Search

In this paper, we show the existence and uniqueness of a solution for a stochastic differential equation driven by an additive noise which is the sum of two fractional Brownian motions with different Hurst parameters. The proofs are based on the techniques of fractional calculus and Girsanov theorem. In particular, we show that the regularization effect of the fractional Brownian motion with the smaller Hurst index dominates. A key challenge in this paper is to extend and apply the Girsanov theorem for two noises given by the sum of two (dependent) fractional Brownian motions by using profound techniques of fractional operator theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonparametric local linear estimator for the drift function of stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H ∈ (1 / 4 , 1). The drift function is one-sided dissipative Lipschitz that ensures the ergodic property of the SDE. We derive the strong consistency of the proposed estimator with proper bandwidth selectors associated with the determined Hurst parameter H. The main tools are the ergodic theorem, Malliavin calculus, and a maximum inequality for It o ̂ –Skorohod integrals. [ABSTRACT FROM AUTHOR]

On the basis of fractional calculus, we introduce an explicit formulation of the integral of controlled paths along Hölder rough paths in terms of Lebesgue integrals for fractional derivatives. The additivity with respect to the interval of integration, a fundamental property of the integral, is not apparent under the formulation because the fractional derivatives depend heavily on the endpoints of the interval of integration. In this paper, we provide a proof of the additivity of the integral under the formulation. Our proof seems to be simpler than those provided in previous studies and is suitable for utilizing the fractional calculus approach to rough path analysis. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the approximate controllability of fractional stochastic delay differential inclusions of order r ∈ (1 , 2). By using fractional calculus, stochastic analysis, the theory of cosine family and Dhage fixed point techniques, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. In particular, the results are established with the assumption that the associated linear part of the system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result. [ABSTRACT FROM AUTHOR]

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a γ -Hölder continuous function 𝜃 with γ ∈ (2 3 , 1). Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to 𝜃 is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the time asymptotic behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination principle for the solutions to forward Kolmogorov equations. The classes of subordinators for which asymptotic analysis may be realized are described. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stochastic elastic equation driven by multiplicative multiparameter fractional noise. By using the Wiener chaos expansion and undetermined coefficient methods, we obtain the existence and uniqueness of the solution in a distribution space. The asymptotic behavior and the Hölder index of the solution are also estimated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. In the end, we give an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper we study the Large Deviations Principle (LDP in abbreviation) for a class of Stochastic Partial Differential Equations (SPDEs) in the whole space , with arbitrary dimension , under random influence which is a Gaussian noise, white in time and correlated in space. The differential operator is a fractional derivative operator. We prove a Large deviations principle for our equation, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion. This approach reduces the proof of LDP to establishing basic qualitative properties for controlled analogues of the original stochastic system. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the approximation theorem as an averaging principle for the solutions to stochastic fractional differential equations of Itô–Doob type with non-Lipschitz coefficients. The simplified systems will be investigated, and their solutions can be approximated to that of the original systems in the sense of mean square and probability, which constitute the approximation theorem. Two examples are presented with a numerical simulation to illustrate the obtained theory. [ABSTRACT FROM AUTHOR]

This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function. [ABSTRACT FROM AUTHOR]

In this paper, the approximate controllability for semilinear stochastic equations in Hilbert spaces is studied. The additive noise is the formal derivative of a fractional Brownian motion in a Hilbert space with the Hurst parameter in the interval (½, 1). Sufficient conditions are established. The results are obtained by using the Banach fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behaviors of parameter estimator in a diagonalizable stochastic evolution equation driven by additive fractional Brownian motion with Hurst parameter H ∈ [½, 1). The moderate deviation for this estimator can be obtained. [ABSTRACT FROM AUTHOR]

This paper is devoted to the determination of the asymptotical optimal input for the estimation of the drift parameter in a partially observed but controlled fractional Ornstein-Uhlenbeck process. Large sample asymptotical properties of the Maximum Likelihood Estimator are deduced using Ibragimov-Khasminskii program and Laplace transform computations. [ABSTRACT FROM AUTHOR]

In this paper we establish precise estimates for the supremum norm for the solution of a dynamical system driven by a Hölder continuous function of order between ⅓ and ½ using the techniques of fractional calculus. As an application we deduce the existence of moments for the solutions to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(⅓, ½) and we obtain an estimate for the supremum norm of the Malliavin derivative. [ABSTRACT FROM AUTHOR]

In this paper, we study long memory phenomenon of functional time series. We consider an operator fractional Brownian motion with values in a Hilbert space defined via operator-valued Hurst coefficient. We prove that this process is a limiting one for polygonal lines constructed from partial sums of time series having space varying long memory. [ABSTRACT FROM AUTHOR]

We consider a stochastic evolution equation driven by a fractional Brownian motion in a separable Hilbert space with Hurst parameter H ∈ (1 / 2 , 1). The coefficient in front of the noise is in general nonlinear. The related integral is a pathwise integral defined by fractional derivatives. The nonlinear coefficients of this equation satisfy weak conditions ensuring only existence of a solution but not uniqueness. This equation generates then a multivalued random dynamical system. We prove the existence of a random attractor for this system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a quasi-linear stochastic heat equation in one dimension on [0,1], with Dirichlet boundary conditions driven by an additive fractional white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n ε N which can approximate the fractional noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued mild solution of the SPDE perturbed by this family of noises converges in law, in the space C([0,T] X [0,1]) of continuous functions, to the solution of the fractional noise driven SPDE. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study parameter estimation problem for diagonalizable stochastic partial differential equations driven by a multiplicative fractional noise with any Hurst parameter H ∈ (0, 1). Two classes of estimators are investigated: traditional maximum likelihood type estimators, and a new class called closed-form exact estimators. Finally the general results are applied to stochastic heat equation driven by a fractional Brownian motion. [ABSTRACT FROM AUTHOR]

We prove that the Wick–Itô–Skorohod integral with respect to a fractional Brownian motion is injective when it is restricted to a class of adapted integrands that satisfy an appropriate integrability condition. [ABSTRACT FROM AUTHOR]