Richard F. Bass, Krzysztof Burdzy
We give a simple proof that in a Lipschitz domain in two dimensions with Lipschitz constant one, there is pathwise uniqueness for the Skorokhod equation governing reflecting Brownian motion.
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Here we study stochastic differential equations with a reflecting boundary condition. We provide sufficient conditions for pathwise uniqueness and non-explosion property of solutions in a framework admitting non-Lipschitz continuous coefficients and non-smooth domains.
We study a class of stochastic differential equations with non-Lipschitzian coefficients.A unique strong solution is obtained and a large deviation principle of Freidln-Wentzell type has been established.
We consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on IR^d for d greater or equal to 1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Hoelder continuity of the solutions near zero is established.
Pathwise uniqueness holds for the Skorokhod stochastic differential equation in $C^{1+γ}$ domains in $\mathbb{R}^d$ for $γ>1/2$ and $d\geq3$.
Pathwise uniqueness for multi-dimensional stochastic McKean--Vlasov equation is established under moderate regularity conditions on the drift and diffusion coefficients. Both drift and diffusion depend on the marginal measure of the solution. For pathwise uniqueness, the drift is assumed to be Dini-continuous in the state variable, while the diffusion must be Lipschitz, continuous in time and uniformly nondegenerate. The setting is classical McKean--Vlasov, that is, coefficients of the equation are represented as integrals over the marginal distributions of the process.
In this paper, we study the continuity of the transition density of the reecting Brownian motion on a general Lipschitz domain. We also provide local estimates for the density. Applying the estimates, we prove that the surface measure on the domain is in the local Kato class of the reecting Brownian motion.
Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
研究具有赫尔德连续系数的退化随机微分方程的路径唯一性问题，利用马利瓦因紧致性准则构造强解，通过对偶山田-渡边论证及弱唯一性得出路径唯一性。
We study the solutions of the stochastic heat equation with multiplicative space-time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is Hölder continuous of index $γ>3/4$, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are Hölder continuous.
Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data
研究由乘性噪声驱动、具粗糙初始数据的多维半线性随机演化方程，在系数全局Lipschitz时建解的适定性等，局部Lipschitz时证最大局部解相关性质，条件针对特定范数，可用于多种方程及处理奇异和非光滑初始数据。
In this paper we establish the strong existence, pathwise uniqueness and a comparison theorem to a stochastic partial differential equation driven by Gaussian colored noise with non-Lipschitz drift, Hölder continuous diffusion coefficients and the spatial domain in finite interval, $[0,1]$, and with Dirichlet, Neumann or mixed nonhomogeneous random conditions imposed on the endpoints. The Hölder continuity of the solution both in time and in space variables is also studied.
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。
In the paper, we consider the no-explosion condition and pathwise uniqueness for SDEs driven by a Poisson random measure with coefficients that are super-linear and non-Lipschitz. We give a comparison theorem in the one-dimensional case under some additional condition. Moreover, we study the non-contact property and the continuity with respect to the space variable of the stochastic flow. As an application, we will show that there exists a unique strong solution for SDEs with coefficients like $x\log|x|$.
We study SPDEs with two reflecting walls $Λ^1$, $Λ^2$ and two singular drifts $\frac{c_1}{(X-Λ^1)^{\vartheta}}$, $\frac{c_2}{(Λ^2-X)^{\vartheta}}$, driven by space-time white noise. First, we establish the existence and uniqueness of the solutions $X$ for $\vartheta\geq 0$. Second, we obtain the following pathwise properties of the solutions $X$. If $\vartheta>3$, then a.s. $Λ^1<X<Λ^2$ for all $t\geq0$; If $0<\vartheta<3$, then $X$ hits $Λ^1$ or $Λ^2$ with positive probability in finite time. Thus $\vartheta=3$ is the critical parameter for $X$ to hit reflecting walls.
A McKean-Vlasov SDE and particle system with interaction from reflecting boundaries
本文研究一维McKean-Vlasov随机微分方程及其平均场粒子系统，其中平均位置约束与边界反射耦合导致反射效应非局部化，作者证明了方程的路径适定性及粒子系统在大粒子数极限下的收敛性。
Pathwise convergence of a novel numerical scheme based on semi-implicit method for stochastic differential-algebraic equations with non-global Lipschitz coefficients
本文针对非全局Lipschitz系数的随机微分代数方程，提出一种半隐式数值格式，通过分裂漂移项处理奇异矩阵，证明其路径收敛速率达1/2-ε，并验证高维有效性。
In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with Hölder continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} ΔX_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $α$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $β$-Hölder continuous for $1<α<2$ and $0<β<1$. For $G\equiv0$ and $H(x)=x^β$, in Mytnik (2002) a weak solution t…
A BSDEs approach to pathwise uniqueness for stochastic evolution equations
本文利用无穷维正倒向随机微分方程系统，证明了一类Hilbert空间中漂移项为Hölder连续的随机发展方程的路径唯一性，并证明解对初值具有Lipschitz依赖性。
We prove pathwise uniqueness and strong existence of solutions for stochastic reaction-diffusion systems with locally Lipschitz continuous reaction term of polynomial growth and Hölder continuous multiplicative noise. Under additional assumptions on the coefficients, we also prove positivity of the solutions.
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。
We consider BSDEs with two reflecting irregular barriers. We give necessary and sufficient conditions for existence and uniqueness of $\mathbb{L}^{p}$ solutions for equations with generators monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$ spaces for $p\ge 1$. We also prove that the solutions can be approximated via penalization method.
For continuous γ, g:[0,1]\to(0,\infty), consider the degenerate stochastic differential equation dX_t=[1-|X_t|^2]^{1/2}γ(|X_t|) dB_t-g(|X_t|)X_t dt in the closed unit ball of R^n. We introduce a new idea to show pathwise uniqueness holds when γand g are Lipschitz and \frac{g(1)}{γ^2(1)}>\sqrt2-1. When specialized to a case studied by Swart [Stochastic Process. Appl. 98 (2002) 131-149] with γ=\sqrt2 and g\equiv c, this gives an improvement of his result. Our method applies to more general contexts as well. Let D be a bounded open set with C^3 boundary and suppose h:\barD\to R Lipschitz on \barD…
Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs in arbitrary moments
该研究针对带随机/确定性系数的McKean-Vlasov随机微分方程，基于伊藤公式推导非利普希茨漂移下的解的存在唯一性、矩估计及带显式指数的稳定性，无需依赖李雅普诺夫函数。
White-noise case stochastic heat equation was derived from Dawson-Watanabe superprocess. The pathwise uniqueness of their solutions with non-Lipschitz coefficients has attracted wide attention and in particular, the square-root coefficient case was listed as a long standing open problem in the famous literature (i.e. Perkins \cite{Perkins02} p217.) This short note gives an affirmative answer to this open problem. Our idea is using the $φ_k(x)$ function constructed by Yamada-Watanable to prove that \beqlb \lab{key} \E_μ| X_{t}(z)-Y_{t}(z) | = \lim_{k \ra \infty} \E_μ|\int_{\r}\frac{1}{\sqrt{ε_k…
Pathwise stability for one-dimensional SDEs driven by Brownian motion and a symmetric stable process
该研究针对由布朗运动和对称α-稳定过程驱动的一维SDE，通过同步耦合建立了定量逐路稳定性估计，明确了不同Hölder指数下的误差率，还得到了含可预测强迫误差和时间一致尾界的估计。
This paper investigates a non-autonomous slow-fast system, which is generalized by stochastic differential equations (SDEs) with locally Lipschitz coefficients, subjected to standard Brownian motion (Bm) and fractional Brownian motion (fBm) with Hurst parameter 1/2<H<1. We concentrate on how to handle both types of integrals with respect to Bm and fBm and the locally Lispchitz continuity. The pathwise approach and the Ito stochastic calculus are combined with the technique of stopping time to establish the averaging principle where the averaged equation is defined. Then, the slow component of …
Pathwise uniqueness in infinite dimension under weak structure conditions
该研究在弱结构条件下，证明了无穷维希尔伯特空间中一类带线性有界算子的SDE弱温和解对初值利普希茨依赖，即轨道唯一性成立，该条件可覆盖一维随机阻尼波动方程及三维内随机阻尼Euler–Bernoulli Beam方程等双曲情形。
We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer Lipschitz. The first type is the equation \label{eq1} X_{t}=\int_{0}^{t}σ(s,X_{s})dW_{s}+\int_{0}^{t}b(s,X_{s})ds+α\max_{0\leq s\leq t}X_{s}. The second type is the equation \label{eq2} {l} X_{t} =\ig{0}{t}σ(s,X_{s})dW_{s}+\ig{0}{t}b(s,X_{s})ds+α\max_{0\leq s\leq t}X_{s}\,\,+L_{t}^{0}, X_{t} \geq 0, \forall t\geq 0. The third type is the equation …
A criterion for the well-posedness of McKean-Vlasov stochastic differential equations
该研究针对满足分布依赖型李雅普诺夫条件的McKean-Vlasov随机微分方程，提出混合Perron-Nagumo准则证明其强解的存在性与路径wise唯一性，且给出了适用该准则但不满足经典条件的实例。
Obliquely reflecting Brownian motion in nonpolyhedral, piecewise smooth cones, with an example of application to diffusion approximation of bandwidth sharing queues
本文给出非多面体分段光滑锥中斜反射布朗运动分布唯一性的充分条件，并验证其在带宽共享网络扩散逼近中的应用，为证明该逼近提供关键步骤。
In this paper we study Backward Stochastic Differential Equations with two reflecting right continuous with left limits obstacles (or barriers) when the noise is given by Brownian motion and a Poisson random measure mutually independent. The jumps of the obstacle processes could be either predictable or inaccessible. We show existence and uniqueness of the solution when the barriers are completely separated and the generator uniformly Lipschitz. We do not assume the existence of a difference of supermartingales between the obstacles. As an application, we show that the related mixed zero-sum d…
We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R}^{n+\ell}$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R}^{n+\ell}$ allows us to geometrically realize the Wright-Fisher diffusion wit…
On the complexity of strong approximation of stochastic differential equations with a non-Lipschitz drift coefficient
本文综述非 Lipschitz 漂移系数 SDE 的强逼近复杂度，指出 Euler-Maruyama 可能非多项式甚至任意慢收敛，并介绍标量方程在分段光滑或分数阶 Sobolev 漂移下的正面结果。
Birth-death processes are time-changed Feller's Brownian motions
本文通过特定时间变换建立 Feller 布朗运动与生灭过程的双向等价关系，并给出参数精确对应与构造性路径表示框架。
Mean Reflected Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Constraints
本文研究带双重均值反射的G-布朗运动驱动的倒向随机微分方程，通过倒向Skorokhod问题与不动点理论证明解的存在唯一性，拓展非Lipschitz系数情形并得到相关性质。
The stochastic Jacobi flow
本文研究实数轴上布朗环汤的占据场条件化问题，构造了雅可比流，并统一了相关分解理论，证明了与Bass-Burdzy流的联系。
Robustness of Optimal Controlled Diffusions with Near-Brownian Noise via Rough Paths Theory
本文利用粗糙路径理论证明受控随机微分方程对近 Brown 噪声的鲁棒性，表明 Brown 模型下的最优 Lipschitz 控制策略在非 Brown 近似噪声系统中仍近似最优。
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