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This paper will prove the normal deviation of the synchronization of stochastic coupled system. According to the relationship between the stationary solution and the general solution, the martingale method is used to prove the normal deviation of the fixed initial value of the multi-scale system, thereby obtaining the normal deviation of the stationary solution. At the same time, with the relationship between the synchronized system and the multi-scale system, the normal deviation of the synchronization is obtained. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with averaging principle for two-time-scale stochastic differential equations (SDEs) with non-Lipschitz coefficients, which extends the existing results: from Lipschitz to non-Lipschitz case. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for coupled system is established, and as a result, the system can be reduced to a single SDEs with a modified coefficient which is also non-Lipschitz. Moreover, it is shown that the slow variable strongly converges to the solution of the corresponding averaging equation. [ABSTRACT FROM AUTHOR]

The synchronization of stochastic differential equations with both additive noise and linear multiplicative noise is investigated in pathwise sense, which generalize the results of [5] and [6]. In our situation, we can deal with the synchronization of the systems with mixed type noise, where one system has the additive noise, another has the linear multiplicative noise. [ABSTRACT FROM AUTHOR]

The existence and uniqueness theorem of solutions provides an effective tool for the model validation of both deterministic and stochastic equations. The objective of this paper is to establish the existence and uniqueness of solutions for a class of Itô-Doob stochastic fractional differential equations under non-Lipschitz condition which is weaker than Lipschitz one and contains it as a special case. The solution is constructed with the aid of Carathéodory approximation. Moreover, the continuous dependence of solutions on the initial value is investigated in view of the stability of solutions in the sense of mean square. Finally, an example is given to illustrate the theory. [ABSTRACT FROM AUTHOR]

The averaging principle for multivalued stochastic differential equations (MSDEs) driven by Brownian motion with Brownian noise is investigated. An averaged MSDEs for the original MSDEs is proposed, and their solutions are quantitatively compared. Under suitable assumptions, it is shown that the solution of the MSDEs converges to that of the original MSDEs in the sense of mean square and also in probability. Two examples are presented to illustrate the averaging principle. [ABSTRACT FROM PUBLISHER]

In this paper, we deal with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equations. Under suitable conditions, we expand the weak error in powers of timescale parameter. We prove that the rate of weak convergence to the averaged dynamics is of order 1. This reveals that the rate of weak convergence is essentially twice that of strong convergence. [ABSTRACT FROM AUTHOR]