Chaman Kumar, Sotirios Sabanis
We extend the taming techniques developed in \cite{konstantinos2014,sabanis2013} to construct explicit Milstein schemes that numerically approximate Lévy driven stochastic differential equations with super-linearly growing drift coefficients. The classical rate of convergence is recovered when the first derivative of the drift coefficient satisfies a polynomial Lipschitz condition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we apply the tamed technique to the Milstein numerical scheme to investigate Neutral Stochastic Delay Differential Equations(NSDDEs) with highly nonlinear coefficients. Under the local Lipschitz condition and Khasminskii condition, the tamed Milstein numerical solution converges strongly to the exact solution.
Weighted power variations of fractional Brownian motion B are used to compute the exact rate of convergence of some approximating schemes associated to one-dimensional stochastic differential equations (SDEs) driven by B. The limit of the error between the exact solution and the considered scheme is computed explicitly.
We propose a new tamed Milstein-type scheme for stochastic differential equation with Markovian switching when drift coefficient is assumed to grow super-linearly. The strong rate of convergence is shown to be equal to $1.0$ under mild regularity (e.g. once differentiability) requirements on drift and diffusion coefficients. Novel techniques are developed to tackle two-fold difficulties arising due to jumps of the Markov chain and the reduction of regularity requirements on the coefficients.
A tamed-adaptive Milstein scheme for stochastic differential equations with low regularity coefficients
针对低正则系数随机微分方程，提出驯化自适应Milstein格式，在增长条件下实现L2范数下(1+α)/2收敛速率，覆盖有限与无限区间。
The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients
针对漂移系数超线性增长且全局单边Lipschitz连续的随机微分方程，提出一种显式强收敛的驯化Milstein方法，其强收敛阶高于驯化Euler方法，并通过自举论证克服了收敛性证明中的困难。
In this paper we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE $$X_t = x_0 + \int_{0}^t{b(X_s)ds}+\int_{0}^t{σ|X_s|^αdW_s}, \;x_0>0,\;σ>0,\; α\in[\tfrac{1}{2},1).$$ Assuming $b(0)/σ^2$ big enough, and $b$ smooth, we prove a strong rate of convergence of order one, recovering the classical result of Milstein for SDEs with smooth diffusion coefficient. In contrast with other recent results, our proof does not relies on Lamperti transformation, and it can be applied to a wide class of drift functions. On the downsi…
A Milstein-type scheme was proposed to improve the rate of convergence of its approximation of the solution to a stochastic differential equation driven by a vector of continuous semimartingales. A necessary and sufficient condition was provided for this rate to be $1/n$ when the SDE is driven by a vector of continuous local martingales, or continuous semimartingales under an additional assumption on their finite variation part. The asymptotic behavior (weak convergence) of the normalized error processes was also studied.
In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms.
Milstein-type methods for strong approximation of systems of SDEs with a discontinuous drift coefficient
针对带分段Lipschitz不连续漂移系数的d维SDE强逼近问题，首次提出适用于任意维数的Milstein型方法，证明其Lₚ误差阶至少为3/4−，还给出σ交换时方法仅依赖有限网格W值的结论，并推导了μ连续时准Milstein格式的误差阶。
Milstein-type schemes for McKean-Vlasov SDEs driven by Brownian motion and Poisson random measure (with super-linear coefficients)
本文研究含超线性系数、由布朗运动与 Poisson 随机测度驱动的 McKean-Vlasov SDE，提出 Milstein 型格式并证明其 $\mathcal{L}^2$ 强收敛速率可任意接近一。
A Tamed Euler Scheme for SDEs with Non-Locally Integrable Drift Coefficient
针对漂移系数非局部可积的SDE常规欧拉格式$L^p$矩无界的问题，提出驯化欧拉格式，证明其以$1/2$速率$L^p$收敛，且适用于带奇异核的两标量粒子系统，为首个该场景下欧拉类格式强收敛证明。
Tamed Euler Schemes for Singular SDEs with Multiplicative Levy Noise
研究具有奇异漂移、乘性布朗及列维噪声的多维随机微分方程的驯服欧拉格式，基于非局部Zvonkin变换等方法，通过误差分解得出强收敛率，能分离跳跃贡献，在跳跃系数消失时恢复布朗型估计。
A Lyapunov-tamed Euler method for singular SDEs
本文提出一种Lyapunov镇定的欧拉方法，用于解决奇异随机微分方程的数值解问题，适用于系数在有界集上爆炸的情况，并证明其收敛阶与标准欧拉方案一致。
Tamed Euler-Maruyama method for SDEs with non-globally Lipschitz drift and multiplicative noise
本文针对具有超线性漂移和乘性噪声的随机微分方程，研究驯服Euler-Maruyama方法，并在部分耗散条件下证明了其在Wasserstein距离和全变差距离下的时间一致收敛速率。
The Milstein scheme for singular SDEs with Hölder continuous drift
针对仅具Hölder连续漂移项的奇异SDE，研究Milstein格式的$L^p$收敛速率，借助噪声正则化尤其是随机缝合法，在扩散项椭圆且正则时得到与加性噪声情形一致的速率，其所需的格式分布精确估计也具独立价值。
A Randomized Milstein Scheme for SDEs with Superlinear Drift Coefficient
本文提出了一种针对具有超线性漂移系数和有限时间正则性的随机微分方程的随机-驯化Milstein方案，实现了最优的强$\mathscr{L}^p$-收敛率。
In this paper, we consider scalar stochastic differential equations (SDEs) with a superlinearly growing and piecewise continuous drift coefficient. Existence and uniqueness of strong solutions of such SDEs are obtained. Furthermore, the classical $L_p$-error rate 1/2 for all $p\in [1, +\infty)$ is recovered for the tamed Euler scheme. A numerical example is provided to support our conclusion.
In this paper, we first establish well-posedness of McKean-Vlasov stochastic differential equations (McKean-Vlasov SDEs) with common noise, possibly with coefficients having super-linear growth in the state variable. Second, we present stable time-stepping schemes for this class of McKean-Vlasov SDEs. Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for an interacting particle system associated with the McKean-Vlasov equation. We prove stability and strong convergence of order $1/2$ and $1$, respectively. To obtain our main results, we employ techniques from calculus …
We combine the rough path theory and stochastic backward error analysis to develop a new framework for error analysis on numerical schemes. Based on our approach, we prove that the almost sure convergence rate of the modified Milstein scheme for stochastic differential equations driven by multiplicative multidimensional fractional Brownian motion with Hurst parameter $H\in(\frac14,\frac12)$ is $(2H-\frac12)^-$ for sufficiently smooth coefficients, which is optimal in the sense that it is consistent with the result of the corresponding implementable approximation of the Lévy area of fractional …
Strong convergence and temporal-spatial regularity for tamed Euler approximations of Lévy-driven SDEs
本文研究了tamed Euler近似法在具有超线性增长漂移和扩散系数的Lévy驱动SDEs中的时间-空间正则性。提出了一种新型tamed Euler型方案，建立了其强收敛性，并推导了关于初始值、初始时间和评估时间的时空正则性估计。
The main purpose of this paper is to give a solution to a long-standing unsolved problem concerning the pathwise strong approximation of stochastic differential equations with respect to the global error in the $L_{\infty}$-norm. Typically, one has average supnorm error of order $\left(\ln k/k\right)^{1/2}$ for standard approximations of SDEs with $k$ discretization points, like piecewise interpolated Ito-Taylor schemes. On the other hand there is a lower bound, which indicates that the order $1/\sqrt{k}$ is best possible for spline approximation of SDEs with $k$ free knots. The present paper …
Convergence of the tamed-Euler-Maruyama method for SDEs with discontinuous and polynomially growing drift
本文针对同时具有多项式增长和不连续漂移系数的SDE，证明了tamed-Euler-Maruyama数值格式具有1/2阶的强收敛性。
A Milstein-type method for highly non-linear non-autonomous time-changed stochastic differential equations
针对系数满足超线性增长与Hölder连续性的高度非线性非自治时间变换SDEs，提出一种Milstein型方法，证明了其有限时间强收敛性并得到收敛阶。
On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficients
本文提出了一类显式Milstein方案，用于近似具有超线性增长漂移和扩散系数的随机微分方程，证明在非常温和的条件下，这些显式方案在$\mathcal L^p$意义下以最优速率收敛于对应的SDE解。
Geometric ergodicity of modified Euler schemes for SDEs with super-linearity
本文提出适用于超线性漂移SDE的修正欧拉格式一般框架，利用精细基本耦合与反射耦合证明其及驯服欧拉格式在多种概率距离下的几何遍历性，并给出驯服欧拉格式不变测度与精确不变测度间的定量Wasserstein误差界。
Tamed-adaptive Euler-Maruyama approximation for SDEs with superlinearly growing and piecewise continuous drift, superlinearly growing and locally Hölder continuous diffusion
针对漂移分段连续、扩散局部Hölder连续的随机微分方程，证明解的存在唯一性，提出驯化自适应Euler-Maruyama格式，并分析其在有限和无限时间区间的$L^1$收敛速率。
On Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay Equations
本文研究了由莱维噪声驱动的随机微分方程的tame欧拉近似方法，分析了局部利普希茨系数下的强收敛性，并探讨了全局利普希茨连续性假设下的收敛速率，进一步扩展到延迟方程。
An explicit Milstein-type scheme for interacting particle systems and McKean--Vlasov SDEs with common noise and non-differentiable drift coefficients
针对含共同噪声与不可微漂移的McKean--Vlasov SDE及相互作用粒子系统，提出显式漂移随机化Milstein格式，在低正则假设下证得1阶强收敛，引入Spijker型范数处理粒子相互作用并讨论了格式复杂度。
We consider the long-time behavior of an explicit tamed Euler scheme applied to a class of stochastic differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses drift nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to th…
A class of implicit Milstein type methods is introduced and analyzed in the present article for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By incorporating a pair of method parameters $θ, η\in [0, 1]$ into both the drift and diffusion parts, the new schemes are indeed a kind of drift-diffusion double implicit methods. Within a general framework, we offer upper mean-square error bounds for the proposed schemes, based on certain error terms only getting involved with the exact solution processes. Such error bounds help us to easily anal…
We consider upper bounds for the approximation error E|g(X)-g(\hat X)|^p, where X and \hat X are random variables such that \hat X is an approximation of X in the L_p-norm, and the function g belongs to certain function classes, which contain e.g. functions of bounded variation. We apply the results to the approximations of a solution of a stochastic differential equation at time T by the Euler and Milstein schemes. For the Euler scheme we provide also a lower bound.
Error of randomized Milstein scheme for scalar SDEs with noisy information about coefficients and Wiener process
本文研究含噪信息下标量SDE的强逼近问题，提出基于含噪漂移、扩散及Wiener路径信息的随机Milstein格式，证明其$L^r$误差界及匹配极小极大下界，验证该格式为极小极大阶最优。
In this article we propose a new explicit Euler-type approximation method for stochastic differential equations (SDEs). In this method, Brownian increments in the recursion of the Euler method are replaced by suitable bounded functions of the Brownian increments. We prove strong convergence rate one-half for a large class of SDEs with polynomial coefficient functions whose local monotonicity constant grows at most like the logarithm of a Lyapunov-type function.
Asymptotic error distribution for tamed Euler method with coupled monotonicity condition
本文研究了具有耦合单调性条件的随机微分方程中tamed Euler方法的渐近误差分布，提出参数化方法并分析其强收敛性及误差分布特性。
A higher order approximation method for jump-diffusion SDEs with discontinuous drift coefficient
本文针对不连续漂移跳扩散 SDE 提出首个基于变换的跳跃适应拟 Milstein 高阶格式，并证明其 $L^p$ 收敛阶为 3/4，特殊情形可达 1 阶。
Weak approximation of kinetic SDEs: closing the criticality gap
本文研究了动力学随机微分方程中通用的 tame Euler-Maruyama 方案的弱收敛性，证明了该方案的边缘密度以 1/2 的速度收敛到 SDE 的边缘密度，且收敛速率与临界性缺口无关。
An Improved Milstein Method for the Numerical Solution of Multidimensional Stochastic Differential Equations
本文提出改进的Milstein方法，用于多维随机微分方程的数值解，通过新的数值技术提高收敛速率并解决随机积分计算问题。
Strong order 1 adaptive approximation of jump-diffusion SDEs with discontinuous drift
本文提出了一种强阶1自适应逼近方案，用于解决具有不连续漂移的跳扩散SDEs问题，通过双适应策略提升收敛性。
Milstein schemes and antithetic multilevel Monte Carlo sampling for delay McKean-Vlasov equations and interacting particle systems
针对带点时滞、漂移项可能超线性增长的McKean-Vlasov方程对应的相互作用粒子系统，推导了Milstein格式并证明其一阶强收敛与矩稳定，还提出无需模拟Lévy面积的对偶多层Milstein格式以实现解的期望泛函的最优复杂度估计。
In this paper, we derive fully implementable first order time-stepping schemes for McKean--Vlasov stochastic differential equations (McKean--Vlasov SDEs), allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretised interacting particle system associated with the McKean--Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with…
Error estimates for tamed Euler and Randomized Euler schemes for SDEs with locally Lipschitz drift with applications to non-logconcave sampling and optimization
针对具有局部Lipschitz超线性增长漂移的随机微分方程，提出KL加速驯服未调整Langevin算法(kTULA)和驯服随机中点方案(tRLMC)，基于移位-组合方法建立局部误差框架，得到有限时间非渐近误差估计，并在对数Sobolev不等式下实现近最优迭代复杂度。
We present an error analysis of weak convergence of one-step numerical schemes for stochastic differential equations (SDEs) with super-linearly growing coefficients. Following Milstein's weak error analysis on the one-step approximation of SDEs, we prove a general conclusion on weak convergence of the one-step discretization of the SDEs mentioned above. As applications, we show the weak convergence rates for several numerical schemes of half-order strong convergence, such as tamed and balanced schemes. Numerical examples are presented to verify our theoretical analysis.
Existence, uniqueness, and $L_p$-approximation results are presented for scalar stochastic differential equations (SDEs) by considering the case where, the drift coefficient has finitely many spatial discontinuities while both coefficients can grow superlinearly (in the space variable). These discontinuities are described by a piecewise local Lipschitz continuity and a piecewise monotone-type condition while the diffusion coefficient is assumed to be locally Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. Moreover, the superlinear nature of the coe…
In recent years, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space has begun. In many of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. For scalar SDEs of that type the best $L_p$-error rate known so far for approximation of the solution at the final time point is $3/4$ in terms of the number of evaluations of the drivin…
An Explicit Euler-type Scheme for Lévy-driven SDEs with Superlinear and Time-Irregular Coefficients
针对带超线性、时间不规则系数的Lévy驱动SDE，提出结合漂移随机化与驯化技术的随机驯化欧拉格式，证明其强收敛率可任意接近0.5，还拓展至随机系数等情形，填补相关研究空白。
In this paper we study strong approximation of the solution of a scalar stochastic differential equation (SDE) at the final time in the case when the drift coefficient may have discontinuities in space. Recently it has been shown in [Müller-Gronbach, T., and Yaroslavtseva, L., On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient, arXiv:1809.08423 (2018)] that for scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient the classical Euler-Maruyama…
Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient
本文证明了在漂移系数分段Lipschitz连续且导数分段Lipschitz连续的标量SDE强逼近中，无需有界性和单调性假设即可达到最优$L^p$误差阶$3/4$，并利用此前仅用于上界的变换技术完成下界证明。
$L^2$-Wasserstein contraction of modified Euler schemes for SDEs with high diffusivity and applications
本文针对漂移项超线性增长且仅在闭球外耗散的SDE，通过同步耦合和拟度量构造，证明了高扩散性下修正Euler格式的$L^2$-Wasserstein收缩，并应用于Poincaré不等式、误差界及强大数律，显著提升了收敛速率。
A Multi-level Monte Carlo simulation for invariant distribution of Markovian switching Lévy-driven SDEs with super-linearly growth coefficients
本文提出结合驯化自适应Euler-Maruyama与多层蒙特卡洛的方法，用于逼近马尔可夫切换Lévy驱动的超线性增长SDE的不变分布。
Non-asymptotic convergence bounds of modified EM schemes for non-dissipative SDEs
本文提出修正欧拉方案用于非耗散SDEs，建立了非渐近收敛界，并探讨了其在瓦瑟斯坦距离下的收敛速率。
Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmet…
In this paper, we introduce adaptive Euler-Maruyama schemes for McKean-Vlasov stochastic differential equations (SDEs) assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity in the state variable for either, while global Lipschitz continuity is required for the measure component only. We prove moment stability of the discretised processes and a strong convergence rate of $1/2$. Several numerical examples, centred around a mean-field model for FitzHugh-Nagumo neurons, illustrate that the standard uniform scheme fails and that t…
Quantitative approximation of stochastic kinetic equations: from discrete to continuum
本文研究带奇异系数的动力学型随机微分方程，证明低正则漂移下系统在强弱意义下良定义，且通用驯服EM格式均以1/2速率收敛。
An Antithetic Multilevel Monte Carlo-Milstein Scheme for Stochastic Partial Differential Equations with non-commutative noise
本文将有限维对偶Milstein格式推广到Hilbert空间值SPDE，提出截断Milstein时间步进与对偶耦合结合的多层蒙特卡洛方法，在不引入额外偏差的前提下显著降低计算复杂度。
Tamed Euler approximation for fully superlinear growth McKean-Vlasov SDE and their particle systems: sharp rates for strong propagation of chaos, convergence and ergodicity
针对全变量超线性增长的 McKean–Vlasov SDE 及粒子系统，论文提出全显式驯服 Euler 格式，证明适定性、维数一致的 N^{-1/2} 混沌传播速率、1/2 强收敛速率及指数遍历性。
Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients
针对系数非全局Lipschitz连续的SDE，提出无Lévy面积的改进Milstein型格式，证明半阶强收敛性，建立对偶MLMC的最优计算成本理论，并通过数值实验验证结论。
On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients
本文研究Lévy过程驱动的McKean-Vlasov随机微分方程的数值逼近，提出了一种驯服自适应Euler-Maruyama格式，并证明了其在非全局Lipschitz连续且超线性增长系数下于有限和无限时间跨度的强收敛性。
We prove the well-posedness of solutions to McKean-Vlasov stochastic differential equations driven by Lévy noise under mild assumptions where, in particular, the Lévy measure is not required to be finite. The drift, diffusion and jump coefficients are allowed to be random, can grow super-linearly in the state variable, and all may depend on the marginal law of the solution process. We provide a propagation of chaos result under more relaxed conditions than those existing in the literature, and consistent with our well-posedness result. We propose a tamed Euler scheme for the associated interac…
Uniform in time convergence of numerical schemes for stochastic differential equations via Strong Exponential stability: Euler methods, Split-Step and Tamed Schemes
本文提出一个充分且必要的准则，证明在局部Lipschitz和严格单调条件下，显式/隐式欧拉、分裂步和驯化欧拉等数值格式对SDE解的概率分布实现时间一致收敛，并通过数值算例验证。
Uniform-in-time error estimates for McKean-Vlasov SDEs with common noise and stochastic algorithms
通过构造反射渐近耦合，对带共同噪声的McKean-Vlasov SDE与相互作用粒子系统诱导的测度值过程，在漂移项仅长程耗散下得到概率距离的均匀时间估计，并应用于数值方法的误差估计和条件混沌传播的量化。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。