APPROXIMATION OF THE INVARIANT DISTRIBUTION FOR A CLASS OF ERGODIC JUMP DIFFUSIONS

. In this article, we approximate the invariant distribution ν of an ergodic Jump Diﬀusion driven by the sum of a Brownian motion and a Compound Poisson process with sub-Gaussian jumps. We ﬁrst construct an Euler discretization scheme with decreasing time steps. This scheme is similar to those introduced in Lamberton and Pag`es Bernoulli 8 (2002) 367-405. for a Brownian diﬀusion and extended in F. Panloup, Ann. Appl. Probab. 18 (2008) 379-426. to a diﬀusion with L´evy jumps. We obtain a non-asymptotic quasi Gaussian (asymptotically Gaussian) concentration bound for the diﬀerence between the invariant distribution and the empirical distribution computed with the scheme of decreasing time step along appropriate test functions f such that f − ν ( f ) is a coboundary of the inﬁnitesimal generator. r -dimensional random vectors with common distribution π on B ( R r ) and ( N t ) t ≥ 0 is a Poisson process, independent of ( Y k ) k ∈ N . The processes ( W t ) t ≥ 0 and ( Z t ) t ≥ 0 are assumed to have the same dimension for the sake of simplicity. Moreover, ( N t ) t ≥ 0 , ( Y k ) k ∈ N and ( W t ) t ≥ 0 are independent and deﬁned on a given ﬁltered probability space (Ω , G , ( G t ) t ≥ 0 , P ) . We assume that b , σ , and κ satisfy a suitable Lyapunov condition (assumption ( L V ) in Sect. 1.3) which ensures the existence of an invariant distribution ν of ( X t ) t ≥ 0 (see [11]). For the sake of simplicity we also assume the uniqueness of the invariant distribution. We refer to [8] under


Setting
Let (X t ) t≥0 be a d-dimensional càdlàg process solution of the stochastic differential equation: where b : R d → R d , σ : R d → R d ⊗ R r and κ : R d → R d ⊗ R r are Lipschitz continuous, (W t ) t≥0 is a Wiener process of dimension r, and (Z t ) t≥0 is a R r -valued compound Poisson process (CPP), Z t = Nt k=1 Y k , where (Y k ) k∈N are i.i.d. r -dimensional random vectors with common distribution π on B(R r ) and (N t ) t≥0 is a Poisson process, independent of (Y k ) k∈N . The processes (W t ) t≥0 and (Z t ) t≥0 are assumed to have the same dimension for the sake of simplicity. Moreover, (N t ) t≥0 , (Y k ) k∈N and (W t ) t≥0 are independent and defined on a given filtered probability space (Ω, G, (G t ) t≥0 , P). We assume that b, σ, and κ satisfy a suitable Lyapunov condition (assumption (L V ) in Sect. 1.3) which ensures the existence of an invariant distribution ν of (X t ) t≥0 (see [11]). For the sake of simplicity we also assume the uniqueness of the invariant distribution. We refer to [8] under irreducibility and Lyapunov conditions for the existence and uniqueness of the invariant distribution for a diffusion driven by Lévy process.
Generally, the invariant distribution ν of (X t ) t≥0 is not explicitly known. Its approximation is then an important issue. The aim of this paper is to construct an appropriate algorithm ν n such that lim n→∞ ν n (f ) = ν(f ) a.s. for all suitable test functions f , and to establish a non-asymptotic concentration bound for the probability of the deviation ν n (f ) − ν(f ).
The algorithm that we define in this article is based on an Euler-like discretization scheme with decreasing time step (γ n ) n≥1 s.t. lim n γ n = 0. Lamberton and Pagès developed such a scheme in [7] for a Brownian diffusion. They showed that the empirical measure of their scheme converges to the invariant measure of the diffusion and that it satisfies the Central Limit Theorem. The decreasing steps allow the empirical measure to directly converge towards the invariant one. If we choose a constant time step γ k = h > 0 in the scheme, the expected ergodic theorem is ν n (f ) where ν h is the invariant distribution of the scheme which is supposed to converge toward the invariant measure of the diffusion (E) when h → 0 (for more details about this approach we refer to [14], [13] and [9]). Next, Panloup [11], [10], adapted the algorithm of [7] to diffusions driven by Lévy processes, he established the convergence and the Central Limit Theorem for the empirical measure in this case.
In the same way as the questions of the convergence of the empirical measure ν n or of its limiting distribution, the natural question is that of the nature of the deviations ν n (f ) − ν(f ) along appropriate test functions f . In the case of the Brownian diffusion this question was considered in [4] and [5]. Note that in the Brownian diffusion case the innovations of the Euler scheme are designed in order to "mimic" Brownian increments, hence it is natural to assume that they satisfy some Gaussian Concentration property (assumption (GC) in Sect. 1.3). In particular this Gaussian Concentration property is satisfied by Gaussian or symmetric Bernoulli law. Taken as an assumption on the Brownian innovations of the scheme, it allows to show a non-asymptotic Gaussian Concentration bound for the probability of the deviations of ν n (f ) from ν(f ). The deviation ν n (f ) − ν(f ) is evaluated along the functions f such that f − ν(f ) is a coboundary of the infinitesimal generator of the diffusion.
When the diffusion contains Lévy jumps, it is not generally expected that these deviations are Gaussian like which is not in accordance with the CLT from [10]. But such a behavior seems natural if we suppose that the driving Lévy process is a Compound Poisson process and the jump size vectors (Y k ) k∈N satisfy a Gaussian Concentration property (GC). In this paper, we focus on this situation which is simple but relevant from the point of view of application.
Some specific stochastic diffusions with Gaussian jumps have already been used in finance, in stochastic volatility models, see e.g. [15], and in interest rate models, see e.g. [6].
In general, for a Euler scheme corresponding to a Jump Diffusion with Lévy jumps, one has to define numerically computable jump vectors designed to "mimic" the increments of the driving Lévy process. In most cases, the increments of a Lévy process are not numerically computable, that is why it is important to propose different ways to approximate these increments according to the nature of the driving Lévy process. In this paper we introduce a scheme (S) particularly suitable in the case where a driven Lévy process is a Compound Poisson. Note that our scheme is close to the scheme (C) of [11]. Like in the previously mentioned articles, we denote time steps (γ k ) k≥1 , and for any n ≥ 0, we define: where X 0 is an R d valued random variables such that X 0 ∈ L 2 (Ω, F 0 , P), (U n ) n≥1 is an i.i.d. sequence of centered random variables matching the moments of the standard Gaussian law on R r up to order three and independent of X 0 . Furthermore, for every n ≥ 1 we put where (B n ) n≥1 are one-dimensional independent Bernoulli random variables, independent of X 0 , (U n ) n≥1 and = Bern(µγ n ), where µ is the intensity of the Poisson process driving the CPP (Z t ) t≥0 . The choice (1.1) of the innovations Z n , n ∈ N is motivated by the following heuristic reasoning: Z n has to "mimic" the increment of the CPP Z t = Nt k=1 Y k on the small-time interval of the length γ n . The probability that the Poisson process (N t ) ≥0 jumps on this interval is equal to 1 − exp(−γ n ) = γ n + o(γ n ), and in this case, it will most probably have only one jump. Hence we approximate the increment ∆N γn of the Poisson process by a {0, 1} random variable with the probability of 1 equal to γ n , and the increment Z γn of the CPP by this jump-detecting Bernoulli variable, multiplied by the size of the jump.
We also introduce the empirical (random) measure of the scheme: for any A ∈ B(R d ) Obviously, to study long time behavior, we have to consider steps (γ k ) k≥1 such that the current time of the scheme Γ n → n +∞.
We recall as well that γ k ↓ k 0. We suppose that both jump amplitudes (Y n ) n≥1 and Brownian innovations (U n ) n≥1 satisfy a Gaussian concentration (see further the assumption (GC)). As we already mentioned, the aim of the paper is to show that this assumption implies a non-asymptotic (quasi) Gaussian Concentration inequality for the probability of the deviations of ν n (f ) from ν(f ) (see Thm. 2.1, Sect. 2 below). The main argument in the proof of Theorem 2.1 is the fact that the (GC) property of jumps' sizes Y k , k ∈ N, permits to show a similar Gaussian Concentration property for the jump innovations Z k , k ∈ N. This result is given in Proposition 1.5. However the Concentration property of jump innovations depends on the dimension of the jump heights. This dependence survives in the main Theorem 2.1 giving a quasi-Gaussian Concentration of the deviation of ν n from ν.
The paper is organized as follows. In Section 1.2, we introduce some useful notations. The assumptions required for our main results are outlined in Section 1.3. In this part, we formulate a quasi-Gaussian concentration property of the jump innovation Z n , the proof is given in Section 2.4. We state in Section 1.4 some already known results connected with the approximation scheme. Our main results are in Section 2, and the proof is located in Section 2.3. Section 3 is dedicated to the analysis of the exponential integrability of Lyapunov function. Technical lemmas are stated in Section 2.2, but their proofs are postponed to Section 4. Eventually, we propose a numerical illustration of our main result in Section 5.

General notations
For two sequences (u n ) n∈N , (v n ) n∈N the notation u n v n means that ∃n 0 ∈ N, ∃C ≥ 1 s.t. ∀n ≥ n 0 , C −1 v n ≤ u n ≤ Cv n .
Henceforth, C will be a non negative constant, and (e n ) n≥1 , (R n ) n≥1 will be deterministic sequences s.t. e n → n 0 and R n → n 1, that may change from line to line.
We denote by I m , m ∈ {d, r} the identity matrix of dimension m. Through the article, for any smooth enough function f : R d → R, for k ∈ N we will denote D k f the tensor of the k th derivatives of f . Namely We will also use the notation [[n, p]], (n, p) ∈ (N 0 ) 2 , n ≤ p, for the set of integers being between n and p.
For k ∈ N 0 , β ∈ (0, 1] and m ∈ {1, d, d × r},C k,β (R d , R m ) and C k,β b (R d , R m ) stand for the standard Hölder spaces 1 . For any function ζ : For a given Borel function f : Moreover, for k ∈ N 0 , we denote Eventually, we define the infinitesimal generator associated with the diffusion (E) which writes for all ϕ ∈ C 2 (R d ) and x ∈ R d : where π stands for the distribution of Y 1 , and A is the infinitesimal generator of the continuous part of the diffusion.

Hypotheses
We assume the following set of hypothesis about the coefficients of the SDE (E) and the parameters of the scheme (S): (C1) The first value of the scheme X 0 is sub-Gaussian: there exists λ 0 ∈ R * + such that (C2) Defining for any x ∈ R d , Σ(x) := σσ * (x), K(x) = κκ * (x), we suppose that (GM) The sequences of random variables (U n ) n≥1 and (Y n ) n≥1 are respectively i.i.d., such that Also, (U n ) n≥1 , (Y n ) n≥1 and X 0 are independent.
(GC) We say that a random variable G ∈ L 1 satisfies the Gaussian concentration property, if for every Lipschitz continuous function g : R r → R and every λ > 0: We assume that U 1 and Y 1 satisfy this Gaussian concentration property.
(L V ) We assume that there exists a non-negative function V : ii) There is C V ∈ (0, +∞) such that for any x ∈ R d : iii) There are α V > 0, β V ∈ R + such that for any x ∈ R d , (U) There is a unique invariant distribution ν to equation (E).
(S) We suppose that the step sequence (γ k ) k≥1 is taken such that γ k k −θ , θ ∈ (0, 1]. For any ≥ 0, we define: We assume that the sequence (γ k ) k≥1 is small enough, namely for every k ≥ 1, where C V is given by the assumption (L V ).
A Gaussian concentration result is natural when a CLT is available (see [10]). When the innovation does not satisfy a Gaussian (or quasi Gaussian) concentration hypothesis, we cannot expect any non-asymptotic Gaussian (or quasi Gaussian) concentration result for the scheme.
The assumption (L V ) together with (C2) ensure, following [10] (Prop. 1) the existence of at least one invariant distribution associated with SDE (E). Note that this Lyapunov assumption (L V ) is equivalent to the similar Lyapunov assumption for the continuous part of the equation (E). Indeed, using second order Taylor expansion, the fact that π(·) = R r yπ(dy) = 0 and π(| · | 2 ) = R r |y| 2 π(dy) = r < ∞, we get that Hence the condition iii) of (L V ) is equivalent that the generator of the diffusion without jumps satisfies Moreover, it is classic to see that this assumption constrains the drift coefficient b to be under a linear map. Indeed, this is the consequence of the fact that the Lyapunov function V has to be lower than the square norm, i.e. there exist constants K,c > 0 such that for any |x| ≥ K, and hence using ii) of (L V ) |b(x)| ≤ √ C Vc |x|.
Remark 1.4. In the assumption (T β ), the condition iii) yields from (1.7) that |ϕ(x)| ≤ C V,ϕ (1 +c|x|) which is consistent with the Lipschitz continuity of ϕ. Whilst the condition ii) is a direct consequence if ϕ is the solution of the Poisson equation: , then the right side of the following identity: is Lipschitz continuous, and so the left side too.
From now on, we identify assumptions (C0), (C1), (C2), (GM), (GC), (L V ), (U), (S) and (T β ) for some β ∈ (0, 1] to (A). Except when explicitly indicated, we assume throughout the paper that assumption (A) is in force. Example: The typical illustration of a diffusion process satisfying the above hypotheses is the Ornstein-Uhlenbeck process with jumps. Precisely, for b(x) = −x, σ = κ = I r , a corresponding Lyapunov function is a second order polynomial function V (x) = 1 + |x| 2 . Then each constant appearing in (L V ) is explicitly known: The corner stone of our analysis is the fact that the R r -valued jump innovations (Z n ) n∈N inherit a quasi-Gaussian concentration property of (Y n ) n∈N : Proposition 1.5 (Quasi Gaussian concentration of the jumps innovation). Let g : R r → R be uniformly Lipschitz continuous function, ε ∈ (0, 1] and Then for any 0 < λ < ε 6[g]1ρ(r) the following inequality holds for every n ∈ N Remark 1.6. Let us point out that the concentration inequality is only valid for λ on a compact set hence the terminology "quasi". This constraint is due to the difficulty to approximate a compound Poisson process which has actually a sub-exponential tail (and not a sub-Gaussian one).
The proof of this proposition is given in Section 2.4.

Existing results
The convergence of the empirical measure ν n of an appropriate Euler scheme with decreasing steps was studied by Lamberton and Pagès [7] and Panloup [11] respectively in the cases of Brownian or Lévy driven diffusion.
The natural next issue concerns the rate of that convergence. In a Brownian diffusion framework, a Central Limit Theorem (CLT) was established by Lamberton and Pagès [7] [10], the author also provided the rate of convergence through a Central Limit Theorem (CLT) for the already mentioned general scheme: is a Lévy process with π as a Lévy measure n √ Γn = 0 then for any function ϕ ∈ C 3,1 (R d , R) we have the following results (with (L) denoting the weak convergence): In the Brownian diffusion context (κ = 0), under some confluence and non-degeneracy or regularity assumptions Honoré, Menozzi and Pagès [4] established suitable derivatives controls for the Poisson problem (e.g. Schauder estimates). With a compound Poisson process, we think that a similar analysis may work. It will be a future research. Let us mention [12] for some Schauder estimates for Poisson equation, with a potential, associated with a SDE purely driven by stable processes but with a constant drift.
In [4], the authors have established a non-asymptotic Gaussian concentration with κ = 0. Namely, they showed that for any ϕ satisfying the condition (T β ), β ∈ (0, 1], there are sequences (c n ) and (C n ) converging to 1, c n ≤ 1 ≤ C n , such that for all n ∈ N, a > 0 and γ k k −θ , θ ∈ ( 1 2+β , 1], Remark that, in [5], a non-asymptotic Gaussian concentration was established with the asymptotically best constants for a particular large deviation called "Gaussian deviations" therein. In other words, for a = o( √ Γ n ): . (1.14) In this present work, we aim to obtain Gaussian deviations bounds like (1.13) for the scheme (S). To do so, we will perform the so-called martingales increments method which was exploited successfully by Frikha and Menozzi [3]. It was also the backbone of the analysis in [4] and [5]. Here, we adapt their techniques for the stochastic differential equation (E) driven by the compound Poisson with jump sizes satisfying Gaussian concentration.
Our techniques do not provide the sharp constant as in [5], this restriction comes from the deteriorating constants (depending on the dimension r) in the quasi Gaussian concentration property of the jump innovation (Z n ) n≥1 stated in Proposition 1.5. In other words, we cannot expect from our techniques to restore sharpness thanks to a suitable annex Poisson equation as in [5].

Result of non-asymptotic quasi-Gaussian concentration
Our main result is stated below.
For every positive sequence (χ n ) n≥1 with lim n→∞ χ n = 0, there are two non-negative sequences (c n ) n≥1 and (C n ) n≥1 , with lim n C n = lim n c n = 1 such that for all n ∈ N, a > 0, satisfying a ≤ χ n √ Γn Γ (2) n , the following bound holds: The proof of Theorem 2.1 is given in Section 2.3.
We have unsurprisingly that σ 2 ϕ ≤ σ 2 ∞ , where σ 2 ϕ is the asymptotic variance of √ Γ n ν n (Aϕ) defined in (1.12). Moreover, the difficulty to adapt a Gaussian Concentration result to compound Poisson process yields that the upper-bound variance σ 2 ∞ depends on the dimension.
Similarly to [5], we have from the proof of Theorem 2.1 that The proof of this corollary is standard, we give it however for the completeness.

Strategy
For the analysis of ν n (Aϕ), we will first perform an appropriate Taylor expansion (Eq. (2.5)). An expansion of this kind is standard in this context, and analogous decompositions were already used in [4], [5] in continuous setting and [11], [10] with a jump component. It can be viewed as a kind of Itô formula for Euler scheme, because it permits to write the difference ϕ(X n ) − ϕ(X 0 ) as a sum of a martingale, a term involving the generator and a remainder term.
Let us define the contributions of the decomposition of ν n (Aϕ) in the following lemma.
Moreover, we define the remainder contributions in the decomposition of ν n (Aϕ).
Lemma 2.4 (Local decomposition of the empirical measure). For all ϕ ∈ C 2 (R d , R), k ∈ N the following decomposition holds: Furthermore, we have the following properties: ii) ∆ ϕ k (X k−1 , U k ) and ∆ ϕ k (X k−1 , Z k ) are martingale increments with respect to F k , namely: The proof of Lemma 2.4 is given in Section 4. Now we introduce the martingales associated to these martingale increments: Summing (2.5) over k we obtain the following global decomposition of the empirical measure: where we denoted Using the definition (2.4) we can write R ϕ n = −L ϕ n + D ϕ 2,b,n + D ϕ 2,Σ,n + D ϕ j,n +Ḡ ϕ n , with The proof of this result is postponed in Section 4. Now we formulate several propositions and lemmas that are used to control the components of the remainder term R ϕ n . The following proposition is the counterpart to the jumps diffusion of the useful Proposition 1 in [4]. Proposition 2.6. Under (A), there is a constant c V := c V ((A)) > 0 such that for any λ ∈ [0, c V ]: The analysis of the crucial Proposition 2.6 is in Section 3.
Remark 2.7. In particular, we easily see that for all λ ∈ [0, c V ] and ξ ∈ [0, 1 2 ]: Note that for κ = 0 (purely continuous case), the integrability of exp λV (X n ) ξ is available until ξ = 1 (see Proposition 1 in [4]). The loss of integrability is the consequence of the bound condition over λ in the Gaussian Concentration result of Proposition 1.5.
The initial term appearing in (2.5) is handled thanks to the result bellow.
V given in Proposition 2.6. The detailed computations to get this result are in Section 4. Next the last remainders are controlled as following: (2.13) (2.14) Inequalities (2.13) The proof of this statement is also in Section 4.
Remark 2.11. The strongest condition over θ comes from this remainder term. Indeed, for θ < 2 3+β , Γ → n 0 which is implied by θ > 1 3 . Now, let us deal with the remainder term D j,n relying on the jump vector (Z k ) k≥1 . Lemma 2.12 (Remainder term due to the jumps). If and ρ(r) defined in (1.9), then we have:

17)
where we recall that e n = e n ((A)), n ≥ 1, is a sequence such that e n → n 0.
The proof of this lemma being one of the most intricate of this article, we postpone it to the end of Section 4.
By Cauchy-Schwarz inequality, we obtain: We recall that all the long of our analysis, C > 0 denotes a generic constant, (R n ) n≥1 and (e n ) n≥1 are generic non-negative sequences, (R n ) n≥1 and (e n ) n≥1 are generic non-negative sequences, depending on the coefficients of the assumption (A), which may change from line to line, such that lim n→∞ R n = 1, lim n→∞ e n = 0. For the term associated with L ϕ n in (2.19), if 2pλ Γn < c V 2C V,ϕ , by Lemma 2.8 we can write: (2.20) From Lemma 2.10, with α n defined in (2.15) and by Young inequality we obtain: In the last equality, R n = exp(α 2 n p n /2) and e n = 1/2p n . Recall that α n → n 0. We choose p = p n → n +∞ such that p n α 2 n → n 0, to obtain R n → n 1. For the term involving D ϕ 2,Σ,n from ( , using Lemma 2.9, we can write (2.23) Finally, Lemma 2.12 yields that if then: Now, let us control the martingale terms thanks to Lemma 2.5. We take 0 < ε < 1, and λ > 0 s.t. qλ/Γ n < Cε 6 κ ∞ ∇ϕ ∞ρ(r) , we recall that ρ(r) is defined in (1.9). Thanks to Lemma 2.5 and the independence of Z n and U n conditionally to F n−1 we can write

(2.27)
We check first the assumptions over λ in all lemmas that we used in this proof. In (2.20) and (2.24) we need pλ n /Γ n < C. And for (2.22), (2.23) and (2.24) we need pλ n /Γ n < C Γ (2) n . Finally qλ n /Γ n < Cε n is required to apply Lemma 2.5. We recall that we choose p → ∞ and q → 1 and ε → 0.
We recall also that from the statement of the theorem a = a(n) (possibly depending on n) s.t. has to be satisfied. Let us calibrate p = p(n) → ∞ depending on χ n → 0 s.t. lim sup n pχ n < C. This pick of p yields (2.28). We can also choose, for C > 0 large enough ε n = C χn Γ (2) n such that all conditions over λ n , p, ε are satisfied with these choices.

1, and λn
√ Γn e n ≤ Cae n . If a ≤ 1, we take C n = R n exp(Cae n ) → n→+∞ 1, otherwise if a > 1 then we set c n = 1 − Cen a 2 → n→+∞ 1. In any case, we write the result:

Proof of the Gaussian property of the jump innovation
Proof of Proposition 1.5. Suppose first that g : R r → R is Lipschitz continuous with [g] 1 ≤ 1. The case of [g] 1 > 1 follows by consideringλ = λ[g] 1 andg = g [g]1 . We suppose w.l.o.g. that g(0) = 0. We recall that thank to the definition (1.1) the law of Z n is the same that the law of B n Y, where B n is a Bernoulli variable with parameter γ n , independent of the random vector Y with distribution π on B(R r ). We establish first that for all ε ∈ (0, 1] and 0 < λ < ε ρ(r) (see (1.9)) Denote for this proof m g := Eg(Y ). Using (GC) property of Y we can write Let us define ). (2.30) Here, the second exponential corresponds to the right-hand side in (2.29). We will show that ∆ exp n < 0. Indeed, let us develop the difference ∆ exp n by power series expansion: In particular, using µγ n ≤ 1 from (S), and ε ≤ 1, we can roughly estimate:
Remark 2.13. Note that r → ρ(r) is increasing, hence the condition we need to put on λ in order to propagate the Gaussian concentration from Y ∈ R r to Z became stronger if the dimension r increase.

Exponential integrability of the square root of Lyapunov function
Like in [4], we control the exponential moments of the Lyapunov function to handle the remainder terms of the decomposition of the empirical measure. But our framework yields more constraints over the analysis. Namely, we cannot directly use exp(CV n ) which is not a priori integrable here. Indeed, let us consider the Compound Poisson processZ t := Nt k=1Ỹ k where (Ỹ k ) k∈N is an i.i.d. sequence of a standard normal variables independent of N t which follows a Poisson law, which is the typical jump random variables that we aim to approximate. Conditionally to N t ,Z t ∼ N (0, N t ). So, if we choose the Lyapunov function to be the standard quadratic map, i.e. for any x ∈ R d , V (x) = |x| 2 + 1. We obtain in fine: this is integrable if almost surely N t < 1 2λ which is not true for λ > 0.
Proof of Proposition 2.6. Preliminarily to the proof of this proposition, we write some useful controls thanks to assumption (L V ), for any x ∈ R d , To begin we check that √ V satisfies assumption (L V ) iii). We have readily that: The first inequality is a consequence of Remark 1.3. Furthermore, for the purely jump part of the infinitesimal generator we write:

using (3.1). The previous inequality and (3.3) implies that:
Next, let us decompose the Lyapunov function √ V with a Taylor expansion similarly to Lemma 2.4. We again use a splitting between the deterministic contributions and those involving the innovation. We write for every n ∈ N: where for any x ∈ R d , the first term is such that: Because b is supposed to be Lipschitz continuous and thanks to (L V ) ii), we readily writes: Whilst the next term is more subtle. Indeed, observe that thanks to (L V ) ii) the following term is bounded: which directly yields again thanks to (L V ) ii) that . (3.9) Hence plugging (3.7) and (3.9) into (3.6) implies that: The second term is handled by (3.2): The third term satisfies the following identity: (3.12) and the last term is: . for which corresponds to assumption (S) and We control the contribution of V 1 3 (X n−1 , U n ) and V 2 3 (U n ) (defined in (3.12)) in the exponential moment of V (X n ) by the Gaussian concentration hypothesis (GC) and V 1 4 (X n−1 , U n , Z n ) (see (3.13)) thanks to Proposition 1.5. We define for all x ∈ R d and λ > 0: Indeed, by (GC), we first write: Next, it is well known that under (GC), for any c < 1 2 , I c := E [exp(c|U n | 2 )] < +∞. So we have for any , by Jensen's inequality: Now, let us deal with the third term I 3 (λ, x). First of all, note that from definition in (3.13) and (3.1) the function z → V 1 4 (X n−1 , U n , z) is κ ∞ √ C V -Lipschitz continuous. Furthermore, we have that Hence, by the Proposition 1.5, and for any 0 < λ < (1.9)), for the corresponding notation of Proposition 1.5 we take ε = 1, and we get: From now on, we assume that for all λ < λ V := min 1, Gathering identities (3.14), (3.15), (3.16), and by the Cauchy-Schwarz inequality, we obtain that for any λ < λ V , where we have defined: So (1 − γ nαV ) ∈ [0, 1) and we deduce by Jensen inequality: .

(3.18)
For any λ > 0, we introduce In particular, we have E[e λ √ V (X0) ] ≤ C V,λ . Let us check by induction that for every n ∈ N: We deduce from (3.18) and by induction assumption that: We pick c V < λ V and the proof is completed.

Proof of the technical lemmas
Thanks to this splitting, we are able to isolate the deterministic, the sub-Gaussian random variable approximating Brownian increments, and the jump contributions. Then we proceed by Taylor expansion up to the order 2 for the function ϕ in the two last terms of the r.h.s. of (4.1), We then write Note finally, that by definition of D k,ϕ j (X k−1 , U k , Z k ) in the previous expansion (4.2): hence after differentiating, we see that u → ψ ϕ k (X k−1 , u) and hence u → ∆ ϕ k (X k−1 , u) are Lipschitz continuous with a modulus bounded by Moreover, from the definition E[∆ ϕ k (X k−1 , U k )|F k−1 ] = 0 and using the definition of Z n we get Proof of Lemma 2.5. We first prove the point ii). For all ε ∈ (0, 1) and 0 < Λ Γn < ε 6[ ∆n(Xn−1,·)]1ρ(r) (ρ(r) set in (1.9)), thanks to Proposition 1.5, we have for every n ∈ N: By definition of ∆ ϕ n (X n−1 , Z n ) in (2.3), and from (4.3) we have: The previous control with (4.4) directly yield: with the constraint 0 < Λ Γn < ε 6 κ ∞ ∇ϕ ∞ ρ(r) . The proof of the point i), is a direct consequence of the previous analysis without using Proposition 1.5, which yields no restriction on λ.
Proof of Lemma 2.8. By assumption (T β ), we know that there exists C V,ϕ > 0 such that for any x ∈ R d , |ϕ(x)| ≤ C V,ϕ 1 + V (x) , so we obtain: The last inequality is obtained by Jensen's inequality for Λ Γn < c V 2C V,ϕ and by Proposition 2.6.
Proof of Lemma 2.9. From the definition (2.4) we can write: From the boundedness of ∇ϕ, Lipschitz property of the mapping x → b(x) (assumption (C 0 )) and Lipschitz property of the mapping x → ∇ϕ(x), b(x) (assumption (T β )), using the assumption (L V ), ii) one derives that: Next, by the Jensen inequality (for the exponential function with 1 Γ (2) n k=1 γ 2 k δ k as a measure), we deduce that: (2) n < 1, and c V is introduced in Proposition 2.6 which readily yields: For the second inequality, we first use a Taylor expansion: Hence, like previously, by Jensen inequality and Proposition 2.6 for Λ Γn ≤ Then, (1 + β)(2 + β)(3 + β) . (4.8) We sum over k to get the result.
In Figure 1, we plot the the curves of g n for θ = 1 3 + 10 −3 , also we perform the simulations for n = 5 × 10 4 , the probability is estimated by Monte Carlo simulation with M C = 10 4 realizations of the random variable | √ Γ n ν n (Aϕ)| in the unbiased case. Let us also introduce the function S ν (a) := − a 2 2 σ 2 ∞ ∇ϕ 2 ∞ + 4 κ 2 ∞ ∇ϕ 2 ∞ , such that g n (a) ≤ c n S ν (a) + log(2C n ). The Figure 1 enhance the fact that g n (a) is indeed under a quadratic form in a. Nevertheless, we see that the result of Theorem 2.1 is not sharp, to obtain such a result we have to avoid the dimension dependency and to provide a more accurate concentration property than the one of Proposition 1.5.