Approximate solutions for a class of doubly perturbed stochastic differential equations

In this paper, we study the Carathéodory approximate solution for a class of doubly perturbed stochastic differential equations (DPSDEs). Based on the Carathéodory approximation procedure, we prove that DPSDEs have a unique solution and show that the Carathéodory approximate solution converges to the solution of DPSDEs under the global Lipschitz condition. Moreover, we extend the above results to the case of DPSDEs with non-Lipschitz coefficients.


Introduction
As the limit process from a weak polymers model, the following doubly perturbed Brownian motion w a sd i s c u s s e db yN o r r i se ta l . [ 1], and it also arises as the scaling limit of some selfinteracting random walks (see [2]). During the past few decades, equation (1.1)h a sa ttracted much interest from many scholars, for example, [3][4][5][6][7][8][9]. Following them, Doney et al. [10] studied the singly perturbed Skorohod equations Using the Picard iterative procedure, they showed the existence and uniqueness of the solution to equation (1.2). Hu et al. [11] discussed the existence and uniqueness of the solution to doubly perturbed neutral stochastic functional equations, while Luo [12]obtained the existence and uniqueness of the solution to doubly perturbed jump-diffusion processes.
In fact, the Picard iterative method is a well-known procedure for approximating the solution of stochastic differential equations (SDEs). However, to obtain the Picard iterative sequence x n (t), one needs to compute x i (t), 0 ≤ i ≤ n -1. And this brings us a lot of calculations on stepwise iterated Ito's integrals. In the early twentieth century, Carathéodory [13] put forward the Carathéodory approximation scheme for ordinary differential equations. In this scheme, Carathéodory defined the approximate solution via a delay equation, and the delay equation can be solved explicitly by successive integrations over intervals of length 1 n . In other words, the Carathéodory approximation scheme avoids calculating x i (t), 0 ≤ i ≤ n -1.
Because of its advantage, this approximation procedure has received great attention, and many people have been devoted to the study of the Carathéodory scheme for SDEs. For example, Bell and Mohammed [14] extended the Carathéodory approximation scheme to the case of SDEs and showed the convergence of the Carathéodory approximate solution. Mao [15,16] considered a class of SDEs with variable delays and studied the Carathéodory approximate solution of delay SDEs. Turo [17] discussed the Carathéodory approximate solution of stochastic functional differential equations (SFDEs) and established the existence theorem for SFDEs. Liu [18] investigated a class of semilinear stochastic evolution equations with time delays and proved that the Carathéodory approximate solution converges to the solution of stochastic delay evolution equations.
Motivated by the above mentioned papers, we will study the Carathéodory approximate scheme of doubly perturbed stochastic differential equations (DPSDEs) To the best of our knowledge, so far little is known about the Carathéodory approximations for equation (1.3), and the aim of this paper is to close this gap. In this paper, we will prove that the Carathéodory approximate solution converges to the solution under the global Lipschitz condition. Moreover, we will replace the global Lipschitz condition by a more general condition proposed by [19,20]

Carathéodory approximation and global Lipschitz DPSDEs
Let ( , F, {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous, while F 0 contains all Pnull sets). Let {w(t)} t≥0 be a one-dimensional Brownian motion defined on the probability space ( , F, P). Let L 2 ([a, b]; R) denote the family of F t -measurable, R-valued processes a |f (t)| 2 dt < ∞ a.s. Consider the following doubly perturbed stochastic differential equations: where α, β ∈ (0, 1), the initial value x(0) = x 0 ∈ R and f :[0,T] × R → R, g :[0,T] × R → R are both Borel-measurable functions. In this paper, we assume that the initial value x 0 is independent of w and satisfies E|x 0 | 2 < ∞. Now, we define the sequence of the Carathéodory approximate solutions x n : [-1, T] → R. For all n ≥ 1, we define Note that x n (t) can be calculated step by step on the intervals [0, 1 n ), [ 1 n , 2 n ),...,etc. To obtain the main results, we give the following conditions.

Assumption 2.1
For any x, y ∈ R and t ∈ [0, T], there exists a positive constant k such that

Assumption 2.2
For any t ∈ [0, T], there exists a positive constantk such that
where C is a constant independent of n.
In the sequel, to prove our main results, we need some useful lemmas.
By the basic inequality |a + b + c| 2 ≤ 3(|a| 2 + |b| 2 + |c| 2 ), one has for any t 1 ∈ [0, T]. Using the Hölder inequality and the Burkholder-Davis-Gundy inequality, we can easily see that Finally, the Gronwall inequality implies that The proof is therefore complete.
where C 2 is a positive constant.
Proof For all n ≥ 1and0≤ s < t ≤ T, it follows from (2.2)that Next, let us consider the following two cases.
Proof of Theorem 2.1 Firstly, we will show that the sequence {x n (t)} is a Cauchy sequence in L 2 ([0, T]; R). For any n > m ≥ 1, it follows that Noting that By the basic inequality and Assumption 2.3,weobtainthat (2.16) Then, using the Hölder inequality and the Burkholder-Davis-Gundy inequality again, we have E sup Hence, . By the Gronwall inequality, we have    x n (t)-x(t) 2 → 0a s n →∞.
In fact, the proof of the convergence of the Carathéodory approximation represents an alternative to the procedure for establishing the existence and uniqueness of the solution to delay DPSDEs. In other words, the Carathéodory approximation scheme is applicable to a class of DPSDEs.

Non-Lipschitz DPSDEs
In this section, we will replace the global Lipschitz condition (2.3) with a more general condition and show that the Carathéodory approximate solution still converges to the true solution of equation (2.1).

Assumption 3.1
For any x, y ∈ R and t ∈ [0, T], there exists a function k(·)suchthat where k(u) is a concave non-decreasing continuous function such that k(0) = 0 and To prove Theorem 3.1, we will need the following Bihari inequality.
whereC 1 is a positive constant.
Proof By the Hölder inequality and the Burkholder-Davis-Gundy inequality, it follows from (2.8)that  Then the Jensen inequality implies that Since k(x) x and k ′ + (x) are non-negative, non-increasing functions, we have that is a non-negative, non-increasing function which implies that ρ is a non-negative, nondecreasing concave function. Note that k(0) = 0, then ρ(0) = 0, and there exists a pair of positive constants a and b such that ρ(u) ≤ a + bu for u ≥ 0.

Remark 3.3
In particular, if we let k(u)=ku, u ≥ 0, we see that the Lipschitz condition (2.3) is a special case of our proposed condition (3.1). In other words, we obtain a more general result than Theorem 2.1.