Super-Brownian motion as the unique strong solution to an SPDE
TLDR
In this paper, a stochastic partial differential equation (SPDE) was derived for super-Brownian motion regarded as a distribution function valued process, and the strong uniqueness for the solution to this SPDE was obtained by an extended Yamada-Watanabe argument.Abstract:
A stochastic partial differential equation (SPDE) is derived for super-Brownian motion regarded as a distribution function valued process. The strong uniqueness for the solution to this SPDE is obtained by an extended Yamada–Watanabe argument. Similar results are also proved for the Fleming–Viot process.read more
Citations
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Journal ArticleDOI
Strong uniqueness for an SPDE via backward doubly stochastic differential equations
TL;DR: In this paper, the authors prove strong uniqueness for a parabolic SPDE involving both the solution v (t, x ) and its derivative ∂ x v ( t, x ) using the Yamada-Watanabe method.
Journal ArticleDOI
Backward doubly stochastic Volterra integral equations and their applications
TL;DR: In this paper, a new class of equations called backward doubly stochastic Volterra integral equations (BDSVIEs) is introduced, and a comparison theorem of the two classes of equations is proved.
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Stochastic equations of super-Lévy processes with general branching mechanism☆
Hui He,Zenghu Li,Xu Yang +2 more
TL;DR: In this article, the process of distribution functions of a one-dimensional super-Levy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time-space Gaussian white noises and Poisson random measures.
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New results on pathwise uniqueness for the heat equation with colored noise
Thomas Rippl,Anja Sturm +1 more
TL;DR: In this article, the authors considered strong uniqueness and the existence of strong solutions for the stochastic heat equation with a multiplicative colored noise term and showed that the noise coefficient is Holder continuous in the solution.
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Existence and pathwise uniqueness to an SPDE driven by α-stable colored noise
Jie Xiong,Xu Yang +1 more
TL;DR: In this paper, a stochastic partial differential equation (SPDE) with a Holder continuous coefficient driven by an α-stable colored noise is studied and the existence of solution is shown by considering the weak limit of a sequence of SDE system which is obtained by replacing the Laplacian operator in the SPDE by its discrete version.
References
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Foundations of modern probability
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.
Journal ArticleDOI
On the uniqueness of solutions of stochastic differential equations
Toshio Yamada,Shinzo Watanabe +1 more
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Stochastic Filtering Theory
TL;DR: The Stochastic Equation of the Optimal Filter (SEF) as discussed by the authors is a generalization of the SEF of the Wiener Process, and it can be expressed in terms of the Ito Formula.
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Backward doubly stochastic differential equations and systems of quasilinear SPDEs
Etienne Pardoux,Shige Peng +1 more
TL;DR: In this paper, a new class of backward stochastic differential equations is introduced, which allows us to produce a probabilistic representation of certain quasilinear SPDE, thus extending the Feynman-Kac formula.