Andreas Neuenkirch, Ivan Nourdin
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.
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The existence and uniqueness are established for McKean-Vlasov SDEs driven by Lévy processes. By using an approximation technique and coupling by change of measures, Harnack inequalities are investigated for McKean-Vlasov SDEs driven by subordinate Brownian motions.
Using coupling by change of measure and an approximation technique, Wang's Harnack inequalities are established for a class of functional SDEs driven by subordinate Brownian motions. The results cover the corresponding ones in the case without delay.
This paper is devoted to studying the properties of the exit times of stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs). In particular, we prove that the exit times of $G$-SDEs has the quasi-continuity property. As an application, we give a probabilistic representation for a large class of fully nonlinear elliptic equations with Dirichlet boundary.
Sufficient and necessary conditions are presented for the comparison theorem of path dependent $G$-SDEs. Different from the corresponding study in path independent $G$-SDEs, a probability method is applied to prove these results. Moreover, the results extend the ones in the linear expectation case.
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.
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.
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, 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 $H\in (1/2,1)$.
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…
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 this paper, we build the equivalence between rough differential equations driven by the lifted $G$-Brownian motion and the corresponding Stratonovich type SDE through the Wong-Zakai approximation. The quasi-surely convergence rate of Wong-Zakai approximation to $G-$SDEs with mesh-size $\frac{1}{n}$ in the $α$-Hölder norm is estimated as $(\frac{1}{n})^{\frac12-}.$ As corollary, we obtain the quasi-surely continuity of the above RDEs with respect to uniform norm.
In this paper we consider a class of stochastic differential equations driven by subordinate Brownian motion with Markovian switching. We use Malliavin calculus to study the smoothness of the density for the solution under uniform Hörmander's type condition.
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.
The path independence of additive functionals for SDEs driven by the G-Brownian motion is characterized by nonlinear PDEs. The main result generalizes the existing ones for SDEs driven by the standard Brownian motion.
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.
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.
In this note, we consider Jensen's inequality for the nonlinear expectation associated with backward SDEs driven by $G$-Brownian motion ($G$-BSDEs for short). At first, we give a necessary and sufficient condition for $G$-BSDEs under which one-dimensional Jensen inequality holds. Second, we prove that for $n>1$, the $n$-dimensional Jensen inequality holds for any nonlinear expectation if and only if the nonlinear expectation is linear, which is essentially due to Jia (Arch. Math. 94 (2010), 489-499). As a consequence, we give a necessary and sufficient condition for $G$-BSDEs under which the $…
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.
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。
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。
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.
Distribution dependent stochastic differential equations have been a very hot subject with extensive studies. On the other hand, under the $G$-expectation framework, stochastic differential equations driven by $G$-Brownian motion (in short form, $G$-SDEs) have received increasing attentions, and the existence and uniqueness of solutions to $G$-SDEs under Lipschitz and non-Lipschitz conditions have been obtained. Based on these studies, it is very natural and also important to investigate the $G$-SDEs which are also distribution dependent. In this paper, we are concerned with the well-posedness…
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, the stability behaviors of stochastic differential equations (SDEs) driven by time-changed Brownian motions are discussed. Based on the generalized Lyapunov method and stochastic analysis, necessary conditions are provided for solutions of time-changed SDEs to be stable in differential senses, such as stochastic stability, stochastically asymptotic stability and globally stochastically asymptotic stability. Also, a connection between the stability of the solution to the time-changed SDEs and that to their corresponding non-time-changed SDEs is revealed by applying the duality th…
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…
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。
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.
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.
Doubly Reflected Backward SDEs Driven by G-Brownian Motions and Fully Nonlinear PDEs with Double Obstacles
本文提出通过单调递减的惩罚反射G-BSDE逼近解的新方法，在已知最弱条件下证明了双重反射G-BSDE解的适定性，并首次揭示其与带双重障碍的完全非线性偏微分方程的关系。
Mean-Field SDEs driven by $G$-Brownian Motion
将平均场SDE推广至$G$-布朗运动驱动的SDE，系数依赖解作为随机变量。
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.
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。
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 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 …
The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the funda…
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 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.
Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator
本文研究由$G$-布朗运动驱动的双反射反向随机微分方程在二次生成函数下的性质，证明了解的存在性和唯一性，并通过罚函数反射建立其与具有双障碍的非线性偏微分方程的关系。
We study the relationship between mixed stochastic differential equations and the corresponding rough path equations driven by standard Brownian motion and fractional Brownian motion with Hurst parameter $H>1/2$. We establish a correction formula, which relates both types of equations, analogously to the Itō-Stratonovich correction formula. This correction formula allows to transfer properties, which are established for one type of equation to the other, and we will illustrate this by considering numerical methods for mixed and rough SDEs
We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a measurable way) on a finite number of sequential observations of the driving Brownian motion. The resulting lower error bounds hold in particular for all methods that are implementable on a computer and use a random number generator to simulate the driving Brownian motion at finitely many points. Our analysis shows that approximation at a single point is strongly con…
We consider non degenerate Brownian SDEs with H{ö}lder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial H{ö}lder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.
Strong solutions for singular SDEs driven by long-range dependent fractional Brownian motion and other Volterra processes
该研究针对长程依赖分数布朗运动等Volterra过程驱动的奇异随机微分方程，突破漂移项需Hölder连续的限制，建立了奇异漂移项下的强解存在唯一性结果，采用概率论证方法。
We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations
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…
In this paper, we show that the integration of a stochastic differential equations driven by G-Brownian motion in R can be reduced to the integration of an ordinary differential equations parametrized by a variable in (Ω,F). We study the sample solutions of G-SDEs by an extention of G-Itô formula. And then we also get a comparison theorem for G-SDEs and its applications.
Distribution-flow dependent SDEs driven by (fractional) Brownian motion and Navier-Stokes equations
该研究提出一类分布流依赖的新型随机微分方程，证明其强/弱解的存在唯一性，揭示其与两类偏微分方程的关联，并证明带分数布朗噪声的二维纳维-斯托克斯方程的全局适定性与光滑性。
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…
In this paper we study the stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs for short). We extend the notion of conditional $G$-expectation from deterministic time to the more general optional time situation. Then, via this conditional expectation, we develop the strong Markov property for $G$-SDEs. In particular, we obtain the strong Markov property for $G$-Brownian motion. Some applications including the reflection principle for $G$-Brownian motion are also provided.
We show that solutions to multidimensional SDEs with Lipschitz coefficients and driven by Brownian motion never reach the set where all coefficients vanish unless the initial position belongs to that set.
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。
We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $H\in (0,1)$. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$B(\cdot,μ) = f\astμ(\cdot) + g(\cdot),\quad f,g\in B^α_{\infty,\infty}, \quad α>1-1/2H,$$ thus extending the results by Catellier and Gubinelli [9] to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the u…
Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime
本文研究了具有奇异漂移和分数布朗运动或莱维噪声的SDE的弱存在性，提出了在最优可积指数条件下成立的结论，并在一维情况下证明了强解的存在。
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。
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。
Gaussian-type density estimates for mixed SDEs driven by correlated fractional Brownian motions
本文研究由相关分数布朗运动混合驱动的多维 SDE 弱解的类高斯密度，在短程与长程相关情形下分别施加奇异与 Hölder 连续漂移，基于 Girsanov 定理与指数 Orlicz 空间框架证明密度存在性并导出高斯型上下界。
Tanaka formula for SDEs driven by fractional Brownian motion
本文为分数布朗运动驱动的非线性随机微分方程推导出 Tanaka 型公式，包含 Skorokhod 积分与 Malliavin 迹修正，并通过新方法建立 L2 收敛，推广至凸泛函。
Constructions of numerous approximate sampling algorithms are based on the well-known fact that certain Gibbs measures are stationary distributions of ergodic stochastic differential equations (SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions it can be shown that the associated SDE is not exponentially ergodic and that related sampling algorithms may perform poorly. A natural idea that has recently been explored in the machine learning literature in this context is to make use of stochastic processes with heavy tails instead of the Brownian motion. In this pape…
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。
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