Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion

Abstract

We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H > 1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration and the classical Itô stochastic calculus. The existence result is based on the Yamada–Watanabe theorem.     

Instrumental Variable Estimation for Linear Stochastic Differential Equations Driven by Fractional Brownian Motion

Abstract

We investigate the asymptotic properties of instrumental variable estimators of the drift parameter for stochastic processes satisfying linear stochastic differential equations driven by fractional Brownian motion.     

Doubly Perturbed Neutral Stochastic Functional Equations Driven by Fractional Brownian Motion

In this paper, we study a class of doubly perturbed neutral stochastic functional equations driven by fractional Brownian motion. Under some non-Lipschitz conditions, we will prove the existence and uniqueness of the solution to these equations by providing a semimartingale approximation of a fractional stochastic integration.     

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

Journal of Theoretical Probability - In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the...     

Large Deviations for Backward Stochastic Differential Equations Driven by G-Brownian Motion

Journal of Theoretical Probability - In this paper, we consider forward–backward stochastic differential equation driven by G-Brownian motion (G-FBSDEs in short) with small parameter...     

Backward Stochastic Differential Equations Driven by G-Brownian Motion with Uniformly Continuous Generators

Journal of Theoretical Probability - The present paper is devoted to investigating the existence and uniqueness of solutions to a class of non-Lipschitz scalar-valued backward stochastic...     

The Regularity of Stochastic Convolution Driven by Tempered Fractional Brownian Motion and Its Application to Mean-field Stochastic Differential Equations

In this paper, some properties of a stochastic convolution driven by tempered fractional Brownian motion are obtained. Based on this result, we get the existence and uniqueness of stochastic mean-field equation driven by tempered fractional Brownian motion. Furthermore, combining with the Banach fixed point theorem and the properties of Mittag-Leffler functions, we study the existence and uniqueness of mild solution for a kind of time fractional mean-field stochastic differential equation driven by tempered fractional Brownian motion.     

单调性条件下G-Brown运动驱动的倒向随机微分方程

研究了由G-Brown运动驱动的倒向随机微分方程■解的存在唯一性问题.其生成元f关于z是Lipschitz连续的,关于y是线性增长且满足单调性条件.     

Neutral Functional Partial Differential Equations Driven by Fractional Brownian Motion with Non-Lipschitz Coefficients

Under a non-Lipschitz condition being considered as a generalized case of Lipschitz condition, the existence and uniqueness of mild solutions to neutral stochas- tic functional differential equations driven by fractional Brownian motion with Hurst parameter 1/2 〈 H 〈 1 are investigated. Some known results are generalized and im- proved.     

Global Attractiveness and Quasi-Invariant Sets of Impulsive Neutral Stochastic Functional Differential Equations Driven by Tempered Fractional Brownian Motion

In this paper, we are concerned with a class of impulsive neutral stochastic functional different equations driven by tempered fractional Brownian motion in the Hilbert space. We obtain the global attracting and quasi-invariant sets of the considered equations driven by tempered fractional Brownian motion B^α,λ(t) with 0<α<1/2 and λ>0. In particular, we give some sufficient conditions which ensure the exponential decay in the p-th moment of the mild solution of the considered ...     

平均场倒向重随机微分方程及其应用

研究了平均场倒向随重机微分方程,得到了平均场倒向重随机微分方程解的存在唯一性.基于平均场倒向重随机微分方程的解,给出了一类非局部随机偏微分方程解的概率解释.讨论了平均场倒向重随机系统的最优控制问题,建立了庞特利亚金型的最大值原理.最后讨论了一个平均场倒向重随机线性二次最优控制问题,展示了上述最大值原理的应用.     

Linear Stochastic Differential Equations Driven by a Fractional Brownian Motion with Hurst Parameter Less than 1/2

Abstract

In this article, we use the chaos decomposition approach to establish the existence of a unique continuous solution to linear fractional differential equations of the Skorohod type. Here, the coefficients are deterministic, the initial condition is anticipating and the underlying fractional Brownian motion has Hurst parameter less than 1/2. We provide an explicit expression for the chaos decomposition of the solution in order to show our results.     

Identification for Linear Stochastic Systems Driven by Fractional Brownian Motion

Abstract

We apply Grenander's method of sieves to the problem of identification or estimation of the “drift” function for linear stochastic systems driven by a fractional Brownian motion (fBm). We use an increasing sequence of finite dimensional subspaces of the parameter space as the natural sieves on which we maximise the likelihood function.     

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In this article, using the limit theory of martingales, we study the moderate deviation for maximum likelihood estimator of unknown parameter in the stochastic partial differential equation driven by additive fractional Brownian motion with Hurst parameter, and the rate function can be calculated. Moreover, we apply our main result to several examples.     

Random Dynamical Systems and Stationary Solutions of Differential Equations Driven by the Fractional Brownian Motion

Abstract

Linear and semilinear stochastic evolution equations with additive noise, where the forcing term is an infinite dimensional fractional Brownian motion are studied. Under usual dissipativity conditions the equations are shown to define random dynamical systems which have unique, exponentially attracting fixed points. The results are applied to stochastic parabolic PDE's. They are also applicable to standard finite-dimensional dissipative stochastic equation driven by fractional Brownian motion.     

Neutral Stochastic Integrodifferential Equations Driven by a Fractional Brownian Motion with Impulsive Effects and Time-varying Delays

This paper deals with the existence,uniqueness and asymptotic behaviors of mild solutions to neutral stochastic delay functional integrodifferential equations with impulsive effects, perturbed by a fractional Brownian motion B ^H , with Hurst parameter \({H \in (\frac{1}{2},1)}\). We use the theory of resolvent operators developed in Grimmer (Trans Am Math Soc 273(1982):333–349, 2009) to show the existence of mild solutions. An example is provided to illustrate the results of this work.     

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??In this paper, we prove the existence and uniqueness of solutions for reflected backward stochastic differential equations driven by a Levy process, in which the reflecting barriers are just right continuous with left limits whose jumps are arbitrary. To derive the above results, the monotonic limit theorem of Backward SDE associated with Levy process is established.     

Discretization of Stationary Solutions of Stochastic Systems Driven by Fractional Brownian Motion

In this article we study the behavior of dissipative systems with additive fractional noise of any Hurst parameter. Under a one-sided dissipative Lipschitz condition on the drift the continuous stochastic system is shown to have a unique stationary solution, which pathwise attracts all other solutions. The same holds for the discretized stochastic system, if the drift-implicit Euler method is used for the discretization. Moreover, the unique stationary solution of the drift-implicit Euler scheme converges to the unique stationary solution of the original system as the stepsize of the discretization decreases. Partially supported by the DAAD, Ministerio de Educación y Ciencia (Spain) and FEDER (European Community) under grants MTM2005-01412 and HA2005-0082, by Junta de Andalucía under the Proyecto de Excelencia P07-FQM-02468, and the DFG-project “Pathwise numerics and dynamics of stochastic evolution equations”.     

The Cocycle Property of Stochastic Differential Equations Driven by G-Brownian Motion

In this paper,solutions of the following non-Lipschitz stochastic differential equations driven by G-Brownian motion:X_t=x+∫~t_0b(s,w,X_s)ds+∫~t_0h(s,ω,X_s)dBs+∫~t_0σ(s,ω,X_s)dB_s are constructed.It is shown that they have the cocycle property.Moreover,under some special non-Lipschitz conditions,they are bi-continuous with respect to t,x.