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.

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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.

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.

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 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.

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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.