Numerical solutions of doubly perturbed stochastic delay differential equations driven by Lèvy process

In this paper, the numerical solutions of doubly perturbed stochastic delay differential equations driven by Lèvy process are investigated. Using the Euler–Maruyama method, we define the numerical solutions, and show that the numerical solutions converge to the true solutions under the local Lipschitz condition. As a corollary, we give the order of convergence under the global Lipschtiz condition.


Introduction
The purpose of this paper is to study the numerical solutions of doubly perturbed stochastic delay differential equations (DPSDDEs) driven by Lèvy process. Such DPSDDEs take the form:

x(δ(s)))d B(s)
where 0 < α, β < 1 and α + β < 1; B(t), t ≥ 0 is a standard Brownian motion and N (dt, dl) is a compensated Poisson random measure; the mappings f : are all Borel-measurable functions; c ∈ (0, +∞] is the maximum allowable jump size; δ(s) stands for the time delay. The perturbed Brownian motion is a limit process from a "weak" polymers model of Norris et al. [14], and it also arises as the scaling limit of some self-interacting random walks (see e.g. [17,18]). This process behaves exactly as a Brownian motion except when it hit its past maximum (or minimum), where it gets an extra 'push'. During the past 30 years, there are numerous papers concerned with Brownian motions perturbed at their extrema (see e.g. [2,[4][5][6]15]). For example, Doney and Zhang [6] obtained the existence and uniqueness of the solutions for the following perturbed nonlinear diffusion process x(s), (1.2) where f and g are Lipschitz continuous functions. Recently, the large deviation principle for Eq. (1.2) was established by Bo and Zhang [2]. Most of DPSDDEs do not have explicit solutions and hence require numerical solutions. However, there are few numerical methods available for DPSDDEs yet. The numerical methods for stochastic differential equations (SDEs) have been well studied (see e.g. [7,12,16]), but this is not the case for stochastic delay differential equations (SDDEs) as it has been pointed out in [8]. Buckwar [3] studied the numerical solution of SDDEs under the global Lipschitz condition, and Mao [13] investigated it under the local Lipschitz condition. Moreover, some results of the numerical solutions of SDEs with jumps were obtained. One can see Li and Chang [10], Wang and Gan [19].
However, few work has been done on the numerical solutions of DPSDDEs. In this paper, we define the numerical solutions by the Euler-Maruyama method, and show that the numerical solutions converge to the true solution under the local Lipschitz condition. As a corollary, we give the order of convergence under the global Lipschitz condition.
The paper is organized as follows. In Sect. 2, we introduce some necessary notations and define the Euler-Maruyama approximate solution to DPSDDEs. In Sect. 3, a number of useful lemmas are presented. In Sect. 4, the convergence of numerical solutions is proved in the sense of mean square. Finally, in Sect. 5, conclusions and discussions on future research topics are given.

Preliminary notation and Euler-Maruyama method
Throughout this paper, we let ( , F, P) be a complete probability space with some filtration {F t } t 0 satisfying the usual conditions (i.e., the filtration is increasing and right continuous while F 0 contains all P-null sets). Let |x| be the Euclidean norm of Let C(a) denote a constant, whose value depends only on a. For simplicity, we denote by a ∨ b = max{a, b} and a ∧ b = min{a, b}.
Let B = (B(t), t ≥ 0) be an m-dimensional standard F t -adapted Brownian motion and N be an independent F t -adapted Poisson random measure defined on R + × (R d − {0}) with compensator N and intensity measure ν, where ν is a Lèvy measure so that N (dt, dy) for some positive constant ρ.
In this paper, we make the following assumptions: (H1) There exists a constant K 1 > 0 such that for all −τ ≤ s < t ≤ 0 (H2) For each n and each 2 ≤ η ≤ p, there exists a positive constant K 2 (n), such that (H3) For 2 ≤ η ≤ p, there exists a positive constant K 3 such that The existence and uniqueness of the solution for Eq. (1.1) can be guaranteed by (H2-H3) (see [11]). Now, we will define the Euler-Maruyama approximate solution of the DPSDDEs (1.1). Let the time-step size ∈ (0, 1) be a fraction of τ, that is = τ N for some sufficiently large integer N . Then the discrete Euler-Maruyama approximate solution is defined bȳ (2.5) denotes the integer part of the real number u . Thus, I [δ(k )] represents the integer part of δ(k ) . Clearly, To define the continuous extension, let us introduce two step processes We define the continuous Euler-Maruyama approximate solution as follows where

t) (s)y(s).
Let . In Theorem 4.1, we shall show that the error of Euler-Maruyama approximate solution converges to zero in L 2 as → 0.

Lemmas and corollary
The key contribution of this paper is to show that the Euler-Maruyama approximate solutions will converge to the true solutions of Eq. (1.1) under the local Lipschitz condition. The proof of the result is rather technical, so we present several lemmas before the main result.
Proof By Theorem 2.11 in [9] (or Theorem 2.4 in [1]), there exists a positive constant C( p), such that In view of Eq. (2.4), we obtain . This completes the proof.
The Hölder inequality and (H3) imply that On the other hand, by virtue of the Burkholder-Davis-Gundy inequality (Theorem 1.7.3 in [12]) and (H3), we have where Combining this with the Gronwall's inequality, we get This implies our claim immediately by letting Proof The proof is similar to Lemma 3.2, here we omit it.
For each n > 0, we define the stopping times and v n = τ n ∧ σ n . (We set inf ∅ = ∞).

Corollary 3.2 Under condition
Proof For any t ∈ [0, T ], we can choose a positive integer k such that t ∈ [k , (k + 1) ) and k = k(ω) is dependent on the sample path. Therefore where C 3 (n) is independent of k. Thus, for any k, we can get E sup k ≤t<(k+1) |y(t) − z 1 (t)| 2 ≤ C 3 (n) and the Corollary follows. (H1) and (H2), if is small enough such that (ρ + 1) ≤ 1, then there exists a positive constant C 4 (n), such that

Lemma 3.5 Under
Proof One just needs to repeat the proof of the Lemma 3.3 as in Mao [13], so we omit the detailed proof.

Main result
The primary aim of this paper is to establish the following main result.
Proof It is obvious that By the Young inequality where a, b, h > 0 and p > 2, we obtain Consequently, We get Moreover, Substituting these inequalities above into Eq. (4.1) leads to Let us now estimate the first term on the right-hand side of Eq. (4.2). Clearly, Thanks to Eqs. (1.1) and (2.6), we derive By the definition of y 1 (t), we have Noting that sup 0≤s≤t∧v n y 1 (s, t ∧ v n ) ≤ sup 0≤s≤t∧v n y(s), we can get  This proves that Consequently, Using the Doob martingale inequality, the local Lipschitz condition (H2) and Corollary 3.2 gives It follows that Note that y(t) = ξ(t) for any t ∈ [−τ, 0] and We get Combining this with Lemmas 3.4 and 3.5, we see that where C 5,1 (n) = 12K 2 (n)(T +8) Now, for any > 0, we can choose a sufficiently small h such that 2 p+1 hC 5 p < 3 , then choose n so large that 2( p−2)C 5 ph 2 p−2 n p < 3 and finally choose sufficiently small such that C 5,2 (n)e C 5,1 (n)T < 3 . So This completes the proof.
Theorem 4.1 shows that under conditions (H1-H3) the Euler-Maruyama approximate solutions strongly converge to the true solutions. However, Theorem 4.1 does not give the order of the convergence. In the following, we will reveal the order of convergence, here we need the global Lipschitz condition instead of the local Lipschitz condition as follows.

Conclusion and outlook
In this paper, the numerical solutions of doubly perturbed stochastic delay differential equations driven by Lèvy process are considered. We define the numerical solutions and show that the numerical solutions converge to the true solutions under the local Lipschitz condition. Moreover, we give the order of convergence under the global Lipschitz condition. In our future works, we will investigate doubly perturbed stochastic delay differential equations with Markovian switching and doubly perturbed neutral SDDEs.