One Brownian Stochastic Flow

Semilinear Backward Doubly Stochastic Differential Equations and SPDEs Driven by Fractional Brownian Motion with Hurst Parameter in (0,1/2)

We study the existence of a unique solution to semilinear fractional backward doubly stochastic differential equation driven by a Brownian motion and a fractional Brownian motion with Hurst parameter less than 1/2. Here the stochastic integral with respect to the fractional Brownian motion is the extended divergence operator and the one with respect to Brownian motion is Itô's backward integral. For this we use the technique developed by R.Buckdahn to analyze stochastic differential equations on the Wiener space, which is based on the Girsanov theorem and the Malliavin calculus, and we reduce …

超布朗运动局部时间的随机微分方程

A stochastic differential equation for local times of super-Brownian motion

本文证明了超布朗运动或由Brownian tree索引的Brownian motion的局部时间满足显式随机微分方程，其证明结合了Brownian snake的excursion theory与超过程理论工具。

Brownian and fractional Brownian stochastic currents via Malliavin calculus

By using Malliavin calculus and multiple Wiener-Itô integrals, we study the existence and the regularity of stochastic currents defined as Skorohod (divergence) integrals with respect to the Brownian motion and to the fractional Brownian motion. We consider also the multidimensional multiparameter case and we compare the regularity of the current as a distribution in negative Sobolev spaces with its regularity in Watanabe space.

Properties of solutions of stochastic differential equations driven by the G-Brownian motion

In this paper, we study the differentiability of solutions of stochastic differential equations driven by the $G$-Brownian motion with respect to the initial data and the parameter. In addition, the stability of solutions of stochastic differential equations driven by the $G$-Brownian motion is obtained.

由分数布朗运动和布朗运动驱动的慢-快McKean-Vlasov随机微分方程的大偏差

Large deviation for slow-fast McKean-Vlasov stochastic differential equations driven by fractional Brownian motions and Brownian motions

本文研究了由布朗运动和分数布朗运动驱动的慢-快McKean-Vlasov随机微分方程，提出乘积空间上的大偏差原理新定义，并在适当假设下利用弱收敛方法证明了该方程的LDP。

Stochastic Calculus with respect to G-Brownian Motion Viewed through Rough Paths

In this paper, we study rough path properties of stochastic integrals of Itô's type and Stratonovich's type with respect to $G$-Brownian motion. The roughness of $G$-Brownian Motion is estimated and then the pathwise Norris lemma in $G$-framework is obtained.

Stochastic Volterra equations driven by fractional Brownian motion with Hurst parameter H > 1/2

In this note we prove an existence and uniqueness result of solution for stochastic Volterra integral equations driven by a fractional Brownian motion with Hurst parameter H > 1/2, showing also that the solution has finite moments. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.

Stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index $H> 1/2$

We consider a mixed stochastic differential equation driven by possibly dependent fractional Brownian motion and Brownian motion. Under mild regularity assumptions on the coefficients, it is proved that the equation has a unique solution.

Bismut formulae and applications for stochastic (functional) differential equations driven by fractional Brownian motions

By using Malliavin calculus, Bismut derivative formulae are established for a class of stochastic (functional) differential equations driven by fractional Brownian motions. As applications, Harnack type inequalities and strong Feller property are presented.

Differentiability of stochastic flow of reflected Brownian motions

We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation.

Stochastic delay fractional evolution equations driven by fractional Brownian motion

In this paper, we consider a class of stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space. Sufficient conditions for the existence and uniqueness of mild solutions are obtained. An application to the stochastic fractional heat equation is presented to illustrate the theory.

Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions

In this paper we show that under some assumptions, for a $d$-dimensional fractional Brownian motion with Hurst parameter $H>1/2$, the density of solution of stochastic differential equation driven by it has a short-time expansion similar to that in the Brownian motion case.

Harnack Inequalities and Applications for Stochastic Differential Equations Driven by Fractional Brownian Motion

In the paper, Harnack inequalities are established for stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H<1/2$. As applications, strong Feller property, log-Harnack inequality and entropy-cost inequality are given.

Application of Stochastic Flows to the Sticky Brownian Motion Equation

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.

Convolution-type stochastic Volterra equations with additive fractional Brownian motion in Hilbert space

We consider convolution-type stochastic Volterra equations with additive Hilbert-valued fractional Brownian motion, $0<H<1$. We find the weak solution to this stochastic Volterra equation, and study its stochastic integral part, the stochastic convolution, which we show to be mean-zero Gaussian. We develop an Itô isometry for stochastic integrals with respect to a Hilbert-valued fractional Brownian motion, and use it to compute the covariance of the stochastic convolution. This formula, which uses fractional integrals and derivatives, generalizes the well-known formula from the case $H=1/2$.

Gradient Bounds for Solutions of Stochastic Differential Equations Driven by Fractional Brownian Motions

We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts formulas on the path space of a fractional Brownian motion.

Existence of density for the solution of stochastic delay differential equations with reflection driven by a fractional Brownian motion

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.

Reflected forward-backward stochastic differential equations driven by G-Brownian motion with continuous monotone coefficients

In this paper, we prove that there exists at least one solution for the reflected forward-backward stochastic differential equation driven by G-Brownian motion satisfying the obstacle constraint with monotone coefficients.

Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion

In this paper we consider stochastic differential equations with non-negativity constraints, driven by a fractional Brownian motion with Hurst parameter $H>\1/2$. We first study an ordinary integral equation where the integral is defined in the Young sense and then we apply this result pathwise to solve the stochastic problem.

由无穷维分数布朗运动驱动的随机波动方程的参数估计

Parametric estimation for stochastic wave equations driven by an infinite dimensional fractional Brownian motion

针对由无穷维分数布朗运动驱动的随机波动方程的参数估计问题展开研究。

Evolution of moments of isotropic Brownian stochastic flows

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.

Stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions

In this paper, we introduce the idea of stochastic integrals with respect to an increasing process in the $G$-framework and extend $G$-Itô's formula. Moreover, we study the solvability of the scalar valued stochastic differential equations driven by $G$-Brownian motion with reflecting boundary conditions (RGSDEs).

Local times of stochastic differential equations driven by fractional Brownian motions

In this paper, we study the existence and (Hölder) regularity of local times of stochastic differential equations driven by fractional Brownian motions. In particular, we show that in one dimension and in the rough case H<1/2, the Hölder exponent (in t) of the local time is 1-H, where H is the Hurst parameter of the driving fractional Brownian motion.

Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion

In this paper, we consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in [10.11], we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in [28]. Then we obtain a generalized dynamic programming principle and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

Representation of G-martingales as stochastic integrals with respect to the G-Brownian motion

The objective of this paper is to derive a representation of symmetric G-martingales as stochastic integrals with respect to the G-Brownian motion. For this end, we first study some extensions of stochastic calculus with respect to G-martingales under the sublinear expectation spaces.

Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion

In this paper we prove, for small Hurst parameters, the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.

Comparison theorem for neutral stochastic functional differential equations driven by G-Brownian motion

In this paper, we investigate suffcient and necessary conditions for the comparison theorem of neutral stochastic functional differential equations driven by G-Brownian motion (G-NSFDE). Moreover, the results extend the ones in the linear expectation case [1] and nonlinear expectation framework [8].

由G-布朗运动驱动的随机偏微分方程及其相关的反向双重随机微分方程

Stochastic PDEs driven by G-Brownian motion and the associated Backward Doubly Stochastic Differential Equations

本文研究了由G-布朗运动驱动的随机偏微分方程的well-posedness，并通过分析方法证明了其弱解的存在性和唯一性，同时建立了与反向双重随机微分方程的关系。

The Full Brownian Web as Scaling Limit of Stochastic Flows

In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all space-time trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg.

Covariance of stochastic integrals with respect to fractional Brownian motion

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>1/2$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart's result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.

Controllability of neutral stochastic functional integro-differential equations driven by fractional Brownian motion

This paper focuses on controllability results of stochastic delay partial functional integro-differential equations perturbed by fractional Brownian motion. Sufficient conditions are established using the theory of resolvent operators combined with a fixed point approach for achieving the required result. An example is provided to illustrate the theory.

Lyapunov-type conditions and stochastic differential equations driven by $G$-Brownian motion

This paper studies the solvability and the stability of stochastic differential equations driven by G-Brownian motion (GSDEs). In particular, the existence and uniqueness of the solution for locally Lipschitz GSDEs is obtained by localization methods, also the stability of such GSDEs are discussed with Lyapunov-type conditions.

Large deviation principles for stochastic dynamical systems with a fractional Brownian noise

We study small noise large deviation asymptotics for stochastic differential equations with a multiplicative noise given as a fractional Brownian motion $B^H$ with Hurst parameter $H>\frac12$. The solutions of the stochastic differential equations are defined pathwise under appropriate conditions on the coefficients. The ingredients in the proof of the large deviation principle, which include a variational representation for nonnegative functionals of fractional Brownian motions and a general sufficient condition for a LDP for a collection of functionals of a fractional Brownian motions, have …

On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients

In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion (GSDEs) with integral-Lipschitz conditions on their coefficients.

Controllability of stochastic impulsive neutral functional differential equations driven by fractional Brownian motion with infinite delay

In this paper we study the controllability results of impulsive neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.

Stochastic integral representation of the $L^{2}$ modulus of Brownian local time and a central limit theorem

The purpose of this note is to prove a central limit theorem for the $L^2$-modulus of continuity of the Brownian local time obtained in \cite{CLMR}, using techniques of stochastic analysis. The main ingredients of the proof are an asymptotic version of Knight's theorem and the Clark-Ocone formula for the $L^2$-modulus of the Brownian local time.

Operators associated with stochastic differential equations driven by fractional Brownian motions

In this paper, by using a Taylor development type formula, we show how it is possible to associate differential operators with stochastic differential equations driven by a fractional Brownian motion. As an application, we deduce that invariant measures for such SDEs must satisfy an infinite dimensional system of partial differential equations.

On the strong Markov property for stochastic differential equations driven by $G$-Brownian motion

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.

Averaging Principle for Backward Stochastic Differential Equations driven both standard and fractional Brownian motions

Stochastic averaging for a class of backward stochastic differential equations driven by both standard and fractional Brownian motions (SFrBSDEs in short), is investigated. An averaged SFrBSDEs for the original SFrBSDEs is proposed, and their solutions are quantitatively compared. Under some appropriate assumptions, the solutions to original systems can be approximated by the solutions to averaged stochastic systems in the sense of mean square and also in probability.

Anticipative backward stochastic differential equations driven by fractional Brownian motion

We study the anticipative backward stochastic differential equations (BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H greater than 1/2. The stochastic integral used throughout the paper is the divergence operator type integral. We obtain the existence and uniqueness of solutions to these equations. A comparison theorem for this type of anticipative BSDEs is also established.

带次微分算子的$G$-布朗运动驱动的倒向随机微分方程的大偏差原理

Large deviation principle for a backward stochastic differential equation driven by $G$-Brownian motion with subdifferential operator

该文针对带次微分算子的$G$-布朗运动驱动倒向随机微分方程，开展其解的大偏差原理研究。

Malliavin differentiability and regularity of densities in semi-linear stochastic delay equations driven by weighted fractional Brownian motion

In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for $d \geq 1$.

Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation

We develop a notion of nonlinear expectation --G-expectation-- generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G-normal distributions. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a multi dimensional G-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Ito's type with respect to our G-Brownian motion and derive the related Ito's formula. We have also obtained the existence and uniqueness of stochastic differential equation under…

Semi-linear Backward Stochastic Integral Partial Differential Equations driven by a Brownian motion and a Poisson point process

In this paper we investigate classical solution of a semi-linear system of backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. By proving an Itô-Wentzell formula for jump diffusions as well as an abstract result of stochastic evolution equations, we obtain the stochastic integral partial differential equation for the inverse of the stochastic flow generated by a stochastic differential equation driven by a Brownian motion and a Poisson point process. By composing the random field generated by the solution of a backward stochastic…

A White Noise Approach to Stochastic Currents of Brownian Motion

In this paper we study stochastic currents of Brownian motion $ξ(x)$, $x\in\mathbb{R}^{d}$, by using white noise analysis. For $x\in\mathbb{R}^{d}\backslash\{0\}$ and for $x=0\in\mathbb{R}$ we prove that the stochastic current $ξ(x)$ is a Hida distribution. Moreover for $x=0\in\mathbb{R}^{d}$ with $d>1$ we show that the stochastic current is not a Hida distribution.

The level-crossing intensity for the density of the image of the Lebesgue measure under the action of a Brownian stochastic flow

In this paper we compute the level-crossing intensity for the density of the image of the Lebesgue measure under the action of a Brownian stochastic flow which is a smooth approximation of the Arratia flow and determine its asymptotic behaviour as the height of the level tends to infinity.

The rate of convergence of estimate for Hurst index of fractional Brownian motion involved into stochastic differential equation

We consider stochastic differential equation involving pathwise integral with respect to fractional Brownian motion. The estimates for the Hurst parameter are constructed according to first- and second-order quadratic variations of observed values of the solution. The rate of convergence of these estimates to the true value of a parameter is established.

Controllability of fractional stochastic neutral functional differential equations driven by fractional Brownian motion with infinite delay

In this paper we study the controllability of fractional neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.

Stochastic flows related to Walsh Brownian motion

We define an equation on a simple graph which is an extension of Tanaka equation and the skew Brownian motion equation. We then apply the theory of transition kernels developped by Le Jan and Raimond and show that all the solutions can be classified by probability measures.

On the p-th mean S-asymptotically omega periodic solution for some Stochastic Evolution Equation driven by Q-Brownian motion

In this paper, we make a slight contribution about the existence (uniqueness) and asymptotic stability of the p-th mean S-asymptotically omega-periodic solutions for some nonautonomous Stochastic Evolution Equations driven by a Q-Brownian motion. This is done using the Banach fixed point Theorem and a Gronwall inequality.

Reflected stochastic differential equations driven by $G$-Brownian motion in non-convex domains

In this paper, we first review the penalization method for solving deterministic Skorokhod problems in non-convex domains and establish estimates for problems with $α$-Hölder continuous functions. With the help of these results obtained previously for deterministic problems, we pathwisely define the reflected $G$-Brownian motion and prove its existence and uniqueness in a Banach space. Finally, multi-dimensional reflected stochastic differential equations driven by $G$-Brownian motion are investigated via a fixed-point argument.

Stability of stochastic differential equations with respect to time-changed Brownian motions

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…

Large deviations for flows of interacting Brownian motions

We establish the large deviation principle (LDP) for stochastic flows of interacting Brownian motions. In particular, we consider smoothly correlated flows, coalescing flows and Brownian motion stopped at a hitting moment.

Backward Stochastic Differential Equations Driven by G-Brownian Motion with Double Reflections

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.