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

Abstract

The Caratheodory approximation for a type of Caputo fractional stochastic differential equations is considered. As is well known, under the Lipschitz and linear growth conditions, the existence and uniqueness of solutions for some type of differential equations can be established. However, this approach does not give an explicit expression for solutions; it is not applicable in practice sometimes. Therefore, it is important to seek the approximate solution. As an extending work for stochastic differential equations, in this paper, we consider Caratheodory’s approximate solution for a type of Caputo fractional stochastic differential equations.

On the Existence and Uniqueness of Solutions for Multidimensional Fractional Stochastic Differential Equations with Variable Order

“…If the functions α(•), N i (•), and i (•, •) are constants, then these special cases have been considered in papers [14,16] (see also paper [10]). Definition 3.…”

“…To treat that, we mainly establish a new set of sufficient conditions for nonlinear functions which generalizes the ones assumed in [14,16]. Then, we construct an iteration sequence involving variable fractional order α(t), which differs from the ones defined in [14,16]. After that, based on our analysis and discussion, we prove that the considered sequence is converging under those conditions to the unique solution of our studied problem (1).…”

“…Several authors [5][6][7][8] have dealt with different research interests for classical SDEs. Then, they extended their studies to the fractional case (FSDEs with constant order α) and investigated many results like existence, uniqueness, and stability for various classes of FSDEs (see [9][10][11][12][13][14][15][16]).…”

“…Motivated by these facts, our purpose is to develop the classical SDEs towards fractional stochastic differential equations involving variable order α(t). In particular, we aim to extend and improve the existence and uniqueness results that appeared in [14,16].…”

“…In this paper, we introduce a new class of Caputo-type nonlinear VOFSDEs (see Equation ( 1)). To treat that, we mainly establish a new set of sufficient conditions for nonlinear functions which generalizes the ones assumed in [14,16]. Then, we construct an iteration sequence involving variable fractional order α(t), which differs from the ones defined in [14,16].…”