Journal ArticleDOI
Existence of strong solutions for Itô's stochastic equations via approximations
Istvan Gyongy,Nicolai V Krylov +1 more
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TLDR
In this paper, it was shown that an Ito's stochastic equation with discontinuous coefficients can be constructed on any probability space by using Euler's polygonal approximations.Abstract:
Given strong uniqueness for an Ito's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on “any” probability space by using, for example, Euler's polygonal approximations. Stochastic equations in ℝd and in domains in ℝd are considered.read more
Citations
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Journal ArticleDOI
Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations
TL;DR: In this paper, it was shown that an implicit variant of Euler-Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschnitz condition.
Journal ArticleDOI
Strong solutions of stochastic equations with singular time dependent drift
Nicolai V Krylov,Michael Röckner +1 more
TL;DR: In this paper, the authors prove the existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time.
Journal ArticleDOI
Well-posedness of the transport equation by stochastic perturbation
TL;DR: In this paper, it was shown that a multiplicative stochastic perturbation of Brownian type is enough to render the linear transport equation well-posed. But it was not shown that multiplicative perturbations alone are sufficient to render a deterministic PDE wellposed.
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Lattice Approximations for Stochastic Quasi-Linear Parabolic Partial Differential Equations driven by Space-Time White Noise II
TL;DR: In this paper, the authors approximate quasi-linear parabolic SPDEs substituting the derivatives with finite differences, and investigate the resulting implicit and explicit schemes, and prove their almost sure convergence when the nonlinear terms are Lipschitz continuous.
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Well-posedness of the transport equation by stochastic perturbation
TL;DR: In this paper, it was shown that a multiplicative stochastic perturbation of Brownian type is enough to render the linear transport equation well-posed under the influece of noise.
References
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Book
Stochastic differential equations and diffusion processes
TL;DR: In this article, Stochastic Differential Equations and Diffusion Processes are used to model the diffusion process in stochastic differential equations. But they do not consider the nonlinearity of diffusion processes.
Journal ArticleDOI
Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00
Book
Multidimensional Diffusion Processes
TL;DR: In this paper, the authors propose extension theorems, Martingales, and Compactness, as well as the non-unique case of the Martingale problem, and some estimates on the transition probability functions.
Book
Controlled Diffusion Processes
TL;DR: In this paper, the theory of Controlled Diffusion Processes with Unbounded Coefficients: The Normed Bellman Equation (NBCE) is used to construct optimal strategies.