Large deviations for multidimensional SDEs with reflection
Zongxia Liang
建立了一类具反射边界条件的多维退化随机微分方程的大偏差原理。结果包括两种情况下，初始条件的适应和预期。
The large deviations principles are established for a class of multidimensional degenerate stochastic differential equations with reflecting boundary conditions. The results include two cases where the initial conditions are adapted and anticipated.
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On the comparison theorem for multidimensional SDEs with jumps
本文给出了多维带跳随机微分方程和矩阵值带跳随机微分方程比较定理成立的一个充要条件.
Strong Convergence of Wong-Zakai Approximations of Reflected SDEs in A Multidimensional General Domain
本文得到了一般多维区域上反射型偏微分方程的Wong-Zakai逼近的强收敛性，从而肯定地回答了[ES]中提出的问题。
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.
Multidimensional McKean-Vlasov SDEs with mean reflection: well-posedness and existence of optimal control
本文研究了带有均值反射的多维麦克肯-瓦拉斯科夫随机微分方程的well-posedness和最优控制的存在性问题。
We study existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock-Varadhan martingale problem associated such equations. The approach we exploit is the one of paracontrolled distributions introduced in [13]. As a result we make sense of the three dimensional polymer measure with white noise potential.
In this paper, we address the problem of existence and uniqueness of a global classical solution to a multidimensional stochastic Burgers equation without gradient-type assumptions on the force or the initial condition. The equation is first transformed to a random PDE, and then solved via the associated forward-backward SDE. Additionally, we obtain a new a priori gradient estimate valid for a large class of second-order quasilinear parabolic PDEs which becomes an important tool in our approach. Also, we study the stochastic Burgers equation in the vanishing viscosity limit.
Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs: a multidimensional Yamada-Watanabe approach
该研究将Yamada-Watanabe假设推广至多维，在不构造Lyapunov函数的情况下，证明了带随机非Lipschitz系数的McKean-Vlasov SDE解的稳定性、唯一性及确定性情形下强解的存在性，还显式算出Lyapunov指数。
This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time t can depend on the whole path of the forward process up to time t. Such a situation appears when solving path-dependent stochastic control problems by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling fiel…
Correction. SDEs with oblique reflections on nonsmooth domains
对《概率年鉴》21（1993）554- 580的更正[http：//projecteuclid.org/euclid.aop/1176989415]
Multidimensional Stable Driven McKean-Vlasov SDEs with Distributional Interaction Kernel: Critical Thresholds and Related Models
本文研究具有分布交互核的稳定驱动McKean-Vlasov随机微分方程的适定性，重点分析初始条件Besov光滑性对密度估计的影响，并获得临界阈值以处理相关物理和生物模型。
The purpose of the article is to address the limiting behavior of the solutions of stochastic differential equations driven by a pointy $d$-dimensional gradient as the intensity of the underlying Brownian motion tends to $0$. By pointy gradient, we here mean that the drift derives from a potential that is ${\mathcal C}^{1,1}$ on any compact subset that does not contain the origin. As a matter of fact, the corresponding deterministic version of the differential equation may have an infinite number of solutions when initialized from $0_{{\mathbb R}^d}$, in which case the limit theorem proved in …
In this paper we prove a new strong uniqueness result and a weak existence result for possibly {\it degenerate} multidimensional stochastic differential equations with Sobolev diffusion coefficients and rough drifts. In particular, examples with Hölder diffusion coefficients are provided to show our results.
Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient
研究了具有不连续漂移和退化扩散系数的多维随机微分方程的欧拉-马尔可夫方法的强收敛性，证明了对于任意小的ε>0，收敛阶为1/4-ε，方法通过估计欧拉-马尔可夫方案与另一种数值方法之间的差异来实现。
We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $σ$, satisfying Krylov--Röckner type assumptions. We prove several stability estimates, comparing solutions driven by different $(b^i,σ^i)$, both for Itô and Stratonovich SDEs, possibly depending on negative Sobolev norms of the difference $b^1-b^2$. We then discuss several applications of these results to McKean--Vlasov SDEs, criteria for strong compactness of solutions and Wong--Zakai type theorems.
Gaussian-type density bounds for solutions to multidimensional backward SDEs and application to gene expression
该研究推导了多维非马尔可夫倒向随机微分方程解各分量密度的高斯型上下界，方法结合Nourdin-Viens公式与随机版Wazewski定理，并将结果应用于估计基因调控网络中蛋白质表达量的分布密度。
Let X be the solution of the multidimensional stochastic differential equationdX(t) = b(t, X(t)) dt + sigma(t, X(t)) dW(t)\, with X(0)=x where W is a standard Brownian motion. We show that when b is measurable and sigma is in an appropriate Sobolev space, the law of X satisfies a uniform quadratic transportation inequality.
Multidimensional Stable driven McKean-Vlasov SDEs with distributional interaction kernel -- a regularization by noise perspective
研究加性稳定噪声和奇异交互核的McKean-Vlasov SDEs的弱与强适定性，通过Lebesgue-Besov空间定量刻画非线性如何超越线性SDE的阈值，并实现噪声尺度缩放阈值。
In this paper, we prove a comparison theorem for multidimensional G-SDEs. Moreover we obtain respectively the sufficient conditions and necessary conditions of the monotonicity and order-preservation for two multidimensional G-diffusion processes. Finally, we give some applications.
Log-Sobolev Inequalities and Exponential Ergodicity for Non-degenerate and Degenerate McKean-Vlasov SDEs
针对部分耗散McKean-Vlasov SDEs在L²-瓦瑟斯坦距离和相对熵下指数遍历性问题，通过建立一致对数-索伯列夫不等式并结合对数-哈纳克不等式与塔拉格兰德不等式，解决该问题并扩展到退化扩散情形。
Weak Error for stable driven SDEs: expansion of the densities
考虑形式为$X_t = x+\int_{0}^{t} b(X_{s-})ds+\int{0}^{t} f(X_{s-})dZ_s$的多维过程，其中$(Z_s)_{s\ge 0}$是对称稳定过程。在适当的系数假设下，上述方程的唯一强解存在一个密度w.r.t.勒贝格测度及其欧拉格式。使用参数化的方法，我们推导出一个误差展开在1阶w.r.t.这些密度差异的时间步长。
A Class of degenerate Stochastic differential equations with non-Lipschitz coefficients
得到了系统在正正交方向上发展的充分条件。我们使用比较定理参数来实现这一点。结果证明了一类非Lipschitz扩散系数的多维退化偏微分方程存在唯一的强解。
Multidimensional SDE with anticipating initial process and reflection
本文建立了具有反射边界和可能期望初始随机变量的多维随机微分方程的强解$ (X, L)$.其关键是通过相应Riemann和的一致收敛性得到Stratonovich积分的替换公式，并证明$ (X, L)$泛函的连续性。
Reflected random walk in higher dimension arises from an ordinary random walk (sum of i.i.d. random variables): whenever one of the reflecting coordinates becomes negative, its sign is changed, and the process continues from that modified position. One-dimensional reflected random walk is quite well understood from work in 7 decades, but the multidimensional model presents several new difficulties. Here we investigate recurrence questions.
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.
Rough SDEs and Robust Filtering for Jump-Diffusions
研究多维相关跳跃扩散模型中条件分布的鲁棒（连续）表示，通过分析带跳跃的粗糙随机微分方程，建立了随机滤波的若干新鲁棒性结果。
We present a general method to construct couplings of stochastic differential equations driven by Lévy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As an application, we establish regularity results for the transition semigroups of the solutions to stochastic differential equations driven by additive Lévy noise.
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。
In this paper, by employing Gaussian type estimate of heat kernel, we establish Krylov's estimate and Khasminskill's estimate for EM algorithm. As applications, by taking Zvonkin's transformation into account, we investigate convergence rate of EM algorithm for a class of multidimensional SDEs under integrability conditions, where the drifts need not to be piecewise Lipschitz and are much more singular in some sense.
We establish well-posedness results for multidimensional non degenerate $α$-stable driven SDEs with time inhomogeneous singular drifts in $\mathbb{L}^r-{\mathbb B}_{p,q}^{-1+γ}$ with $γ<1$ and $α$ in $(1,2]$, where $\mathbb{L}^r$ and ${\mathbb B}_{p,q}^{-1+γ} $ stand for Lebesgue and Besov spaces respectively. Precisely, we first prove the well-posedness of the corresponding martingale problem and then give a precise meaning to the dynamics of the SDE. This allows us in turn to define an ad hoc notion of weak solution, for which well-posedness holds as well. Our results rely on the smoothing p…
Adding constraints to BSDEs with Jumps: an alternative to multidimensional reflections
本文研究了带跳的倒向随机微分方程（BSDEs）的解的全局约束问题。我们研究了这些所谓的约束BSDES跳跃通过惩罚程序的最小解的存在性和唯一性。这种新型的BSDES为[19]中提出的约束BSDES和[14]中引入的具有约束跳跃的BSDES的概念提供了一个很好的和实用的统一框架。更值得注意的是，在[11]和[13]中研究的多维布朗反射边界元的解也可以用一个很好选择的带跳的一维约束边界元来表示，从数值的观点来看，最后一个结果对于高维最优切换问题的求解和更一般的耦合变分不等式系统的求解是非常有希望的
We establish weak well-posedness for critical symmetric stable driven SDEs in R d with additive noise Z, d $\ge$ 1. Namely, we study the case where the stable index of the driving process Z is $α$ = 1 which exactly corresponds to the order of the drift term having the coefficient b which is continuous and bounded. In particular, we cover the cylindrical case when Zt = (Z 1 t ,. .. , Z d t) and Z 1 ,. .. , Z d are independent one dimensional Cauchy processes. Our approach relies on L p-estimates for stable operators and uses perturbative arguments. 1. Statement of the problem and main results W…
In this paper, we propose a weak approximation of the reflection coupling (RC) for stochastic differential equations (SDEs), and prove it converges weakly to the desired coupling. In contrast to the RC, the proposed approximate reflection coupling (ARC) need not take the hitting time of processes to the diagonal set into consideration and can be defined as the solution of some SDEs on the whole time interval. Therefore, ARC can work effectively against SDEs with different drift terms. As an application of ARC, an evaluation on the effectiveness of the stochastic gradient descent in a non-conve…
This paper develops a new efficient scheme for approximations of expectations of the solutions to stochastic differential equations (SDEs). In particular, we present a method for connecting approximate operators based on an asymptotic expansion with multidimensional Malliavin weights to compute a target expectation value precisely. The mathematical validity is given based on Watanabe and Kusuoka theories in Malliavin calculus. Moreover, numerical experiments for option pricing under local and stochastic volatility models confirm the effectiveness of our scheme. Especially, our weak approximati…
Discrete-time approximation of multidimensional BSDEs with oblique reflections
在本文中，我们研究了离散时间近似的多维反射倒向随机微分方程的类型提出的胡和唐[Probab. Theory Related Fields 147（2010）89-121]，并由Hamadène和Zhang [随机过程]推广。120（2010）403-426]。在比较处罚的方法，其次是哈马丹和让布朗[数学。32（2007）182-192]或Elie和Kharroubi [Statist.可能吧Lett. 80（2010）1388-1396]，我们研究了基于斜投影的更自然的方案。我们通过引入和研究多维离散反射边界的概念，提供了对算法误差的控制。在驾驶员不依赖于变量$Z$的特定情况下，网格点上的误差具有$1/2-\varepsilon$、$\varepsilon>0$的量级。
We study four systems and their interactions. First, we formulate a unified system of coupled forward-backward stochastic partial differential equations (FB-SPDEs) with Levy jumps, whose drift, diffusion, and jump coefficients may involve partial differential operators. A solution to the FB-SPDEs is defined by a 4-tuple general dimensional random vector-field process evolving in time together with position parameters over a domain (e.g., a hyperbox or a manifold). Under an infinite sequence of generalized local linear growth and Lipschitz conditions, the well-posedness of an adapted 4-tuple st…
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。
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。
Implementation of Strong Numerical Methods of Orders 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 for Ito SDEs with Non-Commutative Noise Based on the Unified Taylor-Ito and Taylor-Stratonovich Expansions and Multiple Fourier-Legendre Series
该研究基于统一Taylor-Ito、Taylor-Stratonovich展开与多重Fourier-Legendre级数，构建了带非交换噪声的Ito SDE的多阶强数值方法的实现算法，开发了含1-6重随机积分均方逼近功能的Python程序包。
Error distributions for random grid approximations of multidimensional stochastic integrals
本文在非等距随机网格上证明了几个随机积分关于局部布朗半鞅的逼近误差的联合收敛性。收敛所需的条件是被积函数的勒贝格积分一致趋于零，平方变差和协变差过程收敛。本文还提供了一些工具，简化了这些条件的检查，并扩大了结果的范围。这些结果被用来证明一个明确的极限定理随机网格近似的积分的多维SDES的解决方案的基础上，并找到方法来“设计”和优化的近似误差的分布。作为例子，我们简要讨论离散期权套期保值策略。
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。
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.
Large Deviations for Multi-valued Stochastic Differential Equations
本文证明了一类带单调漂移的多值随机微分方程的Freidlin-Wentzell型大偏差原理，特别是其中包含一类在凸区域中具有反射的随机微分方程.
On the performance of the Euler-Maruyama scheme for multidimensional SDEs with discontinuous drift coefficient
研究带不连续漂移系数的多维SDE的强逼近，证明欧拉-丸山格式对所有维数和p≥1达到至少1/2-的L_p误差率。
We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anis…
Large deviation principle for white noise SPDEs with oblique reflection
本文研究凸域中具斜反射的多维反射随机偏微分方程的Fredlin-Wentzell型大偏差原理，利用弱收敛方法和惩罚化方法的充分条件完成证明。
We study the Crank-Nicolson scheme for stochastic differential equations (SDEs) driven by multidimensional fractional Brownian motion $(B^{1}, \dots, B^{m})$ with Hurst parameter $H \in (\frac 12,1)$. It is well-known that for ordinary differential equations with proper conditions on the regularity of the coefficients, the Crank-Nicolson scheme achieves a convergence rate of $n^{-2}$, regardless of the dimension. In this paper we show that, due to the interactions between the driving processes $ B^{1}, \dots, B^{m} $, the corresponding Crank-Nicolson scheme for $m$-dimensional SDEs has a slowe…
A result of A.M. Davie [Int. Math. Res. Not. 2007] states that a multidimensional stochastic equation $dX_t = b(t, X_t)\,dt + dW_t$, $X_0=x$, driven by a Wiener process $W= (W_t)$ with a coefficient $b$ which is only bounded and measurable has a unique solution for almost all choices of the driving Brownian path. We consider a similar problem when $W$ is replaced by a Lévy process $L= (L_t)$ and $b$ is $β$-Hölder continuous in the space variable, $ β\in (0,1)$. We assume that $L_1$ has a finite moment of order $θ$, for some $θ>0$. Using also a new càdlàg regularity result for strong solutions,…
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 …
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。
Enumeration of walks in multidimensional orthants and reflection groups
本文利用普适性将多维正交卦限游走模型关联至多面体域，揭示其变换群与反射群的联系并给出群无限的条件，同时研究节点域特征值以推导往返计数的精确渐近性质。
The Skorohod oblique reflection problem in time-dependent domains
确定性Skorohod问题在构造和分析带反射的扩散过程中起着重要的作用。在这里研究的形式中，多维Skorohod问题是由H。[61]《明史》卷101：“”。L.狮子和A.- S. [42]在他们的著名文章中。随后的几个研究人员的结果导致了大量的文献Skorohod问题的时间无关域。本文对不定常域上的多维Skorohod问题进行了深入的研究。特别地，我们证明了具有斜反射的Skorohod问题的càdlàg解$(x,λ)$的存在性，特别地，假设$D$是一个不定常域（定理1.2）。此外，我们证明了如果$w$是连续的，那么$x$也是连续的（定理1.3）。随后，我们使用已建立的存在性结果构造解的随机微分方程斜反射（定理1.9）在不定常域。在证明这些结果的过程中，我们建立了一些估计的解决方案的Skorohod问题的有界跳跃，此外，几个结果的收敛序列的解决方案的Skorohod问题的设置不定常域。
We extend to the multidimensional case a Wong-Zakai-type theorem proved by Hu and Øksendal in [7] for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô's SDEs, we consider for any finite partition of the time interval $[0,T]$ a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional ca…
Random periodic solutions for stochastic differential equations with non-uniform dissipativity
本文研究非一致耗散随机微分方程随机周期解的存在性与唯一性，通过反射耦合和同步耦合方法，分别处理无记忆及有限/无限时滞情形，证明了分布意义与路径wise意义下的解的存在性。
On Heat kernel Estimtes for Brownian SDEs with Distributional Drift
本文研究了带有分布漂移的布朗运动SDE的热核估计和正则性估计，通过考虑非退化和退化噪声（如动能型模型）在Young regime中的情况，得出了Schauder估计，并证明了奇异SDE的弱well-posed性。
Asymptotic irrelevance of initial conditions for Skorohod reflection mapping on the nonnegative orthant
将由非负正交体上的确定性Skorohod问题所引起的反射映射应用于$[0,\infty)$上的$\mathbb{R}^n$值函数$X$，然后应用于$a+X$，其中$a$是非负常数向量。一个已经开放了15年的问题是，对于任何$a$的选择，随着时间的推移，在什么条件下两个所得调节函数之间的差收敛到零。这又意味着，如果施加足够的随机结构，确保应用于多维过程$X$的反射映射在分布上收敛，那么当它应用于$η+X$时，它也将在分布上收敛，其中$η$是甚至可能依赖于过程$X$的任何几乎肯定有限值的随机向量。在本文中，我们得到了一个有用的等价刻画这个属性。因此，我们能够确定一个自然的充分条件，在给定的数据$X$和常数路由矩阵。还指出了类似的必要条件。我们的分析的一个特殊含义是，在额外的随机假设下，初始条件的渐近无关性不需要存在一个平稳分布。作为我们的（和以前的）结果的直接推论，我们得出结论，在自然稳定条件下，反射Lévy过程以及马尔可夫加性过程具有唯一的平稳分布，并且对于每个初始条件，在分布上收敛于该平稳分布。扩展的充分条件，然后开发的反射映射与漂移和路由系数，可能是时间和状态依赖的一些影响，多维保险模型进行了简要讨论。
Take a multidimensional normally or obliquely reflected diffusion in a smooth domain. Approximate it by solutions of stochastic differential equations without reflection using the penalty method. That is, we approximate the reflection term with an additional drift term. In the existing literature, usually a specific approximating sequence is provided in order to prove existence of a reflected diffusion. In this article, we provide general sufficient conditions on the approximating coefficients.
Reflected Mckean-Vlasov stochastic differential equations with jumps in time-dependent domains
本文研究时间依赖凸区域中带法向反射的McKean-Vlasov随机微分方程，证明了解的存在唯一性、稳定性及混沌传播性质。
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。
In this note we define and study the stochastic process $X$ in link with a parabolic transmission operator $(A,D(A))$ in divergence form. The transmission operator involves a diffraction condition along a transmission boundary. To that aim we gather and clarify some results coming from the theory of Dirichlet forms as exposed in [6] and [14] for general divergence form operators. We show that $X$ is a semimartingale and that it is solution of a stochastic differential equation involving partial reflections in the co-normal directions along the transmission boundary.
Data-driven rules for multidimensional reflection problems
本文研究多维可逆扩散的反射型奇异控制问题，将其等价转化为形状优化问题，提出基于多面体近似的梯度下降算法求解最优域，并在动力学未知时利用非参数统计构造最优域估计器，开发情景式学习算法实现次线性遗憾。
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