sets.
Abstract: * Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices
4,562 citations
12 Oct 1997
TL;DR: The paper reports a reworking of the particle swarm algorithm to operate on discrete binary variables, where trajectories are changes in the probability that a coordinate will take on a zero or one value.
Abstract: The particle swarm algorithm adjusts the trajectories of a population of "particles" through a problem space on the basis of information about each particle's previous best performance and the best previous performance of its neighbors. Previous versions of the particle swarm have operated in continuous space, where trajectories are defined as changes in position on some number of dimensions. The paper reports a reworking of the algorithm to operate on discrete binary variables. In the binary version, trajectories are changes in the probability that a coordinate will take on a zero or one value. Examples, applications, and issues are discussed.
4,478 citations
TL;DR: This work represents the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error.
Abstract: We present a new method for solving stochastic differential equations based on Galerkin projections and extensions of Wiener's polynomial chaos Specifically, we represent the stochastic processes with an optimum trial basis from the Askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error Several continuous and discrete processes are treated, and numerical examples show substantial speed-up compared to Monte Carlo simulations for low dimensional stochastic inputs
4,473 citations
Book•
01 Jan 1984
TL;DR: In this article, the authors introduce the concept of random perturbations in Dynamical Systems with a Finite Time Interval (FTI) and the Averaging Principle.
Abstract: 1.Random Perturbations.- 2.Small Random Perturbations on a Finite Time Interval.- 3.Action Functional.- 4.Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.- 5.Perturbations Leading to Markov Processes.- 6.Markov Perturbations on Large Time Intervals.- 7.The Averaging Principle. Fluctuations in Dynamical Systems with Averaging.- 8.Random Perturbations of Hamiltonian Systems.- 9. The Multidimensional Case.- 10.Stability Under Random Perturbations.- 11.Sharpenings and Generalizations.- References.- Index.
4,070 citations
Book•
01 Dec 1992TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.
4,042 citations