EULER SCHEME FOR FRACTIONAL DELAY STOCHASTIC DIFFERENTIAL EQUATIONS BY ROUGH PATHS TECHNIQUES

In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(1/2,1). In order to prove convergence, we use rough paths techniques. Theoretical bounds are established and numerical simulations are displayed.

Existence and Numerical Solution for a Coupled System of Multi-term Fractional Differential Equations

An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the existence and uniqueness of solutions for the system were derived. Using a fractional predictorcorrector method, a numerical method was presented for the specified system. An example was given to illustrate the obtained results.

STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE:EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

In this paper we study a fractional stochastic heat equation on Rd （d 〉 1） with additive noise /t u（t, x） = Dα/δ u（t, x）＋ b（u（t, x） ） ＋ WH （t, x） where D α/δ is a nonlocal fractional differential operator and W H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition, in the case of space dimension d = 1, we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.

Controllability of a Stochastic Neutral Functional Differential Equation Driven by a fBm

In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When α>1-H, by the technique of Sadovskii’s fixed point theorem, stochastic calculus and the methods adopted directly from deterministic control problems, we study the approximate controllability of the stochastic system.

The Convergence of Euler-Maruyama Method of Nonlinear Variable-Order Fractional Stochastic Differential Equations

In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the method.In particular,when the fractional order is no longer varying,the conclusions obtained are consistent with the relevant conclusions in the existing literature.Finally,the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.

HARNACK TYPE INEQUALITIES FOR SDES DRIVEN BY FRACTIONAL BROWNIAN MOTION WITH MARKOVIAN SWITCHING

In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter H is supposed to be in(1/2, 1). As a direct application, the strong Feller property is presented.

ERROR ESTIMATES OF FINITE ELEMENT METHODS FOR STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS

This paper studies the Galerkin finite element approximations of a class of stochas- tic fractionM differential equations. The discretization in space is done by a standard continuous finite element method and almost optimal order error estimates are obtained. The discretization in time is achieved via the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. We give strong convergence error estimates for both semidiscrete and fully discrete schemes. The proof is based on the error estimates for the corresponding deterministic problem. Finally, the numerical example is carried out to verify the theoretical results.

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 partial differential equations with gradient driven by space-time fractional noises

We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Hölder continuity of the solution to the SPDEs.

The Averaging Principle for Stochastic Fractional Partial Differential Equations with Fractional Noises

The purpose of this paper is to establish an averaging principle for stochastic fractional partial differential equation of order a.>1 driven by a fractional noise.We prove the existence and uniqueness of the global mild solution for the considered equation by the fixed point principle.The solutions for SPDEs with fractional noises can be approximated by the solution for the averaged stochastic systems in the sense of p-moment under some suitable assumptions.

Existence of p-mean Almost Periodic Mild Solution for Fractional Stochastic Neutral Functional Differential Equation

A class of fractional stochastic neutral functional differential equation is analyzed in this paper.With the utilization of the fractional calculations,semigroup theory,fixed point technique and stochastic analysis theory,a sufficient condition of the existence for p-mean almost periodic solution is obtained,which are supported by two examples.

An Averaging Principle for Caputo Fractional Stochastic Differential Equations with Compensated Poisson Random Measure

This article deals with an averaging principle for Caputo fractional stochastic differential equations with compensated Poisson random measure.The main contribution of this article is impose some new averaging conditions to deal with the averaging principle for Caputo fractional stochastic differential equations.Under these conditions,the solution to a Caputo fractional stochastic differential system can be approximated by that of a corresponding averaging equation in the sense ofmean square.

On a semilinear stochastic partial differential equation with double-parameter fractional noises

We study the existence,uniqueness and Hlder regularity of the solution to a stochastic semilinear equation arising from 1-dimensional integro-differential scalar conservation laws.The equation is driven by double-parameter fractional noises.In addition,the existence and moment estimate are also obtained for the density of the law of such a solution.

APPROXIMATE CONTROLLABILITY OF FRACTIONAL IMPULSIVE NEUTRAL STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS AND INFINITE DELAY

This paper is concerned with the approximate controllability of nonlinear fractional impulsive neutral stochastic integro-differential equations with nonlocal conditions and infinite delay in Hilbert spaces under the assumptions that the corresponding linear system is approximately controllable. By the Krasnoselskii-Schaefer-type fixed point theorem and stochastic analysis theory, some sufficient conditions are given for the approximate controllability of the system. At the end, an example is given to illustrate the application of our result.

Sobolev-type Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion with Non-Lipschitz Coefficients

In this paper,we are concerned with the existence and uniqueness of mild solution for a class of nonlinear fractional Sobolev-type stochastic differential equations driven by fractional Brownian motion with Hurst parameter HG(l/2,l)in Hilbert space.We obtain the required result by using semigroup theory,stochastic analysis principle,fractional calculus and Picard iteration techniques with some non-Lipschitz conditions.

Stochastic Differential Equations Driven by Multi-fractional Brownian Motion and Poisson Point Process

In this paper,we study a class of stochastic differential equations with additive noise that contains a non-stationary multifractional Brownian motion(mBm)with a Hurst parameter as a function of time and a Poisson point process of class(QL).The differential equation of this kind is motivated by the reserve processes in a general insurance model,in which between the claim payment and the past history of liability present the long term dependence.By using the variable order fractional calculus on the fractional Wiener-Poisson space and a multifractional derivative operator,and employing Girsanov theorem for multifractional Brownian motion,we prove the existence of weak solutions to the SDEs under consideration,As a consequence,we deduce the uniqueness in law and the pathwise uniqueness.

A superlinear numerical scheme for multi-term fractional nonlinear ordinary differential equations

In this paper,we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations(MTFNODEs).First,the presented problem is equivalently transformed into its integral form with multi-term Riemann-Liouville integrals.Second,the compound product trapezoidal rule is used to approximate the fractional integrals.Then,the unconditional stability and convergence with the order 1+αN−1−αN−2 of the proposed scheme are strictly established,whereαN−1 andαN−2 are the maximum and the second maximum fractional indexes in the considered MTFNODEs,respectively.Finally,two numerical examples are provided to support the theoretical results.

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.

Non-densely defined impulsive neutral stochastic functional differential equations driven by fBm in Hilbert space with infinite delay

We study a class of non-densely defined impulsive neutral stochastic functional differential equations driven by an independent cylindrical fractional Brownian motion （fBm） with Hurst parameter H ∈ （1/2, 1） in the Hilbert space. We prove the existence and uniqueness of the integral solution for this kind of equations with the coefficients satisfying some non-Lipschitz conditions. The results are obtained by using the method of successive approximation.

Stability of stochastic differential equation with linear fractal noise

We study a class of stochastic differential equation with linear fractal noise. By an auxiliary stochastic differential equation, we prove the existence and uniqueness of the solution under some mild assumptions. We also give some estimates of moments of the solution. The exponential stability of the solution is discussed.