In this manuscript, we initiate a study on a class of stochastic fractional differential equations driven by Lévy noise. The existence and uniqueness theorem of solutions to equations of this class is established under global and local Carathéodory conditions. Our analysis makes use of the Carathéodory approximation as well as a stopping time technique. The results obtained here generalize the main results from Pedjeu and Ladde [Chaos, Solitons Fractals 45, 279–293 (2012)], Xu et al. [Appl. Math. Comput. 263, 398–409 (2015)], and Abouagwa et al. [Appl. Math. Comput. 329, 143–153 (2018)]. Finally, an application to the stochastic fractional Burgers differential equations is designed to validate the theory obtained.

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$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.

The aim of this article is to consider a class of neutral Caputo fractional stochastic evolution equations with infinite delay (INFSEEs) driven by fractional Brownian motion (fBm) and Poisson jumps in Hilbert space. First, we establish the local and global existence and uniqueness theorems of mild solutions for the aforementioned neutral fractional stochastic system under local and global Carathéodory conditions by using the successive approximations, stochastic analysis, fractional calculus, and stopping time techniques. The obtained existence result in this article is new in the sense that it generalizes some of the existing results in the literature. Furthermore, we discuss the averaging principle for the proposed neutral fractional stochastic system in view of the convergence in mean square between the solution of the standard INFSEEs and that of the simplified equation. Finally, the obtained averaging theory is validated with an example.

In this manuscript, a new class of impulsive fractional Caputo neutral stochastic differential equations with variable delay (IFNSDEs, in short) perturbed by fractional Brownain motion (fBm) and Poisson jumps was studied. We utilized the Carathéodory approximation approach and stochastic calculus to present the existence and uniqueness theorem of the stochastic system under Carathéodory-type conditions with Lipschitz and non-Lipschitz conditions as special cases. Some existing results are generalized and enhanced. Finally, an application is offered to illustrate the obtained theoretical results.

In this work, we present the analysis of a mixed weighted fractional Brownian motion, defined by ${\eta }_t:=B_t+{\xi }_t$, where B is a Brownian motion and $\xi$ is an independent weighted fractional Brownian motion. We also consider the parameter estimation problem for the drift parameter $\theta >0$ in the mixed weighted fractional Ornstein–Uhlenbeck model of the form $X_0=0;X_t=\theta X_{t}dt+d{\eta }_t$. Moreover, a simulation is given of sample paths of the mixed weighted fractional Ornstein–Uhlenbeck process.

This paper has two parts. In part Ⅰ, existence and uniqueness theorem is established for solutions of neutral stochastic differential equations with variable delays driven by $ G $-Brownian motion (VNSDDEGs in short) under global Carathéodory conditions. In part Ⅱ, a simplified VNSDDEGs for the original one is proposed. And the convergence both in $ L^p $-sense and capacity between the solutions of the simplified and original VNSDDEGs are established in view of the approximation theorems. Two examples are conducted to justify the theoretical results of the approximation theorems.