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Abstract
We investigate the averaging principle for multivalued stochastic differential equations (MSDEs) driven by a random process under non-Lipschitz conditions. We consider the convergence of solutions in L^{p}~(pgeq2) and in probability between the MSDEs and the corresponding averaged MSDEs.
Highlights
1 Introduction Most systems in science and industry are perturbed by some random environmental effects described by stochastic differential equations with Brownian motion, Lévy process, Poisson process, and so on
A series of useful theories and methods have been proposed to explore stochastic differential equations, such as invariant manifolds [1,2,3], averaging principle [3,4,5,6,7,8,9,10,11,12], homogenization principle, and so on. All these theories and methods develop to extract an effective dynamics from these stochastic differential equations, which is more effective for analysis and simulation
The essence of averaging principle is to establish an approximation theorem for a simplified stochastic differential equation that replaces the original one in some sense and the corresponding optimal order convergence
Summary
Most systems in science and industry are perturbed by some random environmental effects described by stochastic differential equations with (fractional) Brownian motion, Lévy process, Poisson process, and so on. A series of useful theories and methods have been proposed to explore stochastic differential equations, such as invariant manifolds [1,2,3], averaging principle [3,4,5,6,7,8,9,10,11,12], homogenization principle, and so on. In [16], the author discussed the existence and uniqueness of a solution in Lp (pth moment) sense for some multivalued stochastic differential equations under a non-Lipschitz condition. We concern the p-moment averaging principle for a multivalued stochastic differential equation under a non-Lipschitz condition, which is different from [14] under the Lipschitz condition. To derive the averaging principle for non-Lipschitz multivalued stochastic differential equation, we need some assumptions given . We will explicitly write the dependence of constants on parameters
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