The averaging principle for stochastic differential equations with Caputo fractional derivative

Averaging principle for a type of Caputo fractional stochastic differential equations

“…In [19], under a different framework, when the existence and uniqueness of solutions are established, the asymptotic distance between two distinct solutions is discussed. To the best of our knowledge, the averaging principle for fractional differential equations still has a big challenge, there are only a few papers [12,20] to investigate the averaging principle for Caputo fractional SDEs. It is worth noting that the averaging principle has been obtained in these two papers by using similar methods but different assumptions.…”

An Averaging Principle for Mckean–Vlasov-Type Caputo Fractional Stochastic Differential Equations

“…On the contrary, the problem of averaging for stochastic fractional order differential equations have received a lot of attention in recent years, and some results [13] have been obtained under averaging condition consistence with the classic case (see [4,5,14]). Noting that the fractional order derivative is a nonlocal operator, therefore, the fractional order differential equation is more effective for describing certain phenomena in the real world (see [15][16][17]).…”

“…where α ∈ (1/2, 1] and B t is a scalar Brownian motion. Nonlinear terms f and g are H-valued functions defined on R + × H × M c 2 (H), and M c 2 (H) denotes a proper subset of probability measure on H. If the terms f and g do not depend on the probability distribution μ(t) of the process X at time t, such equations have been studied by [13] and other authors. If α � 1, the equation becomes a classical Mckean-Vlasov-type stochastic differential equations which have been considered by many authors with different approaches (see [18][19][20]).…”

Caratheodory’s approximation for a type of Caputo fractional stochastic differential equations

“…[13], the authors considered the existence of stable manifolds for a type of stochastic differential equations. The authors of paper [14] considered the averaging principle of a type of stochastic fractional differential under some conditions consistent with the stochastic differential equations. In [15], the existence of global forward attracting set for stochastic lattice systems with a Caputo fractional time derivative in the weak mean-square topology is established.…”