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.     

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.     

Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

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.     

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.     

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.     

Existence of Mild Solutions to Stochastic Delay Evolution Equations with a Fractional Brownian Motion and Impulses

In this article, we study the existence of mild solutions to stochastic impulsive evolution equations with time delays, driven by fractional Brownian motion with the Hurst index H > 1/2 via a new fixed point analysis approach.     

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.     

Estimation for Translation of a Process Driven by Fractional Brownian Motion

Abstract

We investigate the general problem of estimating the translation of a stochastic process governed by a stochastic differential equation driven by a fractional Brownian motion. The special case of the Ornstein-Uhlenbeck process is discussed in particular.     

带分数布朗Brown的随机比例方程半隐式Euler法的数值解(英文)

本文给出并分析了Poisson随机跳测度驱动的带分数Brown运动的随机比例方程半隐式Euler法的数值解,在局部Lipschitz条件下,证明了在均方意义下半隐式Euler数值解收敛到精确解.     

Reaction-Diffusion Equations with Polynomial Drifts Driven by Fractional Brownian Motions

A reaction-diffusion equation on [0, 1] ^d with the heat conductivity κ > 0, a polynomial drift term and an additive noise, fractional in time with H > 1/2, and colored in space, is considered. We have shown the existence, uniqueness and uniform boundedness of solution with respect to κ. Also we show that if κ tends to infinity, then the corresponding solutions of the equation converge to a process satisfying a stochastic ordinary differential equation.     

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”.     

Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion

In this article, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H. We consider two cases. If H>1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution. This result extends those contained in J. Complex. 22(4), 459–474, 2006 and C.R. Acad. Sci. Paris, Ser. I 340(8), 611–614, 2005. When 1/6<H<1/2, the exact rate of convergence of the Crank-Nicholson scheme is determined for a particular equation. Here we show convergence in law of the error to a random variable, which depends on the solution of the equation and an independent Gaussian random variable.     

Mean-Field Backward Stochastic Differential Equations Driven by Fractional Brownian Motion

In this paper, we study a new class of equations called mean-field backward stochastic differential equations(BSDEs, for short) driven by fractional Brownian motion with Hurst parameter H 1/2. First, the existence and uniqueness of this class of BSDEs are obtained. Second, a comparison theorem of the solutions is established. Third, as an application, we connect this class of BSDEs with a nonlocal partial differential equation(PDE, for short), and derive a relationship between the fractional mean-field BSDEs and PDEs.     

Stochastic elastic equation driven by fractional Brownian motion

In this paper, we consider the stochastic elastic equation driven by a cylindrical fractional Brownian motion. The regularities of the solution to the linear stochastic problem corresponding to the stochastic elastic equation are proved. Then, we obtain the existence of the solution using the Picard iteration.     

混合型随机微分方程的传输不等式

该文探讨一类由Wiener过程和Hurst参数1/2＜H＜1分数布朗运动驱动的混合型随机微分方程.通过使用一些变换技巧和逼近方法,这类方程的强解在d2度量和一致度量d∞下的二次传输不等式被建立.     

Stochastic Analysis of the Fractional Brownian Motion

Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.     

分数布朗运动和泊松过程共同驱动下的择好期权定价

本文讨论两资产择好期权的定价问题。在风险中性假设下,建立了两资产价格过程遵循分数布朗运动和带非时齐Poisson跳跃—扩散过程的择好期权定价模型,应用期权的保险精算法,给出了相应的择好期权的定价公式。     

Stochastic evolution equations with fractional Brownian motion

In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed; separate proofs are required for the cases of Hurst parameter above and below 1/2. The particular case of the Laplacian on the circle is discussed in detail. Mathematics Subject Classification (2000): 60H15, 60G15