Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

In this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.     

Stochastic integration for tempered fractional Brownian motion

Tempered fractional Brownian motion is obtained when the power law kernel in the moving average representation of a fractional Brownian motion is multiplied by an exponential tempering factor. This paper develops the theory of stochastic integrals for tempered fractional Brownian motion. Along the way, we develop some basic results on tempered fractional calculus.     

Cylindrical fractional Brownian motion in Banach spaces

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen–Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.     

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. Copyright © 2014 John Wiley & Sons, Ltd.     

Burgers' system with a fractional Brownian random force

We prove the existence of function solutions in mild and distributional sense to Burgers' equation, perturbed by a non-conservative random force given as the formal space–time derivative of a fractional Brownian sheet. In particular, the noise may be chosen to be fractional in time and white in space. We also provide basic regularity results. The methods involve both pointwise and vector-valued considerations.     

Solutions to BSDEs driven by both standard and fractional Brownian motions

The backward stochastic differential equations driven by both standard and fractional Brownian motions (or, in short, SFBSDE) are studied. A Wick-Itô stochastic integral for a fractional Brownian motion is adopted. The fractional Itô formula for the standard and fractional Brownian motions is provided. Introducing the concept of the quasi-conditional expectation, we study some its properties. Using the quasi-conditional expectation, we also discuss the existence and uniqueness of solutions to general SFBSDEs, where a fixed point principle is employed. Moreover, solutions to linear SFBSDEs are investigated. Finally, an explicit solution to a class of linear SFBSDEs is found.     

A Small Noise Asymptotic Expansion for Young SDE Driven by Fractional Brownian Motion: A Sharp Error Estimate With Malliavin Calculus

This article shows an analytically tractable small noise asymptotic expansion with a sharp error estimate for the expectation of the solution to Young’s pathwise stochastic differential equations (SDEs) driven by fractional Brownian motions with the Hurst index H > 1/2. In particular, our asymptotic expansion can be regarded as small noise and small time asymptotics by the error estimate with Malliavin culculus. As an application, we give an expansion formula in one-dimensional general Young SDE driven by fractional Brownian motion. We show the validity of the expansion through numerical experiments.     

Lower classes for fractional Brownian motion

We characterize the lower classes of fractional Brownian motion by an integral test.Work partially supported by an NSF grant. Equipe d'Analyse, Tour 46, U.A. at C.N.R.S. n^o 754, Université Paris VI, 4 place Jussieu, 75230 Paris Cedex 05, and Department of Mathematics, 231 West 18th Avenue, Columbus, Ohio 43210.     

Existence of solutions for fractional neutral functional differential equations driven by fBm with infinite delay

In this paper, we consider a class of fractional neutral stochastic functional differential equations with infinite delay driven by a cylindrical fractional Brownian motion (fBm) in a real separable Hilbert space. We prove the existence of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.     

Transportation inequalities for stochastic delay evolution equations driven by fractional Brownian motion

We discuss stochastic functional partial differential equations and neutral partial differential equations of retarded type driven by fractional Brownian motion with Hurst parameter H>1/2. Using the Girsanov transformation argument, we establish the quadratic transportation inequalities for the law of the mild solution of those equations driven by fractional Brownian motion under the L² metric and the uniform metric.     

Stochastic heat equation driven by fractional noise and local time

The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0,1) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos expansion for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.     

On the two-parameter fractional Brownian motion and Stieltjes integrals for Hölder functions

We extend the Stieltjes integral to Hölder functions of two variables and prove an existence and uniqueness result for the corresponding deterministic ordinary differential equations and also for stochastic equations driven by a two-parameter fractional Brownian motion.     

Asymptotic expansion of the quadratic variation of a mixed fractional Brownian motion

Statistical Inference for Stochastic Processes - We obtain the high-order asymptotic expansion for the distribution of the quadratic variation of the mixed fractional Brownian motion, which is...     

Parametric estimation for cusp-type signal driven by fractional Brownian motion

Abstract

We investigate the asymptotic properties of the maximum likelihood estimator of the drift parameter in a cusp-type signal driven by a fractional Brownian motion.     

Reinsurance control in a model with liabilities of the fractional Brownian motion type

We propose a model for reinsurance control for an insurance firm in the case where the liabilities are driven by fractional Brownian motion, a stochastic process exhibiting long-range dependence. The problem is transformed to a nonlinear programming problem, the solution of which provides the optimal reinsurance policy. The effect of various parameters of the model, such as the safety loading of the reinsurer and the insurer, the Hurst parameter, etc. on the optimal reinsurance program is studied in some detail. Copyright © 2007 John Wiley & Sons, Ltd.     

On $L_p$-solution of fractional heat equation driven by fractional Brownian motion

In this paper, we study the fractional stochastic heat equation driven by fractional Brownian motions of the form $$ du(t,x)=\left(-(-\Delta)^{\alpha/2}u(t,x)+f(t,x)\right)dt +\sum\limits^{\infty}_{k=1} g^k(t,x)\delta\beta^k_t $$ with $u(0,x)=u_0$, $t\in[0,T]$ and $x\in\mathbb{R}^d$, where $\beta^k=\{\beta^k_t,t\in[0,T]\},k\geq1$ is a sequence of i.i.d. fractional Brownian motions with the same Hurst index $H>1/2$ and the integral with respect to fractional Brownian motion is Skorohod integral. By adopting the framework given by Krylov, we prove the existence and uniqueness of $L_p$-solution to such equation.     

Optimization of small deviation for mixed fractional Brownian motion with trend

In this paper, we investigate two-sided bounds for the small ball probability of a mixed fractional Brownian motion with a general deterministic trend function, in terms of respective small ball probability of a mixed fractional Brownian motion without trend. To maximize the lower bound, we consider various ways to split the trend function between the components of the mixed fractional Brownian motion for the application of Girsanov theorem, and we show that the optimal split is the solution of a Fredholm integral equation. We find that the upper bound for the probability is also a function of this optimal split. The asymptotic behaviour of the probability as the ball becomes small is analysed for zero trend function and for the particular choice of the upper limiting function.     

Uniqueness and explosion time of solutions of stochastic differential equations driven by fractional Brownian motion

In this paper, we first study the existence and uniqueness of solutions to the stochastic differential equations driven by fractional Brownian motion with non-Lipschitz coefficients. Then we investigate the explosion time in stochastic differential equations driven by fractional Browmian motion with respect to Hurst parameter more than half with small diffusion.     

On fractional tempered stable motion

Fractional tempered stable motion (fTSm) is defined and studied. FTSm has the same covariance structure as fractional Brownian motion, while having tails heavier than Gaussian ones but lighter than (non-Gaussian) stable ones. Moreover, in short time it is close to fractional stable Lévy motion, while it is approximately fractional Brownian motion in long time. A series representation of fTSm is derived and used for simulation and to study some of its sample paths properties.     

Least squares estimator of Ornstein-Uhlenbeck processes driven by fractional Lévy processes with periodic mean

We deal with the least squares estimator for the drift parameters of an Ornstein-Uhlenbeck process with periodic mean function driven by fractional Lévy process. For this estimator, we obtain consistency and the asymptotic distribution. Compared with fractional Ornstein-Uhlenbeck and Ornstein-Uhlenbeck driven by Lévy process, they can be regarded both as a Lévy generalization of fractional Brownian motion and a fractional generaliza- tion of Lévy process.