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This paper aims to investigate a fractional neutral stochastic functional differential equation (FNSFDE) with non-Lipschitz coefficients. Under the assumptions, we first establish the continuity of the solution in the fractional order of the equation. Furthermore, an Euler-Maruyama (EM) approximation is constructed and then we obtain the strong convergence of the numerical scheme. Specially, if the non-Lipschitz conditions are replaced with the Lipschitz conditions, we shall get a definite convergence rate, which is related to the fractional order of the equation. Finally, we consider the averaging principle for the fractional neutral stochastic equation, which provides us with an easy way to study the properties of the equation. [ABSTRACT FROM AUTHOR]

In the paper, we consider the averaging principle for a class of fractional stochastic evolution equations with random delays modulated by a two‐time‐scale continuous‐time Markov chain under the non‐Lipschitz coefficients, which extends the existing results: from Lipschitz to non‐Lipschitz case, from classical to fractional equations, from constant to random delays. Using α$$ \alpha $$‐order fractional resolvent operator theory and stopping time technique, a general theorem on the existence and uniqueness of mild solutions is obtained first; further, strong averaging principle for fractional stochastic delay evolution equation is investigated, which simplifies the original system. [ABSTRACT FROM AUTHOR]

In this paper we shall prove the Lp (p>2)-strong convergence of multiscale integration scheme for stochastic jump-diffusion systems with two-time-scale, which gives a numerical method for effective dynamical systems. [ABSTRACT FROM AUTHOR]