Mihai Gradinaru, Ivan Nourdin
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.
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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.
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。
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。
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 …
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。
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 …
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。
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…
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。
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。
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.
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。
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。
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…
Limit error distributions of Milstein scheme for stochastic Volterra equations with singular kernels
本文证明奇异核随机Volterra方程的Milstein格式收敛速率$n^{-2H}$最优，并揭示归一化误差的极限分布为随机系数线性Volterra方程的非零解。
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.
We investigate the problem of nonparametric estimation of the trend for stochastic differential equations with delay and driven by a fractional Brownian motion through the method of kernel-type estimation for the estimation of a probability density function.
In this paper, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H.
For degenerate stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H>1/2$, the derivative formulas are established by using Malliavin calculus and coupling method, respectively. Furthermore, we find some relation between these two approaches. As applications, the (log) Harnack inequalities and the hyperbounded property are presented.
Under the key assumption of finite ρ-variation, ρ\in[1,2), of the covariance of the underlying Gaussian process, sharp a.s. convergence rates for approximations of Gaussian rough paths are established. When applied to Brownian resp. fractional Brownian motion (fBM), ρ=1 resp. ρ=1/(2H), we recover and extend the respective results of [Hu--Nualart; Rough path analysis via fractional calculus; TAMS 361 (2009) 2689-2718] and [Deya--Neuenkirch--Tindel; A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion; AIHP (2011)]. In particular, we establish an a.s. rate…
In this note, we take up the study of weak convergence for stochastic differential equations driven by a (Liouville) fractional Brownian motion $B$ with Hurst parameter $H\in(1/3,1/2)$. In the current paper, we approximate the $d$-dimensional fBm by the convolution of a rescaled random walk with Liouville's kernel. We then show that the corresponding differential equation converges in law to a fractional SDE driven by $B$.
In this paper, by using Girsanov's transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with Hurst index $H\in (1/2,1)$.
An estimation of Fisher information bound for distribution-dependent SDEs driven by fractional Brownian motion with small noise
针对小噪声分数布朗运动驱动的分布依赖SDE，本文估计其解在中心极限定理中的Fisher信息收敛速率，并证明该速率为最优阶。
In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian Motion with Hurst parameter H>1/2 and obtain some (functional) convergences properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.
We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition, otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.
In this work, we will show the existence, uniqueness, and weak differentiability of the solution to semi-linear mean-field stochastic differential equations driven by fractional Brownian motion. We prove an extension of the Bismut-Elworthy-Li formula and show some applications in the sensitivity analysis of variance swaps and also the price of derivatives with respect to the initial point.
In this paper, high-order moment, even exponential moment, estimates are established for the Hölder norm of solutions to stochastic differential equations driven by fractional Brownian motion whose drifts are measurable and have linear growth. As applications, we first study the weak uniqueness of solutions to fractional stochastic differential equations. Moreover, combining our estimates and the Fourier transform, we establish the existence of density of solutions to equations with irregular drifts.
We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander's condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that they admit a unique stationary solution that is physical in the sense that it does not "look into the future". The main technical result required for the analysis is a bound on the moments of the inverse of…
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…
The Onsager-Machlup action functional for degenerate SDEs driven by fractional Brownian motion
该研究针对分数布朗运动驱动的退化 SDE，在适当条件下建立 Onsager–Machlup 作用泛函的显式形式及相应分数 Euler–Lagrange 方程，并通过示例验证结果。
Euler scheme for SDEs driven by fractional Brownian motions: Malliavin differentiability and uniform upper-bound estimates
本文研究了由分数布朗运动驱动的随机微分方程Euler格式的Malliavin可微性，并推导出步长一致且逐路径的上界估计。
It is known that the transition probabilities of a solution to a classical Itô stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coeffcients determined by the corresponding SDE. Time-fractional Kolmogorov type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distribut…
Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod's selection theorem.
On the density of singular SDEs with fractional noise and applications to McKean-Vlasov equations
该研究针对带分数布朗噪声和奇异漂移的SDE解的分布，量化其空间正则性与时间可积性、证明其具有高斯尾密度，并据此得到卷积型McKean-Vlasov方程解存在唯一性的新结果。
Strong solutions to SDEs with singular drifts driven by fractional Brownian motions
本文研究了由分数布朗运动驱动的具有奇异漂移的随机微分方程的强解存在性，扩展了经典结果并证明了 Sobolev 偏微分同胚的随机流存在性。
In this paper we study a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H\in(1/2,1). We prove the well-posedness of this type equations, and then establish a general result on the Bismut formula for the Lions derivative by using Malliavin calculus. As applications, we provide the Bismut formulas of this kind for both non-degenerate and degenerate cases, and obtain the estimates of the Lions derivative and the total variation distance between the laws of two solutions.
Wellposedness and averaging principle for conditional distribution dependent SDEs driven by standard Brownian motions and fractional Brownian motions
本文研究了由标准Brown运动和分数阶Brown运动驱动的条件分布依赖型随机微分方程，通过不动点定理证明了解的存在唯一性，并建立了其解可由平均方程逼近的平均原理。
In this paper, we study small-time asymptotic behaviors for a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H\in(1/2,1)$ and magnitude $\ep^H$. By building up a variational framework and two weak convergence criteria in the factional Brownian motion setting, we establish the large and moderate deviation principles for this type equations. Besides, we also obtain the central limit theorem, in which the limit process solves a linear equation involving the Lions derivative of the drift coefficient.
On mixed fractional SDEs with discontinuous drift coefficient
本文研究由标准与分数布朗运动共同驱动且漂移不连续的混合分数SDE，通过建立适用于导数绝对连续函数的广义Itô法则并证明分布密度存在，得到解的存在唯一性。
Exponential Euler method for stiff SDEs driven by fractional Brownian motion
本文针对由分数布朗运动驱动的刚性随机微分方程，利用 Malliavin 导数技术证明指数欧拉方法可达到收敛阶 1，优于已有接近 Hurst 参数 H 的结果，并达到数值模拟最优速率。
In this paper we obtain Gaussian-type lower bounds for the density of solutions to stochastic differential equations (SDEs) driven by a fractional Brownian motion with Hurst parameter $H$. In the one-dimensional case with additive noise, our study encompasses all parameters $H\in(0,1)$, while the multidimensional case is restricted to the case $H>1/2$. We rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools.
In this paper, we will focus - in dimension one - on the SDEs of the type dX_t=s(X_t)dB_t+b(X_t)dt where B is a fractional Brownian motion. Our principal motivation is to describe one of the simplest theory - from our point of view - allowing to study this SDE, and this for any Hurst index H between 0 and 1. We will consider several definitions of solution and we will study, for each one of them, in which condition one has existence and uniqueness. Finally, we will examine the convergence or not of the canonical scheme associated to our SDE, when the integral with respect to fBm is defined usi…
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…
Regularities for distribution dependent SDEs with fractional noises
本文研究了由两个独立分数阶噪声驱动的分布依赖型 SDE 的正则性，利用向后耦合论证与分数阶导数算子分析，在非退化与退化情形下建立了 Lions 导数的 log-Harnack 不等式与 Bismut 公式。
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…
Strong rate of convergence for the Euler-Maruyama scheme of additive fractional SDEs with Lipschitz drift
本文研究了加性分数随机微分方程Euler-Maruyama方案的强收敛速率，证明了在不同Hurst参数范围内收敛速率的差异，并利用随机缝合论证控制噪声误差。
Distribution dependent SDEs with multiplicative fractional noise
本文研究了由分数阶布朗运动与分布相依乘性噪声驱动的分布相依随机微分方程的适定性，引入概率测度路径的 Hölder 空间并结合压缩映射原理与分数阶微积分工具，进一步建立了方程的大偏差与中偏差原理。
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 Itô 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 …
Uniform pathwise stability of additive singular SDEs driven by fractional Brownian motion
本文研究了分数布朗运动驱动的加性奇异随机微分方程的长期行为，通过统一路径估计方法证明了解的均匀路径稳定性与不变测度的唯一性。
Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion
本文推导了由分数布朗运动驱动的麦肯-瓦拉索夫随机微分方程的福克-普兰克方程，并建立了分数费曼-科克表示，揭示了分数噪声与均场相互作用对系统结构的影响。
Strong solutions of fractional Brownian sheet driven SDEs with integrable drift
本文研究了由Hurst参数小于1/2的分数布朗运动驱动的随机微分方程的强解存在性，利用Malliavin微积分和扇形局部非确定性理论证明了唯一可微强解的存在。
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…
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