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

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.


Introduction
Recently, stochastic fractional differential equations and stochastic fractional partial differential equations have attracted more and more attention. It turns out that differential equations involving derivatives of non-integer orders have memory properties, which are called non-local properties. Because of the non-local property of the Caputo fractional derivatives in time, Caputo fractional differential equations are important to model and describe problems in many disciplines, such as engineering, physics, and chemistry. For more details, see [1][2][3][4][5][6][7].
Compared with the work on deterministic fractional differential equations, the study of stochastic fractional differential equations is still in its infancy. However, the majority of work is concerned about the existence and uniqueness of solutions; see [8][9][10][11][12]. Until quite recently, there were some authors who considered some types of Caputo fractional stochastic differential equations and Caputo fractional stochastic partial differential equations by different approaching. For example, in Ref. [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. In [16], the asymptotic distance between two distinct solutions is considered under a temporally weighted norm. Its worth mentioning that the Euler-Maruyama type approximate results for Caputo fractional stochastic differential equations have been established by [17]. For more related work, see [12,[18][19][20][21][22].
The Caratheodory approximation scheme was first considered by Caraheodory for ordinary differential equations, then Bell, Mohammad and Mao extended it to the stochastic differential equations case; see [23]. To the best of our knowledge, there is no work paying attention to the Caratheodory approximation for the Caputo fractional stochastic differential equation. In this paper, we will consider the Caratheodory approximation for the following type of Caputo fractional stochastic differential equation: where α ∈ ( 1 2 , 1). For more details see Sect. 2. The aim of this paper is to extend the Caratheodory approximate results for Eq. (1.1).
This article is organized as follows. In Sect. 2 we will give some assumptions and basic results that we need. The existence and uniqueness of solution will be discussed in Sect. 3. In the last section, we will consider the Caratheodory approximation for the Caputo fractional stochastic differential equations.
Throughout this paper, the letter C will denote positive constants whose value may change in different occasions. We will write the dependence of a constant on parameters explicitly if it is essential.

Preliminaries
We impose the following assumptions to guarantee the existence and uniqueness of solution, H denote a Hilbert space, its norm is denoted by | · |. H1: Lipschitz condition: Let t ≥ 0 and constant k > 0, such that, for all x, y ∈ H, H2: Growth condition: Let t ≥ 0 and constant k > 0, such that, for all x ∈ H, The following generalization of Gronwall's lemma for singular kernels is needed for us to establish our results; see [15,24]. Lemma 2.1 Suppose b ≥ 0, β > 0 and a(t) is a nonnegative function locally integrable on 0 ≤ t < T (some T ≤ +∞), and suppose u(t) is nonnegative and locally integrable on where (·) is the Gamma function.

Well-posedness
In this section, we consider the existence and uniqueness of solution for the following equation under conditions H1 and H2: where B t is a scalar Brownian motion, f and g are H-value functions.
) and satisfies the following integral equation: The existence and uniqueness of solutions for Eq. (3.1) have been considered by our previous work [25]. Similar problem also considered by [16] under different framework. To make this paper self-contained, we just give the main part of the proof for the following theorem. Proof We prove the theorem by the contraction mapping principle. Using conditions H1 and H2, Lemma 2.1, we can derive that be the Banach space of all F t -adapted processes.
For any t ∈ [0, T] and X t ∈ S, define a mapping as follows: It is easy to verify that Denote β = 2α -1 > 0, by the Cauchy-Schwartz inequality, Itô's isometry formula and condition H1, we have Using mathematical induction methods, we can deduce the following fact: For n = 1, by simple calculation we get which satisfies Eq. (3.3) with n = 1. Now, assuming that Eq. (3.3) is satisfied for n = j, we claim that it is also correct for n = j + 1. We have To get the estimate for n = j + 1, we only need to consider the following integral: (1z) β-1 t β-1 t jβ z jβ t dz where B(·, ·) is the Beta function. Combining this result with Eq. (3.4) we have Then we arrive at the following estimate for all n: If we can prove for sufficient large n, then the theorem holds. Consider the following series of positive terms: We will show that as n → +∞, which guarantees that Eq. (3.7) holds. Thanks to the d' Alembert discriminant method, we only need to justify lim n→∞ ( 2k(T+1) (α) 2 ) (β)T β (nβ) ((n + 1)β)) < 1.

Caratheodory's approximate solutions
In this section, we consider the Caratheodory approximation for stochastic fractional differential equations. Similar to the stochastic differential equations approach, we try to give the definition of Caratheodory's approximate solutions for stochastic fractional differential equations as follows.
Proof From the simple arithmetic inequality By the Cauchy-Schwarz inequality and condition H2, we can estimate the term I 2 as follows: Similarly, with Itô's isometry formula and condition H2, we have an estimate for the stochastic integral term: Combining the estimate for I 1 , I 2 , I 3 , we arrive at where we denote and Note that, for t 1 ≤ t 2 , we have Then sup 0≤r≤t E x n (r) 2 ≤ r 1 + r 2 t 0 (ts) (2α-1)-1 sup 0≤r≤s E x n (r) 2 ds.

Lemma 4.2 Under the condition H2, for all n
has been defined in Lemma 4.1.
For J 12 , we have the following result: For J 2 , taking the Itô isometry formula and condition H2 into account, using similar estimate methods to J 1 , it can be shown that Combining all the deduced estimates, we have This completes the proof.
Using the Cauchy-Schwartz inequality and the condition H1, we have the following estimate for I 11 : Similarly, for I 12 , we have Also, we can divide I 2 into two parts as follows: By the Itô isometry formula, we get and Combining with the estimate for I 1 and I 2 , it is derived that E x(t)x n (t) 2 ≤ k(T + 1) (α) 2 Applying Lemma 2.1, we obtain E x(t)x n (t) 2 ≤ q 2 1 + E 2α-1,1 q 1 (2α -1)T 2α-1 =: C n 2α-1 .
This completes the proof.
Remark 4.1 When α = 1, i.e. Eq. (1.1) becomes a stochastic differential equation, the convergent rate of the scheme in Theorem 4.1 coincides with the well-known convergent rate of the classical Caratheodory results; see [23].