TCI for SDEs with irregular drifts

带非正则漂移项的随机微分方程的极限定理

Limit theorems for SDEs with irregular drifts

本文研究Hölder连续及分段连续漂移项SDEs加性泛函的极限定理与收敛速率，通过反射耦合与测试函数得到解过程的（拟）Wasserstein距离指数收缩性以支撑结论。

Path-dependent SDEs with jumps and irregular drift: well-posedness and Dirichlet properties

We discuss a concept of path-dependent SDE with distributional drift with possible jumps. We interpret it via a suitable martingale problem, for which we provide existence and uniqueness. The corresponding solutions are expected to be Dirichlet processes, nevertheless we give examples of solutions which do not fulfill this property. In the second part of the paper we indeed state and prove significant new results on the class of Dirichlet processes.

A functional limit theorem for irregular SDEs

Let $X_1, X_2, \ldots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y^N_{k+1} = Y^N_{k} + a_N(Y^N_k) X_{k+1}$, where $a_N: \mathbb R \to \mathbb R_+$. We show, under mild assumptions on the law of $X_i$, that one can choose the scale factor $a_N$ in such a way that the process $(Y^N_{\lfloor N t \rfloor})_{t \in \mathbb R_+}$ converges in distribution to a given diffusion $(M_t)_{t \in \mathbb R_+}$ solving a stochastic differential equation with possibly irregular coefficients, as $N \to \infty$. To this end we embed the…

Degenerate Irregular SDEs with Jumps and Application to Integro-Differential Equations of Fokker-Planck type

We investigate stochastic differential equations with jumps and irregular coefficients, and obtain the existence and uniqueness of generalized stochastic flows. Moreover, we also prove the existence and uniqueness of $L^p$-solutions or measure-valued solutions for second order integro-differential equation of Fokker-Planck type.

Strong Solutions of Mean-Field SDEs with irregular expectation functional in the drift

We analyze multi-dimensional mean-field stochastic differential equations where the drift depends on the law in form of a Lebesgue integral with respect to the pushforward measure of the solution. We show existence and uniqueness of Malliavin differentiable strong solutions for irregular drift coefficients, which in particular include the case where the drift depends on the cumulative distribution function of the solution. Moreover, we examine the solution as a function in its initial condition and introduce sufficient conditions on the drift to guarantee differentiability. Under these assumpt…

具有非正则漂移项的随机微分方程 Euler 方法的中心极限定理

A central limit theorem for the Euler method for SDEs with irregular drifts

本文研究弱正则漂移下多维 SDE 的 Euler-Maruyama 逼近误差，证明重标度涨落的中心极限定理，并分别用混合 Young-Itô 方程和变换后 SDE 刻画极限。

Stochastic flow for SDEs with jumps and irregular drift term

We consider non-degenerate SDEs with a $β$-Holder continuous and bounded drift term and driven by a Levy noise $L$ which is of $α$-stable type. If $α\in [1,2)$ and $β\in (1 - \fracα{2},1) $ we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise $L$. In our previous paper $L$ was assumed to be non-degenerate, $α$-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes of temperated stable processes.

具有不规则系数的SDEs的双因果最优传输

Bicausal optimal transport for SDEs with irregular coefficients

本文研究了具有不规则系数的SDEs的约束最优传输问题，通过双因果耦合方法计算适应性Wasserstein距离，并提出了基于变换的半隐式数值方案，首次证明了指数增长和不连续漂移SDEs的强收敛性。

Stochastic Flows of SDEs with Irregular Coefficients and Stochastic Transport Equations

In this article we study (possibly degenerate) stochastic differential equations (SDE) with irregular (or discontiuous) coefficients, and prove that under certain conditions on the coefficients, there exists a unique almost everywhere stochastic (invertible) flow associated with the SDE in the sense of Lebesgue measure. In the case of constant diffusions and BV drifts, we obtain such a result by studying the related stochastic transport equation. In the case of non-constant diffusions and Sobolev drifts, we use a direct method. In particular, we extend the recent results on ODEs with non-smoot…

Weak convergence of Euler scheme for SDEs with singular drift

In this paper, we investigate the weak convergence rate of Euler-Maruyama's 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.

具有不连续和多项式增长漂移的SDE的tamed-Euler-Maruyama方法的收敛性

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。

The Euler-Maruyama Scheme for SDEs with Irregular Drift: Convergence Rates via Reduction to a Quadrature Problem

We study the strong convergence order of the Euler-Maruyama scheme for scalar stochastic differential equations with additive noise and irregular drift. We provide a general framework for the error analysis by reducing it to a weighted quadrature problem for irregular functions of Brownian motion. Assuming Sobolev-Slobodeckij-type regularity of order $κ\in (0,1)$ for the non-smooth part of the drift, our analysis of the quadrature problem yields the convergence order $\min\{3/4,(1+κ)/2\}-ε$ for the equidistant Euler-Maruyama scheme (for arbitrarily small $ε>0$). The cut-off of the convergence …

针对具有超线性且时间不规则系数的Lévy驱动SDE的显式欧拉型格式

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。

Optimal bounds for the densities of solutions of SDEs with measurable and path dependent drift coefficients

We consider a process given as the solution of a stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. Explicit and optimal bounds for the Lebesgue density of that process at any given time are derived. The bounds and their optimality is shown by identifying the worst case stochastic differential equation. Then we generalise our findings to a larger class of diffusion coefficients.

Regularity properties of jump diffusions with irregular coefficients

In this paper we investigate the regularity properties of strong solutions to SDEs driven by Lévy processes with irregular drift coefficients. Under some mild conditions, we show that the singular SDE has a unique strong solution for each starting point and the family of all the solutions forms a stochastic flow. Moreover, the Malliavin differentiability of the strong solutions is also obtained. As an application, we also prove a path-by-path uniqueness result for some related random ODEs.

Hölder continuous densities of solutions of SDEs with measurable and path dependent drift coefficients

We consider a process given as the solution of a one-dimensional stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. Hölder continuity of the Lebesgue density of that process at any given time is achieved using a different approach than the classical ones in the literature. Namely, the Hölder regularity of the densities is obtained via a control problem by identifying the stochastic differential equation with the worst global Hölder constant. Then we generalise our findings to a larger class of diffusion coefficients. The…

Singular SDEs with critical non-local and non-symmetric Lévy type generator

In this work, by using Levi's parametrix method we first construct the fundamental solution of the critical non-local operator perturbed by gradient. Then, we use the obtained estimates to prove the existence and uniqueness of strong solutions for stochastic differential equation driven by Markov process with irregular coefficients, whose generator is a non-local and non-symmetric Lévy type operator.

由Lévy噪声驱动的具有不规则漂移的SDE的Euler格式的强收敛速率

Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise

本文研究了由Lévy噪声驱动且具有不规则Hölder漂移的多维SDE的Euler-Maruyama格式的强收敛速率，利用随机缝缀论证等技术在更宽参数范围内获得了不依赖于p的收敛速率，并推导出几乎必然收敛速率及非光滑泛函逼近的收敛速率。

具有不规则分布依赖扩散系数的平均场随机微分方程

Mean field stochastic differential equations with a diffusion coefficient with irregular distributional dependence

研究扩散系数不连续依赖分布的平均场随机微分方程，通过Lamperti型变换转化为确定性系数方程，证明存在唯一性并讨论失效情形。

Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients

In this article we prove the existence and uniqueness for degenerate stochastic differential equations with Sobolev (possibly singular) drift and diffusion coefficients in a generalized sense. In particular, our result covers the classical DiPerna-Lions flows and, we also obtain the well-posedness for degenerate Fokker-Planck equations with irregular coefficients. Moreover, a large deviation principle of Freidlin-Wenzell type for this type of SDEs is established.

Strong completeness for a class of stochastic differential equations with irregular coefficients

We prove the strong completeness for a class of non-degenerate SDEs, whose coefficients are not necessarily uniformly elliptic nor locally Lipschitz continuous nor bounded. Moreover, for each $t$, the solution flow $F_t$ is weakly differentiable and for each $p>0$ there is a positive number $T(p)$ such that for all $t<T(p)$, the solution flow $F_t(\cdot)$ belongs to the Sobolev space $W_{\loc}^{1,p}$. The main tool for this is the approximation of the associated derivative flow equations. As an application a differential formula is also obtained.

Zero noise limits using local times

We consider a well-known family of SDEs with irregular drifts and the correspondent zero noise limits. Using (mollified) local times, we show which trajectories are selected. The approach is completely probabilistic and relies on elementary stochastic calculus only.

Distribution dependent SDEs driven by additive fractional Brownian motion

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…

带乘性噪声的退化分布依赖SDE的弱存在性——一种路径正则化方法

Weak Existence for Degenerate Distribution Dependent SDEs with multiplicative Noise -- a pathwise regularization approach

本研究将现有弱存在性技术拓展至分布依赖框架，结合路径平均与局部时分解方法，证明了带退化乘性噪声和奇异系数的分布依赖随机微分方程弱解的存在性，揭示了不规则噪声对随机系统退化相关分析难题的正则化作用。

具有Hölder连续漂移系数的SDE的随机化Euler-Maruyama方法

Randomised Euler-Maruyama method for SDEs with Hölder continuous drift coefficient

本文研究了随机化Euler-Maruyama方法在求解具有时间与空间Hölder连续不规则漂移的加性非时齐SDE时的性能，利用随机sewing引理证明其强收敛阶高于标准EM方法。

完全次临界机制下分数阶SDE的解理论

Solution theory of fractional SDEs in complete subcritical regimes

本文研究了由分数布朗运动驱动的SDEs，其漂移系数在尺度意义上可任意接近临界值，并提出包含强存在性、逐路径唯一性、解流存在性、Malliavin可微性及$ρ$-不规则性的完整解理论，进而可处理McKean-Vlasov等方程。

由长程依赖分数布朗运动及其他Volterra过程驱动的奇异随机微分方程的强解

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。

Stochastic differential equations with irregular coefficients:~mind the gap!

Numerical methods for stochastic differential equations with non-globally Lipschitz coefficients are currently studied intensively. This article gives an overview of our work for the case that the drift coefficient is potentially discontinuous complemented by other important results in this area. To make the topic accessible to a broad audience, we begin with a heuristic on SDEs and a motivation.

由 $α$-稳定 Lévy 过程驱动的具有 Hölder 连续漂移系数的 SDE 的随机化 Euler-Maruyama 方法

Randomised Euler-Maruyama Method for SDEs with Hölder Continuous Drift Coefficient Driven by $α$-stable Lévy Process

本文研究了由 $α$-稳定 Lévy 过程驱动且漂移项 Hölder 连续的 SDE 的随机化 Euler-Maruyama 方法，证明了其在 $L^p$-范数下的强收敛阶优于标准 EM 方法，并推广了已有高斯噪声驱动情形的结论。

噪声局部耦合技术及其在不规则系数SDEs强逼近误差下界中的应用

The local coupling of noise technique and its application to lower error bounds for strong approximation of SDEs with irregular coefficients

本文研究非Lipschitz系数SDEs的强逼近误差下界，提出噪声局部耦合技术，证明了分段Lipschitz漂移和非可加扩散下最终时间逼近$L^p$误差率至多3/4，局部有界漂移和局部Lipschitz扩散下全局逼近误差率至多1/2。

$L^p(Ω)$-Difference of One-Dimensional Stochastic Differential Equations with Discontinuous Drift

We consider a one-dimensional stochastic differential equations (SDE) with irregular coefficients. The purpose of this paper is to estimate the $L^p(Ω)$-difference of SDEs using the norm of the difference of coefficients, where the discontinuous drift coefficient satisfies a one-sided Lipschitz condition and the diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. As an application, we consider the stability problem.

关于具有奇异和无界系数的SDE的导数估计

Derivative estimates for SDEs with singular and unbounded coefficients

本文提出统一的PDE-概率框架，用于估计与具有奇异和无界系数的随机微分方程相关马尔可夫半群的点wise梯度和Hessian估计，通过局部结构假设和积分正则性条件，获得短时正则化估计和长时衰减界。

On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift

The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $α$-Hölder drift in the recent literature the rate $α/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $α>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.

Strong solutions of a stochastic differential equation with irregular random drift

We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form $$\mathrm{d} X= u(ω,t,X)\, \mathrm{d} t + \frac12 σ(ω,t,X)σ'(ω,t,X)\,\mathrm{d} t + σ(ω,t,X) \, \mathrm{d}W(t), $$ where the drift coefficient $u$ is random and irregular. The random and regular noise coefficient $σ$ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that $u$ belongs to $L^p(Ω; L^\infty([0,T];\dot{H}^1(\mathbb{R})))$ for any finite $p\ge 1$, $\mathbb{E}\left|u(t)-u(0)\right|_{\dot{H}^1(\mathbb{R})}^2 \to 0$ as $t\d…

关于McKean-Vlasov随机微分方程的Euler-Maruyama格式的收敛性

On the convergence of the Euler-Maruyama scheme for McKean-Vlasov SDEs

本文基于Wasserstein空间中向后Kolmogorov方程的适定性，分析Euler-Maruyama格式逼近McKean-Vlasov SDE的强、弱收敛率，在不规则系数和光滑系数两种情形下分别给出关于粒子数N和网格尺寸h的收敛率，并证明光滑情形下弱误差率最优。

在希尔伯特空间中具有不规则漂移的随机微分方程的唯一性

Uniqueness for stochastic differential equations in Hilbert spaces with irregular drift

本文提出一种新方法，研究在希尔伯特空间中具有不规则漂移的随机微分方程的强存在性和唯一性，扩展了Da Prato和Flandoli的工作。

神经随机Volterra方程：学习路径依赖动态

Neural stochastic Volterra equations: learning path-dependent dynamics

本文提出神经随机Volterra方程，用于学习具有路径依赖性的动态过程，并在多个SVEs上验证了其有效性。

通过复合泊松过程对随机Volterra方程进行强逼近

Strong approximation for stochastic Volterra equations by compound Poisson processes

本文提出通过复合泊松过程对具有奇异系数的随机Volterra方程进行强逼近，克服了传统Euler-Maruyama方法在时间不规则系数和奇异性下的局限性。

具有有界变分漂移的系统随机最优控制：最大原理方法

Stochastic Optimal Control for Systems with Drifts of Bounded Variation: A Maximum Principle Approach

本文研究非线性系统在随机微分方程中具有不规则漂移的控制问题，通过最大原理建立理论框架，并应用到保险盈余模型中，提出最优走廊型资本调整策略。