Andrey Pilipenko
Some topological properties of stochastic flow $φ_t(x)$ generated by stochastic differential equation in a ${\mathbb R}^d_+$ with normal reflection at the boundary are investigated. Sobolev differentiability in initial condition is received. The absolute continuity of the measure-valued process $μ\circφ_t^{-1}$, where $μ\llλ^d,$ is studied.
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Exponential ergodicity of Stochastic Evolution Equations with reflection
针对无穷维球内带反射的随机发展方程，本文采用耦合方法证明其指数遍历性，并将该结论应用于带反射的随机Navier-Stokes方程。
Large Deviation Principle of Stochastic Evolution Equations with reflection
该研究运用弱收敛方法，证明了无限维球中带反射随机发展方程的大偏差原理。
Reflection of Stochastic Evolution Equations in Infinite Dimensional Domains
本文证明了无限维球中带反射的随机演化方程解的存在性与唯一性，其框架可推广至随机Navier-Stokes方程。
Large deviation principles for multiscale stochastic Burgers equations with reflection
研究具有反射的多尺度随机伯格斯方程，利用弱收敛方法，通过惩罚方法等关键技术工具，严格建立慢分量的大偏差原理。
A non-negative Markovian solution is constructed for a class of stochastic generalized porous media equations with reflection. To this end, some regularity properties and a comparison theorem are proved for stochastic generalized porous media equations, which are interesting by themselves. Invariant probability measures and ergodicity of the solution are also investigated.
We establish transportation cost inequalities, with respect to the uniform and $L_2$-metric, on the path space of continuous functions, for laws of solutions of stochastic differential equations with reflections. We also consider the case of stochastic differential equations involving local times. Harnack inequalities for the associated semigroups are also established.
In this note we prove the existence of a density for the law of the solution for 1-dimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.
In this paper, we consider a class of Mckean-Vlasov stochastic differential equation with oblique reflection over an non-smooth time dependent domain. We establish the existence and uniqueness results of this class, address the propagation of chaos and prove a Fredlin-Wentzell type large deviations principle (LDP). One of the main difficulties is raised by the setting of non-smooth time dependent domain. To prove the LDP, a sufficient condition for the weak convergence method, which is suitable for Mckean-Vlasov stochastic differential equation, plays an important role.
Diffusion approximations for interacting stochastic systems with reflection and control
研究一类具有反射和控制的交互随机系统的扩散近似，通过扩散缩放建立到Ornstein-Uhlenbeck型反射随机微分方程组的分布收敛，并用数值例子展示在人群动力学和神经群体动力学中的应用。
Stochastic Differential Equations Driven by G-Brownian Motion with Mean Reflections
本文研究G-布朗运动驱动的均值反射随机微分方程，通过分析G-期望下的均值反射Skorokhod问题证明方程适定性，提出两种解的构造方法以辅助压缩映射法应用。
In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. The uniqueness can be proved by a priori estimates and the existence is obtained via a penalization method.
Backward Stochastic Differential Equations with Double Mean Reflections
本文研究带两个非线性平均反射的 BSDE，基于带非线性约束的倒向 Skorokhod 问题，用压缩映射证明解的存在唯一性，并在线性约束下给出惩罚平均场 BSDE 逼近。
Asymptotic couplings by reflection are constructed for a class of non-linear monotone SPDES (stochastic partial differential equations). As applications, the gradient/Hölder estimates as well as the exponential convergence are derived for the associated Markov semigroup. The main results are illustrated by stochastic generalized porous media equations, stochastic $p$-Laplacian equations, and stochastic generalized fast-diffusion equations. We emphasize that the gradient estimate is studied at the first time for these equations.
Uniform large deviation principles and averaging principles for stochastic Burgers type equations with reflection
本文针对带反射的随机Burgers型方程，利用等连续一致Laplace原理和时间离散化方法，证明了有界集上的一致大偏差原理、紧集上的一致大偏差原理以及平均原理。
In this paper, we consider a class of multi-dimensional stochastic delay differential equations with jump reflection. Based on existence and uniqueness of the strong solution to the equation, we prove that the Markov semigroup generated by the segment process corresponding to the solution admits a unique invariant measure on the Skorohod space when the coefficients of equation satisfy a class of monotone conditions. Finally, we establish a relationship between the regulator and the local time of the solution and discuss a local time property at large time under the stationary setting.
The present paper is devoted to the study of the well-posedness of mean field BSDEs with mean reflection and nonlinear resistance. By the contraction mapping argument, we first prove that the mean-field BSDE with mean reflection and nonlinear resistance admits a unique deterministic flat local solution on a small time interval. Moreover, we build the global solution by introducing a two-step approach, which is a combination of stitching method and fixed point method. We further provide an application to the super-hedging problem with risk constraint.
We consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $K$ with nonempty interior and regular boundary $Σ$ in a Hilbert space $H$. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov equation with Neumann boundary condition on $Σ$.
Backward Stochastic Differential Equations with Nonlinear Expectation Reflection
本文提出了一种带有非线性期望反射的反向随机微分方程，用于在风险约束下对 contingent claims 进行超对冲。
Mean Field Backward Stochastic Differential Equations with Double Mean Reflections
本文研究了具有双重均值反射的均值场反向随机微分方程的可解性与唯一性，通过压缩映射、惩罚方法和固定点论证等方法，探讨了生成元和约束依赖于分布的MFBSDEs的数学性质。
Mean-Field Backward Stochastic Differential Equations with Nonlinear Resistance and Double Mean Reflections
本文研究了具有双均值反射和非线性阻力的均场反向随机微分方程，通过期望解的约束和补偿项建立存在性和唯一性，探讨了Lipschitz生成元和二次生成元的两种情况，以及补偿项绝对连续时的well-posedness。
Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection
本文在最小假设下建立带反射的非自治多尺度随机Burgers方程的一般 averaging 原理，通过周期或渐近条件得到两个系数不依赖小参数的独立平均方程，并给出实例验证。
In this paper, we introduce a definition of BV functions in a Gelfand triple which is an extension of the definition of BV functions in [2] by using Dirichlet form theory. By this definition, we can consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $Γ$ in a Hilbert space $H$. We prove the existence and uniqueness of a strong solution of this problem when $Γ$ is a regular convex set. The result is also extended to the non-symmetric case. Finally, we extend our results to the case when $Γ=K_α$, where $K_α={f\in…
We consider a stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space--time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution. We propose a method based on infinite dimensional integration by parts formulae, obtaining existence and uniqueness of a strong solution for all continuous nonnegative initial…
In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochastic differential games with reflection. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games with nonlinear cost functionals defined by doubly controlled reflected backward stochastic differential equations.
Conditional Expectation Backward Stochastic Differential Equations and Related Backward Stochastic Differential Equations with Conditional Reflection
本文提出条件期望BSDEs，其驱动项依赖解值及特定子σ-代数的条件期望，建立了其well-posedness，并提供无需子滤波器左连续性假设的条件反射BSDEs解的构造方法。
Backward doubly stochastic differential equations with or without reflection under weak conditions
本文在弱条件下研究反向双随机微分方程（BDSDEs）的well-posedness，探讨了生成元f在y和z上的增长条件对存在性、唯一性和比较原理的影响。
Stochastic billiards with Markovian reflections in generalized parabolic domains
该研究针对带马尔可夫反射的广义抛物域随机台球，通过转化为半带马尔可夫链，在近临界广义Lamperti漂移情形下给出常返性分类，拓展了有限状态分量的相关前期结果。
This paper is concerned with the switching game of a one-dimensional backward stochastic differential equation (BSDE). The associated Bellman-Isaacs equation is a system of matrix-valued BSDEs living in a special unbounded convex domain with reflection on the boundary along an oblique direction. In this paper, we show the existence of an adapted solution to this system of BSDEs with oblique reflection by the penalization method, the monotone convergence, and the a priori estimates.
The present paper is devoted to the study of diagonally quadratic backward stochastic differential equation with oblique reflection. Using a penalization approach, we show the existence fo a solution by providing some delicated a priori estimates. We further obtain the uniqueness by verifying the first component of the solution is indeed the value of a switching probelm for quadratic BSDEs. Moreover, we provide an extension for the solvability and apply our results to study a risk-sensitive switching problem for functional stochastic differential equations.
Uniqueness results of a nonlinear stochastic diffusion-convection equation with reflection
本文研究带反射的非线性伪单调随机扩散演化问题解的唯一性，指出因缺乏一般 Itô 公式无法得到一般唯一性结果，但在解或反射具有更高正则性时可证明比较原理。
A nonlinear stochastic diffusion-convection equation with reflection
本文研究了任意空间维度下具有反射的非线性伪单调随机扩散-对流演化问题，证明了带非负初值的方程解及防止解取负值的反射测度的存在性，并通过研究抛物容度下解的拟处处定义代表元证明了测度的极小性条件。
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.
We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche and Zambotti, we use a method based on infinite dimensional equations, approximation by regu…
In this note, we prove the Freidlin-Wentzell's large deviation principle for BSDEs with one-sided reflection.
We investigate stochastic quantization as a method to go from a classical PDE (with stochastic time $λ$) to a corresponding quantum theory in the limit $λ\to\infty$. We test the method for a linear PDE satisfied by the free scalar field. We begin by giving some background about the importance of establishing the property of {\em reflection positivity} for the limit $λ\to\infty$. We then prove that the measure determined through stochastic quantization of the free scalar field violates reflection positivity (with respect to reflection of the physical time) for every $λ<\infty$. If a non-linear …
The aim of this short note is to fill in a gap in our earlier paper [16] on 2BSDEs with reflections, and to explain how to correct the subsequent results in the second paper [15]. We also provide more insight on the properties of 2RBSDEs, in the light of the recent contributions [13, 23] in the so--called $G-$framework.
Transportation cost inequalities for mean reflection SPDEs with white noise
本文研究了均值反射随机偏微分方程的解在统一范数下的二次运输成本不等式，提出了一种新的方程类型，其中补偿反射部分不依赖于路径，而是依赖于解的分布。
We consider a stochastic partial differential equation with two logarithmic nonlinearities, with two reflections at 1 and -1 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche, Goudenège and Zambotti, we obtain existence and uniqueness of solution for initial conditions in …
In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimations. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As appl…
We consider a one-dimensional Stochastic Differential Equation with reflection where we allow the drift to be merely bounded and measurable. It is already known that such equations have a unique strong solution. Recently, it has been shown that non-reflected SDE's with discontinuous drift possess more regularity than one could expect, namely they are Malliavin differentiable and weakly differentiable w.r.t. the initial value. In this paper we show that similar results hold for one-dimensional SDE's with reflection. We then apply the results to get a Bismut-Elworthy-Li formula for the correspon…
In this paper, the strong solutions $ (X, L)$ of multidimensional stochastic differential equations with reflecting boundary and possible anticipating initial random variables is established. The key is to obtain some substitution formula for Stratonovich integrals via a uniform convergence of the corresponding Riemann sums and to prove continuity of functionals of $ (X, L)$.
In this paper we first investigate zero-sum two-player stochastic differential games with reflection with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straight-forward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions of the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs hea…
Gatheral double stochastic volatility model with Skorokhod reflection
针对 Gatheral 双均值回复随机波动率模型中波动率可长期贴近零的问题，提出引入 Skorokhod 反射的修正模型，在保留灵活性的同时避免波动率趋近零。
We prove a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations with monotone drifts, which in particular contains a class of SDEs with reflection in a convex domain.
A coupling by reflection of a time-inhomogeneous diffusion process on a manifold are studied. The condition we assume is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow. As in time-homogeneous cases, our coupling provides a gradient estimate of the diffusion semigroup which yields the strong Feller property. To construct the coupling via discrete approximation, we establish the convergence in law of geodesic random walks as well as a uniform non-explosion type estimate.
We introduce a new type of reflected backward stochastic differential equations (BSDEs) for which the reflection constraint is imposed on its main solution component, denoted as $Y$ by convention, but in terms of its conditional expectation $\mathbb{E}[Y_t|\mathcal{G}_{t}]$ on a general sub-filtration $\{\mathcal{G}_{t}\}.$ We thus term such equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier, and the recent developed BSDE with a mean reflection constraint, as its two special and extreme cases: …
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。
This paper studies a system of multi-dimensional reflected backward stochastic differential equations with oblique reflections (RBSDEs for short) in infinite horizon associated to switching problems. The existence and uniqueness of the adapted solution is obtained by using a method based oa combination of penalization, verification method and contraction property.
Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case
该研究针对非凸条件下带随机梯度的广义哈密顿蒙特卡洛方法，设计时间离散耦合方法证明Wasserstein距离收缩速率，推导经验均值集中界并给出多度量收敛性与数值偏差估计。
In this work, we consider the outer Stefan problem for the short-time prediction of the spread of a volatile asset traded in a financial market. The stochastic equation for the evolution of the density of sell and buy orders is the Heat Equation with a non-smooth noise in the sense of Walsh, posed in a moving boundary domain with velocity given by the Stefan condition. This condition determines the dynamics of the spread, and the solid phase $[s^-(t),s^+(t)]$ defines the bid-ask spread area wherein the transactions vanish. We introduce a reflection measure and prove existence and uniqueness of…
We consider systems of backward stochastic differential equations with càdlàg upper barrier $U$ and oblique reflection from below driven by an increasing continuous function $H$. Our equations are defined on general probability spaces with a filtration satisfying merely the usual assumptions of right continuity and completeness. We assume that the pair $(H(U),U)$ satisfies a Mokobodzki--type condition. We prove the existence of a solution for integrable terminal conditions and integrable quasi--monotone generators. Applications to the optimal switching problem are given.
Mean-field BSDEs with non-Lipschitz coefficients and double mean reflections
研究生成元非Lipschitz连续的双平均反射平均场倒向随机微分方程，利用Skorokhod问题与先验估计，建立解的存在唯一性。
We determine the Lyapunov spectrum and stable manifolds of some stochastic flows on the Poincaré group associated to Dudley's relativistic processes.
Strong measurable continuous modifications of stochastic flows
本文提出一种新构造方法，在局部紧可分度量空间中为度量图的可测映射随机流生成强可测连续修正。
In this paper, we establish an existence and uniqueness result for system of quasilinear stochastic partial differential equations (SPDEs for short) with reflection in a convex domain in R^k by analytical approach. The method is based on the approximation of the penalized systems of SPDEs.
In this article we consider multiplicative operator-valued white noise functionals related to a stochastic flow. A generalization of the Krylov-Veretennikov expansion is presented. An analog of such expansion for the Arratia flow is derived.
In this paper we consider the asymptotic behaviour of all moments of the interparticle distance and of all mixed moments of an isotropic Brownian stochastic flow which serves as a smooth approximation of the Arratia flow.
Maximal Solutions and Stochastic Free Boundary Formulations for Stochastic Willmore and Surface Diffusion Flows on $\R^2$
本文研究了随机Willmore流和表面扩散流在R²上的最大解问题，通过将问题转化为随机Stefan问题并应用相关理论，证明了局部强解的存在性和唯一性。
In this article we prove that stochastic differential equation (SDE) with Sobolev drift on compact Riemannian manifold admits a unique $ν$-almost everywhere stochastic invertible flow, where $ν$ is the Riemannian measure, which is quasi-invariant with respect to $ν$. In particular, we extend the well known DiPerna-Lions flows of ODEs to SDEs on Riemannian manifold.
We show how the theory of stochastic flows allows to recover in an elementary way a well known result of Warren on the sticky Brownian motion equation.
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