In this paper, the discrete parameter expansion is adopted to investigate the estimation of heat kernel for Euler-Maruyama scheme of SDEs driven by {alpha}-stable noise, which implies krylovs estimate and khasminskiis estimate. As an application, the convergence rate of Euler-Maruyama scheme of a class of multidimensional SDEs with singular drift( in aid of Zvonkins transformation) is obtained.
In this article, we consider the so-called modified Euler scheme for stochastic differential equations (SDEs) driven by fractional Brownian motions (fBm) with Hurst parameter $frac13<H<frac12$. This is a first-order time-discrete numerical approximation scheme, and has been recently introduced by Hu, Liu and Nualart in order to generalize the classical Euler scheme for It^o SDEs to the case $H>frac12$. The current contribution generalizes the modified Euler scheme to the rough case $frac13<H<frac12$. Namely, we show a convergence rate of order $n^{frac12-2H}$ for the scheme, and we argue that this rate is exact. We also derive a central limit theorem for the renormalized error of the scheme, thanks to some new techniques for asymptotics of weighted random sums. Our main idea is based on the following observation: the triple of processes obtained by considering the fBm, the scheme process and the normalized error process, can be lifted to a new rough path. In addition, the Holder norm of this new rough path has an estimate which is independent of the step-size of the scheme.
In this paper, we investigate the weak convergence rate of Euler-Maruyamas approximation for stochastic differential equations with irregular drifts. Explicit weak convergence rates are presented if drifts satisfy an integrability condition including discontinuous functions which can be non-piecewise continuous or in fractional Sobolev space.
In this paper we study the asymptotic properties of the power variations of stochastic processes of the type X=Y+L, where L is an alpha-stable Levy process, and Y a perturbation which satisfies some mild Lipschitz continuity assumptions. We establish local functional limit theorems for the power variation processes of X. In case X is a solution of a stochastic differential equation driven by L, these limit theorems provide estimators of the stability index alpha. They are applicable for instance to model fitting problems for paleo-climatic temperature time series taken from the Greenland ice core.
We establish Harnack inequality and shift Harnack inequality for stochastic differential equation driven by $G$-Brownian motion. As applications, the uniqueness of invariant linear expectations and estimates on the $sup$-kernel are investigated, where the $sup$-kernel is introduced in this paper for the first time.
In this paper, the existence and uniqueness of the distribution dependent SDEs with H{o}lder continuous drift driven by $alpha$-stable process is investigated. Moreover, by using Zvonkin type transformation, the convergence rate of Euler-Maruyama method is also obtained. The results cover the ones in the case of distribution independent SDEs.
Xing Huang
,Yongqiang Suo
,Chenggui Yuan
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(2021)
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"Estimate of Heat Kernel for Euler-Maruyama Scheme of SDEs Driven by {alpha}-Stable Noise and Applications"
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Yongqiang Suo
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