Journal ArticleDOI
Strong convergence of split-step theta methods for non-autonomous stochastic differential equations
TLDR
The strong convergence of the split-step theta methods for non-autonomous stochastic differential equations under a linear growth condition on the diffusion coefficient and a one-sided Lipschitz condition is proved.Abstract:
In this paper, we first prove the strong convergence of the split-step theta methods for non-autonomous stochastic differential equations under a linear growth condition on the diffusion coefficient and a one-sided Lipschitz condition on the drift coefficient. Then, if the drift coefficient satisfies a polynomial growth condition, we further get the rate of convergence. Finally, the obtained results are supported by numerical experiments.read more
Citations
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Dissertation
On the theory of superconductivity
TL;DR: In this article, it was shown that a metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.
Journal ArticleDOI
The truncated Milstein method for stochastic differential equations with commutative noise
TL;DR: Inspired by the truncated EulerMaruyama method developed in Mao (2015), this paper proposed a truncated Milstein method with strong convergence rate close to 1 for a class of highly non-linear stochastic differential equations with commutative noise.
Posted Content
Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
Matteo Beccari,Martin Hutzenthaler,Arnulf Jentzen,Ryan Kurniawan,Felix Lindner,Diyora Salimova +5 more
TL;DR: This work proves that full- Discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations.
Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
Matteo Beccari,Martin Hutzenthaler,Arnulf Jentzen,Ryan Kurniawan,Felix Lindner,Diyora Salimova +5 more
TL;DR: In this article, it was shown that the divergence phenomenon also holds for stochastic partial differential equations with superlinearly growing nonlinearities, such as the Allen-Cahn equations.
Journal ArticleDOI
Discrete gradient methods and linear projection methods for preserving a conserved quantity of stochastic differential equations
TL;DR: The relationship of the two classes of methods for preserving a conserved quantity is proved, which is, the constructed linear projection methods can be considered as a subset of the constructed discrete gradient methods.
References
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Book
Numerical Solution of Stochastic Differential Equations
Peter E. Kloeden,Eckhard Platen +1 more
TL;DR: In this article, a time-discrete approximation of deterministic Differential Equations is proposed for the stochastic calculus, based on Strong Taylor Expansions and Strong Taylor Approximations.
Journal ArticleDOI
An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations
TL;DR: The article is built around $10$ MATLAB programs, and the topics covered include stochastic integration, the Euler--Maruyama method, Milstein's method, strong and weak convergence, linear stability, andThe stochastics chain rule.
Dissertation
On the theory of superconductivity
TL;DR: In this article, it was shown that a metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.
Numerical Solution of Stochastic Differential Equations
TL;DR: In this article, the main concepts and techniques necessary for someone who wishes to carry out numerical experiments involving stochastic differential equations (SDEs) are described and compared. And the convergence of Euler-Maruyama and Milstein and Taylor approximate solutions are compared.
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Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations
The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients
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