Particle representations for a class of nonlinear SPDEs
Thomas G. Kurtz,Jie Xiong +1 more
TLDR
In this paper, an innite system of stochastic dierential equations for the locations and weights of a collection of particles is considered, and the particles interact through their weighted empirical measure, V, and V is shown to be the unique solution of a nonlinear stochiastic partial die-rential equation (SPDE).About:
This article is published in Stochastic Processes and their Applications.The article was published on 1999-09-01 and is currently open access. It has received 193 citations till now. The article focuses on the topics: Lebesgue measure & Stochastic partial differential equation.read more
Citations
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Large Deviations for Stochastic Processes
Jin Feng,Thomas G. Kurtz +1 more
TL;DR: The general theory of large deviations: Large deviations and exponential tightness Large deviations for stochastic processes, large deviations for Markov processes and semigroup convergence, and nonlinear semiigroup convergence using viscosity solutions is discussed in this article.
Book
Nonlinear Markov Processes and Kinetic Equations
TL;DR: In this article, the authors developed the interplay between probability and analysis in nonlinear Markov evolution, and used probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior.
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Integration of Brownian vector fields
Yves Le Jan,Olivier Raimond +1 more
TL;DR: Using the Wiener chaos decomposition, the authors showed that strong solutions of non-Lipschitzian stochastic differential equations are given by random Markovian kernels.
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Dynamic Programming for Optimal Control of Stochastic McKean--Vlasov Dynamics
Huyên Pham,Xiaoli Wei +1 more
TL;DR: In this paper, the optimal control of general stochastic McKean-Vlasov equation under common noise is studied. But the authors focus on the control of the value function in the Wasserstein space of probability measures, which is proved from a flow property of the controlled state process.
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Approximate McKean–Vlasov representations for a class of SPDEs
Dan Crisan,Jie Xiong +1 more
TL;DR: In this article, the solution of a class of linear stochastic partial differential equations is approximated using Clark's robust representation approach, and the ensuing approximations coincide with the t
References
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Limit Theorems for Stochastic Processes
Jean Jacod,Albert N. Shiryaev +1 more
TL;DR: In this article, the General Theory of Stochastic Processes, Semimartingales, and Stochastically Integrals is discussed and the convergence of Processes with Independent Increments is discussed.
Book
Markov Processes: Characterization and Convergence
TL;DR: In this paper, the authors present a flowchart of generator and Markov Processes, and show that the flowchart can be viewed as a branching process of a generator.
Journal ArticleDOI
On the uniqueness of solutions of stochastic differential equations
Toshio Yamada,Shinzo Watanabe +1 more
Book
Stochastic Differential Equations in Infinite Dimensional Spaces
Gopinath Kallianpur,Jie Xiong +1 more
Book ChapterDOI
Weak convergence of stochastic integrals and differential equations
Thomas G. Kurtz,Philip Protter +1 more
TL;DR: In this paper, the authors define a notion of an integral J: H.dW, where H is a stochastic process; or more generally an indefinite integral J; H.w.d W, o t < 00.