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Abstract In this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients.

Abstract Let u ( t , x ) $u(t,x)$ be the solution to a stochastic heat equation ∂ ∂ t u = 1 2 ∂ 2 ∂ x 2 u + ∂ 2 ∂ t ∂ x X ( t , x ) , t ≥ 0 , x ∈ R $$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\in { \mathbb{R}} $$ with initial condition u ( 0 , x ) ≡ 0 $u(0,x)\equiv 0$ , where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution u ( t , x ) $u(t,x)$ . In fact, it is well known that the solution is a Gaussian process such that the process t ↦ u ( t , x ) $t\mapsto u(t,x)$ is a bi-fractional Brownian motion with Hurst indices H = K = 1 2 $H=K=\frac{1}{2}$ for every real number x. However, the many properties of the process x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ are unknown. In this paper we consider the generalized quadratic covariations of the two processes x ↦ u ( ⋅ , x ) , t ↦ u ( t , ⋅ ) $x\mapsto u(\cdot ,x),t\mapsto u(t,\cdot )$ . We show that x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with “Itô’s integral”, but it is not a semimartingale. Moreover, some generalized Itô formulas and Bouleau–Yor identities are introduced.

Abstract In this research, we study the existence and uniqueness results for a new class of stochastic fractional differential equations with impulses driven by a standard Brownian motion and an independent fractional Brownian motion with Hurst index 1/2

Abstract This paper presents the periodic averaging principle for impulsive stochastic dynamical systems driven by fractional Brownian motion (fBm). Under non-Lipschitz condition, we prove that the solutions to impulsive stochastic differential equations (ISDEs) with fBm can be approximated by the solutions to averaged SDEs without impulses both in the sense of mean square and probability. Finally, an example is provided to illustrate the theoretical results.

Abstract By using stochastic analysis, fractional analysis, compact semigroups and the Schauder fixed-point theorem, we discuss the approximate boundary controllability of a nonlocal Hilfer fractional stochastic differential system with fractional Brownian motion and a Poisson jump. In addition, we establish the sufficient conditions for exact null controllability for the same problem. Finally, an example is given to illustrate the results obtained.

Abstract Most stochastic age-dependent capital systems cannot be solved explicitly, so it is necessary to develop numerical methods and study the properties of numerical solutions. In this paper, we consider a class of stochastic age-dependent capital systems with Poisson jumps and fractional Brownian motion (fBm) and investigate the convergence of the split-step θ-method (SSθ) for this system. It is proved that the numerical approximation solutions converge to the analytic solutions for the equations, and the order of approximation is also provided. Finally, a numerical experiment is simulated to illustrate that the SSθ method has better accuracy than the Euler method.

In this paper, we consider fuzzy stochastic differential equations (FSDEs) driven by fractional Brownian motion (fBm). These equations can be applied in hybrid real-world systems, including randomness, fuzziness and long-range dependence. Under some assumptions on the coefficients, we follow an approximation method to the fractional stochastic integral to study the existence and uniqueness of the solutions. As an example, in financial models, we obtain the solution for an equation with linear coefficients. [ABSTRACT FROM AUTHOR]

Abstract Let U, V and W be three Hilbert spaces and let BH $B^{H}$ be a W-valued fractional Brownian motion with Hurst index H∈(12,1) $H\in(\frac{1}{2},1)$. In this paper, we consider the approximate controllability of the Sobolev-type fractional stochastic differential equation {Dtαc[Lx(t)]=Ax(t)+f(t,xt)+Bu(t)+σ(t)ddtBH(t),t∈(0,T],x(t)=ϕ(t),t∈(−∞,0], $$\textstyle\begin{cases} {}^{\mathrm{c}}D^{\alpha}_{t}[Lx(t)]=Ax(t)+f(t,x_{t})+ Bu(t)+\sigma(t)\frac {d}{dt}B^{H}(t), &t\in(0,T], \\ x(t)=\phi(t), &t\in(-\infty,0], \end{cases} $$ where Dαc ${}^{\mathrm{c}}D^{\alpha}$ is the Caputo fractional derivative of order α∈(1−H,1) $\alpha\in(1-H,1)$, the time history xt:(−∞,0]→xt(θ)=x(t+θ) $x_{t}:(-\infty,0]\rightarrow x_{t}(\theta)=x(t+\theta)$ with t>0 $t>0$ belonging to the phase space Bh $\mathscr{B}_{h}$, the control function u(⋅) $u(\cdot)$ is given in L2([0,T],V) $L^{2}([0,T],V)$, B is a bounded linear operator from V into U. Under some suitable conditions, we show that the system is approximately controllable on [0,T] $[0,T]$ and we give an example to illustrate the theory.

Abstract In this paper, by employing the fractional power of operators, semigroup theory, and fixed point strategy we obtain some new criteria ensuring the existence and exponential stability of a class of impulsive neutral stochastic integrodifferential equations driven by a fractional Brownian motion. We establish some new sufficient conditions that ensure the exponential stability of mild solution in the mean square moment by utilizing an impulsive integral inequality. Also, we provide an example to show the efficiency of the obtained theoretical result.

Let u (t , x) be the solution to a stochastic heat equation ∂ ∂ t u = 1 2 ∂ 2 ∂ x 2 u + ∂ 2 ∂ t ∂ x X (t , x) , t ≥ 0 , x ∈ R with initial condition u (0 , x) ≡ 0 , where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution u (t , x) . In fact, it is well known that the solution is a Gaussian process such that the process t ↦ u (t , x) is a bi-fractional Brownian motion with Hurst indices H = K = 1 2 for every real number x. However, the many properties of the process x ↦ u (⋅ , x) are unknown. In this paper we consider the generalized quadratic covariations of the two processes x ↦ u (⋅ , x) , t ↦ u (t , ⋅) . We show that x ↦ u (⋅ , x) admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with "Itô's integral", but it is not a semimartingale. Moreover, some generalized Itô formulas and Bouleau–Yor identities are introduced. [ABSTRACT FROM AUTHOR]

In this research, we study the existence and uniqueness results for a new class of stochastic fractional differential equations with impulses driven by a standard Brownian motion and an independent fractional Brownian motion with Hurst index 1 / 2 < H < 1 under a non-Lipschitz condition with the Lipschitz one as a particular case. Our analysis depends on an approximation scheme of Carathéodory type. Some previous results are improved and extended. [ABSTRACT FROM AUTHOR]

Abstract The paper is devoted to investigating a class of neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion. By establishing two new impulsive integral inequalities which improve the inequalities established by Li (Neurocomputing 177:620-627, 2016) and Long et al. (Stat. Probab. Lett. 82(9):1699-1709, 2012), attracting and quasi-invariant sets of the system are obtained. Moreover, exponential stability of the mild solution is established with sufficient conditions.

This paper presents the periodic averaging principle for impulsive stochastic dynamical systems driven by fractional Brownian motion (fBm). Under non-Lipschitz condition, we prove that the solutions to impulsive stochastic differential equations (ISDEs) with fBm can be approximated by the solutions to averaged SDEs without impulses both in the sense of mean square and probability. Finally, an example is provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider a class of nonlocal fractional stochastic differential equations driven by fractional Brownian motion with Hurst index H > 1 2 $H>\frac{1}{2}$ . Sufficient conditions for the existence and uniqueness of mild solutions are obtained. Finally, an example is presented to illustrate our obtained results.

Abstract In this paper, we are concerned with a class of second-order neutral stochastic functional differential equations driven by a fractional Brownian motion with Hurst parameter 1 / 2 < ħ < 1 $1/2

The paper is devoted to investigating a class of neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion. By establishing two new impulsive integral inequalities which improve the inequalities established by Li (Neurocomputing 177:620-627, 2016) and Long et al. (Stat. Probab. Lett. 82(9):1699-1709, 2012), attracting and quasi-invariant sets of the system are obtained. Moreover, exponential stability of the mild solution is established with sufficient conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlocal fractional stochastic differential equations driven by fractional Brownian motion with Hurst index $H>\frac{1}{2}$ . Sufficient conditions for the existence and uniqueness of mild solutions are obtained. Finally, an example is presented to illustrate our obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of second-order neutral stochastic functional differential equations driven by a fractional Brownian motion with Hurst parameter $1/2<\hbar <1$ on the Hilbert space. By combining some stochastic analysis theory and new integral inequality techniques, we identify the global attracting sets of the equations under investigation. Some sufficient conditions ensuring the exponential decay of mild solutions in the pth moment to the stochastic systems are obtained. Last, an example is presented to illustrate our theory in the work. [ABSTRACT FROM AUTHOR]

We study the stability problem of mild solutions of impulsive stochastic differential equations driven by a fractional Brown motion with finite time-varying delay. The Hurst parameter H of the fractional Brown motion belongs to $(\frac{1}{2},1)$ . In terms of fractional power of operators and semigroup theory, we obtain sufficient conditions that guarantee the stability of the mild solution of such a equation in two cases: the impulse depends on current states of the system and the impulse depends not only on current states but also on historical states of the system. We give two examples illustrating the theorems. [ABSTRACT FROM AUTHOR]

In this paper we study some stability criteria for some semilinear integral equations with a function as initial condition and with additive noise, which is a Young integral that could be a functional of fractional Brownian motion. Namely, we consider stability in the mean, asymptotic stability, stability, global stability, and Mittag-Leffler stability. To do so, we use comparison results for fractional equations and an equation (in terms of Mittag-Leffler functions) whose family of solutions includes those of the underlying equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of mild solutions for nonlocal stochastic integro-differential equations driven by fractional Brownian motions with Hurst parameter $H>\frac{1}{2}$ in a Hibert space. Sufficient conditions for the existence of mild solutions are derived by means of the Leray-Schauder nonlinear alternative. A special case of this result is given and an example is provided to illustrate the effectiveness of the proposed result. [ABSTRACT FROM AUTHOR]