Euler approximations with varying coefficients: the case of superlinearly growing diffusion coefficients
TLDR
In this article, a new class of explicit Euler schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, is proposed, and it is shown, under very mild conditions, that these explicit schemes converge in probability and in $\mathcal{L}p$ to the solution of the corresponding SDEs.Abstract:
A new class of explicit Euler schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, is proposed in this article. It is shown, under very mild conditions, that these explicit schemes converge in probability and in $\mathcal{L}^p$ to the solution of the corresponding SDEs. Moreover, rate of convergence estimates are provided for $\mathcal{L}^p$ and almost sure convergence. In particular, the strong order $1/2$ is recovered in the case of uniform $\mathcal{L}^p$-convergence.read more
Citations
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TL;DR: In this article, a perturbation theory for stochastic differential equations (SDEs) was developed, by which they mean both stochastastic ordinary differential equations and stochastically partial differential equations.
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Solving stochastic differential equations and Kolmogorov equations by means of deep learning.
TL;DR: A numerical approximation method is derived and proposed which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region without suffering from the curse of dimensionality.
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On Tamed Euler Approximations of SDEs Driven by L\'evy Noise with Applications to Delay Equations
TL;DR: The taming techniques for explicit Euler approximations of stochastic differential equations driven by Levy noise with superlinearly growing drift coefficients are extended and rate of convergence results are obtained in agreement with classical literature.
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Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment, and stability
TL;DR: In this paper, a truncated Euler-Maruyama (TEM) scheme is proposed to solve SDEs under global Lipschitz conditions for both drift and diffusion coefficients.
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An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz Coefficients
TL;DR: A modified explicit Euler-Maruyama discretisation scheme that allows us to prove strong convergence, with a rate, is presented, and under some regularity and integrability conditions, the optimal strong error rate is obtained.
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Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations
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Numerical Approximations of Stochastic Differential Equations With Non-globally Lipschitz Continuous Coefficients
TL;DR: In this article, moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method were established.