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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.


Papers
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Journal ArticleDOI
TL;DR: In this paper, it has been shown that the Hausdorff dimension of the chordal SLE is equal to Min(2, 1 + κ/8) with probability one.
Abstract: Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ ≥ 0. We prove that, with probability one, the Haus-dorff dimension of γ is equal to Min(2, 1 + κ/8). Introduction. It has been conjectured by theoretical physicists that various lattice models in statistical physics (such as percolation, Potts model, Ising model, uniform spanning trees), taken at their critical point, have a continuous confor-mally invariant scaling limit when the mesh of the lattice tends to 0. Recently, Oded Schramm [15] introduced a family of random processes which he called Stochastic Loewner Evolutions (or SLE), that are the only possible conformally invariant scaling limits of random cluster interfaces (which are very closely related to all above-mentioned models). An SLE process is defined using the usual Loewner equation, where the driving function is a time-changed Brownian motion. More specifically, in the present paper we will be mainly concerned with SLE in the upper-half plane (sometimes called chordal SLE), defined by the following PDE:

294 citations

Book
01 Jan 1995
TL;DR: Students considering more advanced research in probability theory will benefit from this wide-ranging survey of the subject that provides them with a foretaste of the subjects many treasures.
Abstract: The purpose of this book is to provide the reader with a solid background and understanding of the basic results and methods in probability theory before entering into more advanced courses. The first six chapters focus on some central areas of what might be called pure probability theory: multivariate random variables, conditioning, transforms, order variables, the multivariate normal distribution, convergence. A final chapter is devoted to the Poisson process as a means both to introduce stochastic processes and to apply many of the techniques introduced earlier in the text. Students are assumed to have taken a first course in probability, though no knowledge of measure theory is assumed. Throughout, the presentation is thorough and includes many examples that are discussed in detail. Thus, students considering more advanced research in probability theory will benefit from this wide-ranging survey of the subject that provides them with a foretaste of the subjects many treasures. The present second edition offers updated content, one hundred additional problems for solution, and a new chapter that provides an outlook on further areas and topics, such as stable distributions and domains of attraction, extreme value theory and records, and martingales. The main idea is that this chapter may serve as an appetizer to the more advanced theory.

294 citations

Journal ArticleDOI
TL;DR: In this paper, the problem of controlling a stochastic process with unknown parameters over an infinite horizon with discounting is considered, where agents express beliefs about unknown parameters in terms of distributions.
Abstract: The problem of controlling a stochastic process, with unknown parameters over an infinite horizon, with discounting is considered. Agents express beliefs about unknown parameters in terms of distributions. Under general conditions, the sequence of beliefs converges to a limit distribution. The limit distribution may or may not be concentrated at the true parameter value. In some cases, complete learning is optimal; in others, the optimal strategy does not imply complete learning. The paper concludes with examination of some special cases and a discussion of a procedure for generating examples in which incomplete learning is optimal. Copyright 1988 by The Econometric Society.

293 citations

Proceedings ArticleDOI
03 Jun 1991
TL;DR: A highly parallel incremental stochastic minimization algorithm is presented which has a number of advantages over previous approaches and the incremental nature of the scheme makes it dynamic and permits the detection of occlusion and disocclusion boundaries.
Abstract: A novel approach to incrementally estimating visual motion over a sequence of images is presented. The authors start by formulating constraints on image motion to account for the possibility of multiple motions. This is achieved by exploiting the notions of weak continuity and robust statistics in the formulation of a minimization problem. The resulting objective function is non-convex. Traditional stochastic relaxation techniques for minimizing such functions prove inappropriate for the task. A highly parallel incremental stochastic minimization algorithm is presented which has a number of advantages over previous approaches. The incremental nature of the scheme makes it dynamic and permits the detection of occlusion and disocclusion boundaries. >

293 citations

Book
01 Dec 1990
TL;DR: In this article, the Martingale Problem is revisited and the authors present a new approach to the problem of finding the optimal control and value functions for a large number of SDEs.
Abstract: 1 Weak Convergence- 0 Outline of the Chapter- 1 Basic Properties and Definitions- 2 Examples- 3 The Skorohod Representation- 4 The Function Space Ck [0, T]- 5 The Function Space Dk [0, T]- 6 Measure Valued Random Variables and Processes- 2 Stochastic Processes: Background- 0 Outline of the Chapter- 1 Martingales- 2 Stochastic Integrals and Ito's Lemma- 3 Stochastic Differential Equations: Bounds- 4 Controlled Stochastic Differential Equations: Existence of Solutions- 5 Representing a Martingale as a Stochastic Integral- 6 The Martingale Problem- 7 Jump-Diffusion Processes- 8 Jump-Diffusion Processes: The Martingale Problem Formulation- 3 Controlled Stochastic Differential Equations- 0 Outline of the Chapter- 1 Controlled SDE's: Introduction- 2 Relaxed Controls: Deterministic Case- 3 Stochastic Relaxed Controls- 4 The Martingale Problem Revisited- 5 Approximations, Weak Convergence and Optimality- 4 Controlled Singularly Perturbed Systems- 0 Outline of the Chapter- 1 Problem Formulation: Finite Time Interval- 2 Approximation of the Optimal Controls and Value Functions- 3 Discounted Cost and Optimal Stopping Problems- 4 Average Cost Per Unit Time- 5 Jump-Diffusion Processes- 6 Other Approaches- 5 Functional Occupation Measures and Average Cost Per Unit Time Problems- 0 Outline of the Chapter- 1 Measure Valued Random Variables- 2 Limits of Functional Occupation Measures for Diffusions- 3 The Control Problem- 4 Singularly Perturbed Control Problems- 5 Control of the Fast System- 6 Reflected Diffusions- 7 Discounted Cost Problem- 6 The Nonlinear Filtering Problem- 0 Outline of the Chapter- 1 A Representation of the Nonlinear Filter- 2 The Filtering Problem for the Singularly Perturbed System- 3 The Almost Optimality of the Averaged Filter- 4 A Counterexample to the Averaged Filter- 5 The Near Optimality of the Averaged Filter- 6 A Repair and Maintainance Example- 7 Robustness of the Averaged Filters- 8 A Robust Computational Approximation to the Averaged Filter- 9 The Averaged Filter on the Infinite Time Interval- 7 Weak Convergence: The Perturbed Test Function Method- 0 Outline of the Chapter- 1 An Example- 2 The Perturbed Test Function Method: Introduction- 3 The Perturbed Test Function Method: Tightness and Weak Convergence- 4 Characterization of the Limits- 8 Singularly Perturbed Wide-Band Noise Driven Systems- 0 Outline of the Chapter- 1 The System and Noise Model- 2 Weak Convergence of the Fast System- 3 Convergence to the Averaged System- 4 The Optimality Theorem- 5 The Average Cost Per Unit Time Problem- 9 Stability Theory- 0 Outline of the Chapter- 1 Stability Theory for Jump-Diffusion Processes of Ito Type- 2 Singularly Perturbed Deterministic Systems: Bounds on Paths- 3 Singularly Perturbed Ito Processes: Tightness- 4 The Linear Case- 5 Wide Bandwidth Noise- 6 Singularly Perturbed Wide Bandwidth Noise Driven Systems- 10 Parametric Singularities- 0 Outline of the Chapter- 1 Singularly Perturbed Ito Processes: Weak Convergence- 2 Stability- References- List of Symbols

292 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023159
2022355
2021985
20201,151
20191,119
20181,115