{'fraction_non_alphanumeric': 0.059275758011208504, 'fraction_numerical': 0.022417013938784307, 'mean_word_length': 3.960441910192445, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Strong convergence of the tamed and the semi-tamed Euler schemes for stochastic differential equations with jumps under non-global Lipschitz condition

We consider the explicit numerical approximations of stochastic differential equations (SDEs) driven by Brownian process and Poisson jump. It is well known that under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution of such SDEs without jumps, while implicit Euler method converges but requires much computational efforts. We investigate the strong convergence, the linear and nonlinear exponential stabilities of tamed Euler and semi-tamed methods for stochastic differential equation driven by Brownian process and Poisson jumps, both in compensated and non compensated forms. We prove that under non-global Lipschitz condition and superlinearly growing drift term, these schemes converge strongly with the standard one-half order. Numerical simulations to substain the theoretical results are provided.


Introduction
In this work, we consider jump-diffusion Itô's stochastic differential equations (SDEs) of the form : dX(t) = f (X(t − ))dt + g(X(t − ))dW (t) + h(X(t − ))dN(t), X(0) = X 0 , t ∈ [0, T ], T > 0. (1) Here W t is a m-dimensional Brownian motion, f : R d −→ R d , d ∈ N satisfies the one-sided Lipschitz condition and the polynomial growth condition, the functions g : R d −→ R d×m and h : R d −→ R d satisfy the globally Lipschitz condition, and N t is a one dimensional poisson process with parameter λ. Extension to vector-valued jumps with independent entries is straightforward. The one-sided Lipschitz function f can be decomposed as f = u + v, where the function u : R d −→ R d is the global Lipschitz continuous part and v : R d −→ R d is the non-global Lipschitz continuous part. Using this decomposition, we can rewrite the jumpdiffusion SDEs (1) in the following equivalent form dX(t) = u(X(t − ) + v(X(t − )) dt + g(X(t − ))dW (t) + h(X(t − ))dN(t). ( This decomposition will be used only for semi-tamed schemes. Equations of type (1) arise in a range of scientific, engineering and financial applications [11,10,6]. Most of such equations do not have explicit solutions and therefore one requires numerical schemes for their approximations. Their numerical analysis has been studied in [4,9,12,13] with implicit and explicit schemes where strong and weak convergence have been investigated. The implementation of implicit schemes requires significantly more computational effort than the explicit Euler-type approximations as Newton method is usually required to solve nonlinear systems at each time iteration in implicit schemes. The standard explicit method for approximating SDEs of type (1) is the Euler-Maruyama method [13]. Recently it has been proved (see [14,1]) that the Euler-Maruyama method often fails to converge strongly to the exact solution of nonlinear SDEs of the form (1) without jump term when at least one of the functions f and g grows superlinearly. To overcome this drawback of the Euler-Maruyama method, numerical approximation which computational cost is close to that of the Euler-Maruyama method and which converge strongly even in the case the function f is superlinearly growing was first introduced in [2] and strong convergence was investigated. Further investigations have been performed in the litterature (see for example [15,16,8] and references therein), where in [15] the time step ∆t in [2] is replaced by his power ∆t α , α ∈ (0, 1/2] in the denominator of the taming drift term. Recently the work in [15] has been extended for SDEs driven by compensated Levy noise in [3,17]. In opposite to [3,17], following the breakthrough idea in [2], we present new numerical schemes by extending their tamed scheme and the corresponding semi-tamed scheme developed in [19] to SDEs (1) driven by Brownian process and Poisson jump. Furthermore, we prove the strong convergence of the corresponding numerical approximations, both in compensated and non compensated forms. The extensions are not straightforward as several technical lemmas are needed. The linear and nonlinear mean-square stabilities of these schemes are provided in the accompanied paper [22].
The paper is organized as follows. Section 2 presents the classical result of existence and uniqueness of the solution X of (1). The compensated and non compensated tamed schemes and semi-tamed scheme are presented in Section 3. The proof of strong convergence of compensated tamed scheme is provided at Section 4, while the proof of strong convergence of semi-tamed scheme and non compensated tamed scheme are provided respectively in Section 5 and Section 6. We end in Section 7 by providing some numerical simulations to sustain our theoretical results.

Notations, assumptions and well posedness
Throughout this work, (Ω, F , P) denotes a complete probability space with a filtration (F t ) t≥0 .
For all x, y ∈ R d , we denote by x, y = Ax for all A ∈ R m×d and X L p (Ω,R) := (E X p ) 1/p , for all p ∈ [1, +∞) and all (F t )− adapted process X. We use also the following convention : We first ensure that SDEs (1) is well posed. The following assumption is needed.
Assumption 2.1. We assume that: (A. 2) The functions g, h and u satisfy the following global Lipschitz condition 3) The function f satisfies the following one-sided Lipschitz condition (A.4) The function f satisfies the following superlinear growth condition where C and c are positives constants.

Numerical Schemes and main results
We consider the SDEs (1) in the current non compensated form. Applying the tamed Euler scheme (as in [2]) in the drift term of (1) yields the following scheme that we will call non compensated tamed scheme (NCTS) where ∆t = T /M is the time step-size, M ∈ N is the number of time subdivisions, ∆W M n = W (t n+1 ) − W (t n ) and ∆N M n = N(t n+1 ) − N(t n ). Applying the semi-tamed Euler scheme (as in [19]) in the non globally Lipschitz part v of the drift term of (2) yields the following scheme that we will call semi-tamed scheme (STS) Recall that the compensated poisson process N (t) := N(t) − λt is a martingale and satisfies the the following properties We can easily check that N 2 (t) − λt is also a martingale, and therefore its quadratic variation is given by [N , N] t = λt.
We can therefore rewrite the jump-diffusion SDEs (1) in the following equivalent form where Note that as f , the function f λ satisfies the one-sided Lipschitz condition (A.3) and the superlinear growth (A.4). Applying the tamed Euler scheme in the drift term of (6) as in [2] yields the following updated scheme for jump SDEs (1) that we will call compensated tamed scheme (CTS) where ∆N Note that if the equivalent model (2) is putting in the compensated form, and the semi-tamed is applied on the non globally Lipschitz part v of the drift term f , we will obtain the same scheme as in (4).
The main result of this work is given in the following theorem.
4. Proof of Theorem 3.1 for Before giving the proof of Theorem 3.1, some preparatory results are needed. Here we consider the compensated tamed scheme (CTS) given by (7).

Preparatory results
Throughout this work the following notations will be used with slight modification in the next The aims of this section is to update all lemmas used in [2] and provide typically lemmas for Poisson jump.
Following closely [2, Lemma 3.1], we have the following main lemma.
where D M n and Ω M n are given in (12).
≤ T , using Assumption 2.1 on the functions g, h and f λ , following closely [2], the following estimation holds on Using Cauchy-Schwartz inequality and Hölder inequality, after some simplications, we have on Ω, for all M ∈ N and all n ∈ {0, 1, · · · , M − 1}.
Our proof is concluded by induction exactly as in [2, Lemma 3.1]. Details can be found in [21].
The proofs of the following lemmas can be found in [2,21].
The following inequality holds M} be the process defined in (12). The following inequality holds Since ∆N n follows a poisson law with parameter λ∆t, it follows that Lemma 4.6. The following inequality holds So from inequality (20) and using Lemma 4.5 it follows that Iterating this last inequality M times leads to for all M ∈ N, all p ∈ (1, ∞) and all z ∈ {−1, 1}.
Lemma 4.8. The following inequality holds Proof. Using the independence and the stationarity of ∆N M k , along with Lemma 4.5, it follows that Inspired by [2, Lemma 3.5], we have the following estimation.
Using Hölder's inequality, it follows that By assumption A 1 is bounded. Along the same lines as above, we prove that A 5 is bounded.
Since each of the terms A 1 , A 2 , A 3 , A 4 and A 5 is bounded, this complete the proof of Lemma 4.9.
The following lemma is an extension of [2, Lemma 3.6]. Here, we include the jump part. Proof. Using the subadditivity of the probability measure and the Markov's inequality, it follows that for all q > 1.
Multiplying both sides of the above inequality by M p leads to For q > max{2pc, 2p + 2}, we have M p+1−q/2 < 1, M p−q/2c < 1 and M p+1−q < 1. It follows for this choice of q that Using Lemma 4.9 and the fact that W T and N T are independents of M, it follows that The following lemma can be found in [7,Theorem 48 pp 193]   The proof of the following lemma can be found in [2,Lemma 3.7] or [21].
where C p is a positive constant depending on p and λ.
Using the definition of X L p (Ω,R d ) for any random variable X, it follows that Using Minkowski's inequality in its integral form yields This completes the proof of the lemma. ≤ sup This completes the proof of the lemma. Proof. Let's first represent the numerical approximation Y M n in the following appropriate form for all M ∈ N and all n ∈ {0, · · · , M}.
Using the inequality it follows that : .
Using Lemma 4.13 and Lemma 4.15, it follows that From g i (0) 2 ≤ g(0) 2 and the global Lipschitz condition satisfied by g and h, we obtain So using (26), we obtain Using the inequality (a + b + c) 2 ≤ 3a 2 + 3b 2 + 3c 2 , it follows that : for all p ∈ [1, ∞). Using the fact that Applying Gronwall lemma to (27) leads to Taking the square root and the supremum in the both sides of (28) leads to Unfortunately, (29) is not enough to conclude the proof of the lemma due to the term M in the right hand side. Using the fact that (Ω M n ) n is a decreasing sequence and by using Hölder's inequality, we obtain : Using inequality (29) yields for all p ≥ 1. From the relation it follows using Lemma 4.10 that for all p ≥ 1.
Proof. As f λ satisfies the polynomial growth condition, for all x ∈ R d we have • If x ≤ 1, then CK x ≤ CK, hence • If x ≥ 1, then x ≤ x c+1 , hence So it follows from (36) and (37) that Using inequality (38) and Lemma 4.16, it follows that for all p ∈ [1, ∞). In other hand, using the global Lipschitz condition satisfied by g and h, it follows that Using once again Lemma 4.16, it follows from (39) that for all p ∈ [1, ∞). Using the same argument as for g the following holds for all p ∈ [1, +∞). This complete the proof of Lemma 4.17.
In the sequel, for all s ∈ [0, T ] we denote by ⌊s⌋ the greatest grid point less than s. sup Proof. Using Lemma 4.14, Lemma 4.12 and the time continuous approximation (10), it follows that sup for all M ∈ N.
Using the inequalities (44), a ≤ a − b + b for all a, b ∈ R d and Lemma 4.16 it follows for all p ∈ [1, +∞) and all M ∈ N. Further, using the polynomial growth condition for all x, y ∈ R d , it follows using Hölder inequality that Using (45) and (40), the following inequality holds for all p ∈ [1, ∞).
Now we are ready to give the proof of Theorem 3.1.

Main part of the proof for
Recall that for s ∈ [0, T ], ⌊s⌋ denote the greatest grid point less than s. The time continuous solution (10) can be written into its integral form as bellow for all s ∈ [0, T ] almost surely and all M ∈ N.
Let's estimate first the quantity X s − Y M s 2 , where X s is the exact solution of (1).
Using the relation dN u = dN u − λdu, it follows that The function k : R d −→ R, x −→ x 2 is twice differentiable. Applying Itô's formula for jumps process ( [18, pp. 6-9]) to the process X s − Y M s leads to Using again the relation dN u = dN u + λdu leads to In the next step, we give some useful estimations of A 1 , A 2 , A 3 and A 6 .
Using the one-sided Lipschitz condition satisfied by f λ leads to Moreover, using the inequality 2 a, b ≤ 2 a b ≤ a 2 + b 2 for all a, b ∈ R d leads to a, b ∈ R d and the global Lipschitz condition we have Using Lemma 4.14 and the inequality (a + b) 4 ≤ 16a 4 + 16b 4 , it follows that , for all p ≥ 2.
Using the inequality Using Hölder's inequality, it follows that Using the inequality 2ab ≤ a 2 + b 2 for a, b ∈ R leads to Using the inequalities (a+b) 4 ≤ 4a 4 +4b 4 and √ a + b ≤ √ a+ √ b for a, b ∈ R + , we obtain Using the global Lipschitz condition, leads to Using the same estimations as for B 21 , it follows that : Taking the supremum under the integrand in the last term of the above inequality and using the fact that C p is a generic constant leads to Inserting (58) and (59) into (57) gives Using again Lemma 4.14 leads to Using the same argument as for B 21 , we obtain Taking the L p norm in both side of (55), inserting inequalities (56), (60), (61) and using Minkowski's inequality in its integral form leads to The functions λh and f in the numerical solution of the scheme NCTS given by (3) (or (66)) satisfy respectively the same conditions as the u λ and v in the numerical solution of the STS given by (4). Hence, it follows from the proof in Section 5 that there exists a constant C p > 0 such that for all p ∈ [1, ∞).

Numerical simulations
In this section, we present some numerical experiments to illustrate our theoretical strong convergence. We consider the stochastic differential equation with initial condition X 0 = 1. N t is the scalar Poisson process with parameter λ = 1. Here It is obvious to check that u, v, g and h satisfy