Gilles Pagès

Clément Rey

Recursive computation of the invariant distribution of Markov and Feller processes

Keywords: Ergodic theory, Markov processes, Invariant measures, Limit theorem, Stochastic approximation AMS MSC 2010: 60G10, 47A35, 60F05, 60J25, 60J35, 65C20

 and based on simulation algorithms of stochastic schemes with decreasing step can be used to build invariant measures for general Markov and Feller processes. We also propose applications in three dierent congurations: Approximation of Markov switching Brownian diusion ergodic regimes using Euler scheme, approximation of Markov Brownian diusion ergodic regimes with Milstein scheme and approximation of general diusions with jump components ergodic regimes.

Introduction

In this paper, we propose a method for the computation of invariant measures of Markov processes (denoted ν). In particular, we study a sequence of empirical stochastic measures (ν n ) n∈N * which can be recursively computed using a discrete process simulated with a sequence of vanishing step γ = (γ n ) n∈N and transition semigroups (Q γ n ) n∈N * . We show that lim n→∞ ν n f = νf a.s., for a class of test functions f . The recursive algorithm which is employed to build (ν n ) n∈N * considering that (Q γ n ) n∈N * is given, has been introduced in the seminal paper [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF].

Invariant mesasures are crucial in the study of the long term behavior of stochastic dierential systems. We invite the reader to refer to [START_REF]Stochastic stability of dierential equations[END_REF] and [START_REF] Ethier | Markov processes[END_REF] for an overview of the subject. The construction of invariant measure for stochastic systems has already been widely explored in the literature. In [START_REF] Soize | The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions[END_REF], the author provides a computation of the invariant distribution for some solutions of Stochastic Dierential Equations but in many cases there is no explicit formula for ν. A rst approach consists in studying the convergence of the semigroupoup of the Markov process (denoted (P t ) t 0 ) with innitesimal generator A towards the invariant measure ν as it is done in [START_REF] Ganidis | Convergence rate of some semi-groups to their invariant probability[END_REF] for the variation topology. If (P t ) t 0 can be computed, one can approximate ν controlling only the error between (P t ) t 0 and ν. If the process with semigroup (P t ) t 0 can be simulated, we can use a Monte Carlo method to estimate (P t ) t 0 producing a second term in the error analysis. When the process with semigroup (P t ) t 0 can not be simulated with reasonnable time, a solution consists in simulating an approximation for the semigroup (P t ) t 0 , using (Q γ n ) n∈N * (given a step sequence (γ n ) n∈N ). The semigroup (Q γ n ) n∈N * is supposed to weakly converge towards (P t ) t 0 . A natural construction rely on numerical homogeneous schemes ((γ n ) n∈N is constant and equal to the time step γ 0 ). This approach induced two more terms to control in the approximation e-mails : gilles.pages@upmc.fr, clement.rey@upmc.fr This research beneted from the support of the "Chaire Risques Financiers.

1 INTRODUCTION 2 of ν in addition to the error between (P t ) t 0 and ν: The rst one is due to the approximation of (P t ) t 0 by (Q γ n ) n∈N * and the second one is due to the Monte Carlo error involved in the computation of the law of the process simulated with (Q γ n ) n∈N * .

Nevertheless, for Brownian diusions, many eorts have been done in order simplify this problem. In [START_REF] Talay | Second-order discretization schemes of stochastic dierential systems for the computation of the invariant law[END_REF], the author suggests an elegant procedure to simplify this last approach. He considers the case where the process simulated with (Q γ n ) n∈N * (where (γ n ) n∈N is still constant) has an invariant measure ν γ0 . In a rst step, he shows that lim n→∞ ν n f = ν γ0 f , and then he proves that lim γ0→0 ν γ0 = ν. Consequently, he gets rid of the Monte Carlo approximation (since there is no estimation procedure for the computation of (P t ) t 0 or (Q γ n ) n∈N * ), and there are only two terms to treat in the error. He manages to control this error under a uniform ellipticity condition that is not necessary in our work. He also extended these results in [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF].

Another approach has been proposed in [START_REF] Basak | Weak convergence of recursions[END_REF] and avoid asymptotic analysis with respect to the size of the time step. In this paper, the authors prove directly that the random discrete process simulated with (Q γ n ) n∈N * , with (γ n ) n∈N vanishing to 0, converges weakly toward ν. Therefore, there are two terms to treat in the error: The rst one is due to this convergence and the second one to the Monte Carlo error involved in the computation of the law of the process simulated with (Q γ n ) n∈N * . The reader may notice that in all those cases, strong ergodicity assumptions are required for the process with innitesimal generator A.

Inspired among others by the ideas from [START_REF] Talay | Second-order discretization schemes of stochastic dierential systems for the computation of the invariant law[END_REF] and [START_REF] Basak | Weak convergence of recursions[END_REF], in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF], the authors designed a recursive algorithm with decreasing step and showed that the sequence (ν n ) n∈N * built with a discrete process that can be simulated using a sequence of vanishing step γ = (γ n ) n∈N and transition semigroups (Q γ n ) n∈N * directly converges towards ν. This initial paper treated the case where (Q γ n ) n∈N * is the transition semigroup of an inhomogeneous Euler scheme with decreasing step associated to a strongly mean reverting ergodic Brownian diusion process. In this paper, they introduce the recursive algorithm to build the sequence of random measures (ν n ) n∈N * given (Q γ n ) n∈N * (which is the procedure that is used in every work we mention from now and is also the one we use in this paper). Moreover, they prove that lim n→∞ ν n f = νf a.s. for a class of test functions which is larger than the domain (denoted D(A)) of A and contains test functions with polynomial growth. They also obtained rates and limit gaussian laws for the convergence of (ν n (f )) n∈N * for test functions f which can be written f = Aϕ. Finally they do not require that the invariant measure ν is unique controversially to the results obtained in [START_REF] Talay | Second-order discretization schemes of stochastic dierential systems for the computation of the invariant law[END_REF] and [START_REF] Basak | Weak convergence of recursions[END_REF] for instance. In the case where ν is an invariant distribution of a stochastic diusion, many complementary works to [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF] have been led. The authors extended their rst results in [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF], where they achieve convergence towards invariant measures for Euler scheme of Brownian diusions using weak mean reverting assumptions for the dynamical stochastic system. Thereafter, in its thesis [START_REF] Lemaire | Estimation récursive de la mesure invariante d'un processus de diusion[END_REF], the author extended the class of function for which we have lim n→∞ ν n f = νf a.s. from test functions f with polynomial growth to test functions with exponential growth. Finally, in [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic dierential equation driven by a lévy process[END_REF], the author generalized those results to the construction of invariant measures for Levy diusion processes still using the algorithm from [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF]. He thus opened the door to treat not only approximation of Brownian diusions' ergodic regime but also a larger class of processes.

The aim of this paper is to show that the algorithm presented in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF] enables to approximate invariant measures (when there exists without being necessarily unique) for general Markov and Feller processes.

We present a general framework adapted to the construction of invariant measures for Markov processes under general mean reverting assumption (which includes weak mean reverting assumptions). Then, we provide some applications for three dierent congurations always under weak mean reverting assumptions. The rst one treats the case of the Euler scheme for Markov Switching diusions for test functions with polynomial growth. This particular case has already been studied in [START_REF] Mei | Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic dierential equations with markov switching[END_REF] under strong ergodicity assumptions (which includes among others strong mean reverting assumption). Then, we prove convergence for the Milstein scheme for test functions with polynomial or exponential growth.

Finally, we consider the Euler scheme for general diusion processes with jump and test functions with polynomial growth. In particular, this last results involves Levy processes (as in [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic dierential equation driven by a lévy process[END_REF]) but also piecewise deterministic Markov processes or diusions processes with censored jump.

In a rst step we present some general useful results in this study. Then we present the general and abstract framework in order to obtain convergence toward an invariant measure of a general Markov and Feller processes. The end of this paper is devoted to the miscellaneous examples mentioned above. Theorem 2.1. (Echeverria Weiss). Let E be a locally compact and separable metric space and let A be a linear operator of C 0 (E) satisfying the positive maximum principle 1 and such that D(A) is dense in

C 0 (E). If ν ∈ P(E) satises ∀f ∈ D(A), E Af dν = 0, (1)
then there exists a stationary solution to the martingale problem (A, ν).

For our approach, one main advantage of this result is that the only property we will have to prove of obtain the existence of a stationary solution to the martingale problem (A, ν) is [START_REF] Basak | Weak convergence of recursions[END_REF]. Indeed, for Feller processes, we have the following useful result Proposition 2.1. Let A be the generator of a Feller semigroup. The space D(A) is dense in C 0 (E). Moreover, A satises the positive maximum principle.

The proof of this result can be found in [START_REF] Revuz | of Grundlehren der Mathematischen Wissenschaften[END_REF] (Chapter VII, Proposition 1.3 and Proposition 1.5) or [START_REF] Ethier | Markov processes[END_REF] (Chapter IV, Theorem 2.2). Consequently, this paper will be devoted to the construction of a measure ν, and then to the proof of (1) with this measure. Using the results mentioned in this section, property (1) is sucient to prove that ν is an invariant measure for the process with innitesimal generator A. To be more concrete, in this paper, the measure ν will be built as the limit of a sequence of random measure (ν n ) n∈N * that we specify in the sequel. When (1) holds for this limit, we say that the sequence (ν n ) n∈N * converges towards an invariant measure of the Feller process with generator A. In order to obtain [START_REF] Basak | Weak convergence of recursions[END_REF] for this measure, we will employ the following well known results. Theorem 2.2. (Chow). Let (M n ) n∈N * be a real valued martingale with respect to some ltration F = (F n ) n∈N . Then 3

Convergence to invariant distribution -A general approach

This section presents a general approach inspired from the seminal work in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF] to construct (ν n ) n∈N * and prove that it converges towards an invariant measure of a Markov and Feller with innitesimal generator

A as soon as it is built with a sequence of approximating semigroup of that Markov and Feller process.

Presentation of the framework

In this part, we present the recursive algorithm in order to build (ν n ) n∈N * and also the general hypothesis that are required to obtain convergence towards an invariant distribution of a Markov and Feller process.

In other words, we give some general assumptions on (ν n ) n∈N * in order to obtain (1) for lim n→∞ ν n .

Construction of the random measures

In this paper we consider a locally compact and separable metric space E. We introduce a sequence of nite transition measures P n (x, dy), n ∈ N * from E to itself. This means that for each xed x and n, P n (x, dy) is a probability measure on (E, B(E)) with the Borel σ-eld and, for each bounded measurable function f , the mapping

x → P n f (x) := E f (y)P n (x, dy)
is Borel measurable. We also suppose that P n f ∈ C 0 (E) for every measurable function f ∈ C 0 (E) and n ∈ N * . Now we associate the sequence P to a time grid with vanishing steps. Let γ :

= (γ n ) n∈N * such that ∀n ∈ N * , 0 γ n γ < ∞, lim n→∞ γ n = 0 and lim n→∞ Γ n = +∞ (2)
with the notations Γ 0 = 0 and Γ n = n k=1 γ k . From now, we will use the notation P γ n = P n .

Denition 3.1. We dene the family of discrete linear operator (P γ n ) n∈N * from C 0 (E) to itslef in the following way.

P γ 0 f (x) = f (x), P γ n+1 f (x) = P γ n P γ n+1 f (x) = P γ n R d f (y)P γ n+1 (x, dy).
Remark 3.1. If we dene more generally (P γ n,m ) n,m∈N;n m by

P γ n,n f (x) = f (x), ∀n, m ∈ N * , n m, P γ n,m+1 f (x) = P γ n,m P γ m+1 f (x),
we have the following semiroup property:

for n, m, k ∈ N, n m k, P γ n,k f = P γ n,m P γ m,k f .
We consider now a second sequence of nite transition probability measures Q γ n (x, dy), n ∈ N * . Moreover, we introduce the corresponding semigroup Q γ dened in a similar way as P γ with P γ replaced by Q γ . Finally, we assume that there exists a continuous Feller semigroup (P t ) t such that for every t ∈ {Γ n , n ∈ N}, then P Γn = P γ n . We do not make such assumption for (Q γ n ) n∈N . In our framework, Q γ is thus considered as the approximation discrete semigroup of (P t ) t on the time grid {Γ n , n ∈ N}. In the sequel we will denote by A the innitesimal generator of P and ν will denote an invariant measure for A. We now propose the construction of (ν n ) n∈N * which inspired from [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF]. We dene the Markov process X := (X n ) n∈N in the following way

X 0 ∈ E, P(X n+1 ∈ dy|X n ) = Q γ n+1 (X n , dy) (3)
The main dierence with [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF] is that we do not suppose that Q γ is the semigroup associated to the Euler scheme of a Brownian diusion process. In this study, we simply consider approximations of Markov processes that can be simulated. At this point, we are going to dene a weighted empirical measure with X. This construction is totally similar to the one in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF] but with the Euler scheme replaced by (X n ) n∈N . First, we introduce the weights. Let η :

= (η n ) n∈N * such that ∀n ∈ N * , η n 0, lim n→∞ H n = ∞, (4)
with the notation H n = H η,n = n k=1 η k . Now we present the algorithm initially introduce in [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF]. First, for x ∈ E, let δ x denote the Dirac mass at point x. For every n ∈ N * , we dene the random weighted empirical random measures as follows

ν η n (dx) = 1 H n n k=1 η k δ X k-1 (dx). (5)
The aim of this paper is to show that lim n→∞ ν η n f = νf a.s., for a class of test functions f . This will hold as soon as (ν n ) n∈N * is tense, (1) is satised with ν replaced by lim n→∞ ν n and f

(x) = o |x|→∞ g(x)
with sup n∈N * ν n (g) < ∞.

Assumptions on the random measures

In this section, we present the hypothesis that we require in order to prove that convergence. Those assumptions are related to the increments of the approximation semigroup Q γ , also called pseudo generator of Q γ . More particularly a rst assumption concerns the recursive control of this pseudo generator while the second describe its connection to the innitesimal generator A. We begin with some denitions.

Let us dene the family of linear operators

Ãγ := ( Ãγ n ) n∈N * from C 0 (E) to itslef, in the following way ∀f ∈ C 0 (E), x ∈ E, n ∈ N * , Ãγ n f = Q γ n f -f γ n . (6)
The reader may notice that Ãγ is also called the pseudo generator of the semigroup Q γ . In order to obtain our results, it is necessary to introduce some hypothesis concerning the stability of the semigroup Q γ . A key point in our approach, as it is the case for most studies concerning invariant distributions, is the existence of a Lyapunov function. We say that V is a Lyapunov function if

L V V : E → [v * , +∞), v * > 0 and lim |x|→∞ V (x) = ∞. (7)
A classical interest of Lyapunov functions is is to show the existence and sometimes uniqueness of the invariant measure for the process with innitesimal generator A. We invite the reader to refer to the large literature on the subject for more details: See for instance [START_REF]Stochastic stability of dierential equations[END_REF], [START_REF] Ethier | Markov processes[END_REF] or [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF].

Recursive control

In our framework, we introduce a well suited stability assumption for the pseudo generator in order to obtain existence (in weak sense) of the limit of of the sequence of random measures (ν n ) n∈N . This will be done using a tightness property. We now give this assumption that will be mentioned from now as the recursive control of the pseudo generator Ãγ :

Let v * > 0, V : E → [v * , ∞), ψ, φ ∈ [v * , ∞) → R + , such that Ãγ n ψ • V exists for every n ∈ N * .
Then we assume that there exists α > 0 and β ∈ R + , such that

I Q,V (ψ, φ) ∃n 0 ∈ N * , ∀n n 0 , x ∈ E, Ãγ n ψ • V (x) V -1 (x)ψ • V (x)(β -αφ • V (x)). ∃λ ∈ [0, 1), C λ 0, ∀y v * , ψ(y)(β -λαφ(y)) C λ y. ( 8
)
Let us notice that the second part of the assumption I Q,V (ψ, φ) is satised as soon as lim y→∞ φ(y) = ∞. The function φ controls the mean reverting property. In particular, we say that we have strong mean reverting property if φ = I d and that we have weak mean reverting property when φ(y) = y a , a ∈ (0, 1) for every y ∈ [v * , ∞). The function ψ is referred in this paper as the test function and is related to the set of functions f for which we have lim n→∞ ν η n (f ) = ν(f ), when ν is the unique invariant measure of the process with innitesimal generator A. This assumption is crucial to prove the tightness of the sequence (ν η n ) n∈N * and consequently to obtain the existence of a limit point (not necessarily unique) for this sequence.

Innitesimal approximation

This part presents the assumption that enable to show that any limit point of the sequence (ν η n ) n∈N * is an invariant measure for the Markov and Feller process with innitesimal generator A. We aim to estimate the distance between an invariant measure of P and ν η n (see ( 5)) for n large enough. In order to do it, we introduce an additional hypothesis concerning the distance between Ãγ , the pseudo generator of Q γ , and A, the innitesimal generator of P . We assume that

E( Ãγ , A) ∃n 0 ∈ N * , ∀n n 0 , ∀f ∈ D(A), ∀x ∈ E | Ãγ n f (x) -Af (x)| Λ f (x, γ n ), (9)
where Λ f : R d × R + → R + can be decomposed in the following way: Let q ∈ N. We introduce g = (g 1 , . . . , g q ) and Λf = ( Λf,1 , . . . , Λf,q ) two families of functions with g i : R d → R + and Λf,i :

E × R + × R Ni × G i → R + .
Moreover, for every i ∈ {1, . . . , q}, we introduce a positive nite measure π i dened on a measurable space (G i , G i ), and a family of random processes

(U i (x, Θ i )) Θi∈Gi taking values in R Ni , N i ∈ N * . We suppose that we have ∀x ∈ R d , ∀t ∈ [0, γ n0 ], Λ f (x, t) = q i=1 Gi E[ Λf,i (x, t, U i (x, Θ i ) t , Θ i )]π i (dΘ i )g i (x) with sup x∈E,t∈[0,γn 0 ],i∈{1,.,q} Gi E[ Λf,i (x, t, U i (x, Θ i ) t , Θ i )]π i (dΘ i ) < ∞.
Using this decomposition, we assume that for every couple of functions ( Λf,i , g i ), i ∈ {1, . . . , q}, one the following assumption holds, that is E loc ( Λf,i , g i ) or E ergo ( Λf,i , g i ).

I) Locally compact case We say that E loc ( Λf,i , g i ) holds if g i is locally compact and: for every

u ∈ R Ni , for every Θ i ∈ G i and every compact subset K of E we have lim t→0 sup x∈K Λf,i (x, t, u, Θ i ) = 0. (10)
Moreover, we assume that there exists t 0 > 0 and a compact subset

K 0 of E such that ∀x ∈ E \ K 0 , ∀t ∈ [0, t 0 ], ∀u ∈ R Ni , ∀Θ i ∈ G i , Λf,i (x, t, u, Θ i ) = 0. (11) II) Case sup n∈N * ν η n (g i ) < ∞ a.s. We say that E ergo ( Λf,i , g i ) holds if g i is locally compact, sup n∈N * ν η n (g i ) < ∞
and one of the following properties holds:

For every compact subset K of E and all Θ i ∈ G i , we have

lim t→0 sup x∈K Λf,i (x, t, U i (x, Θ i ) t , Θ i ) = 0 a.s., and (12)
lim |x|→∞ sup t∈[0,γn 0 ] Λf,i (x, t, U i (x, Θ i ) t , Θ i ) = 0 a.s.
or the following holds instead: For every

Θ i ∈ G i , lim t→0 sup x∈E Λf,i (x, t, U i (x, Θ i ) t , Θ i )g i (x) = 0 a.s, (13)
The reader may notice that the measures π i , i ∈ {1, . . . , q} are not supposed to be probability measures. However, in many cases, those measures are built using some probability measures. This representation assumption is related to the fact that the transition functions Q γ n (x, dy), x ∈ E can be represented using random variables (which does not depend from γ) through the variable Θ i and using random processes through (U i (x, Θ i ) t ) t 0 . This approach is well adapted to stochastic approximations that can be associated to a time grid such as numerical schemes for stochastic dierential equation with a Brownian part or/and a Jump part. This concludes the part concerning the assumption and we can focus on the main results concerning this abstract approach.

Convergence

Almost sure tightness

From the recursive control assumption, we obtain the tightness of (ν η n ) n∈N * . This is one of the purpose of the following Theorem. We recall that tightness implies that the sequence has at least one limit point.

Another interest of this result is that the functions φ and ψ are not specied explicitly and then this framework apply to many diverse congurations.

Theorem 3.1.

Let s ∈ (1, 2], v * > 0, V : E → [v * , ∞), ψ, φ : [v * , ∞) → R + .
We assume that ψ is lower bounded, that I Q,V (ψ, φ) (see [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF]) hold and that

P-a.s. sup n∈N * - 1 H n n k=1 η k Ãγ k (ψ • V ) 1/s (X k-1 ) < ∞. (14)
Then

P-a.s. sup n∈N * ν η n (V -1 .φ • V.(ψ • V ) 1/s ) < ∞. (15)
Finally, if L V (see [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF]) holds, and the function x → x -1 φ(x)ψ(x) 1/s tends to innity as x goes to innity, then the sequence (ν η n ) n∈N * is tight. Consequently, if the sequence (ν η n ) n∈N * has a unique weak limit ν then for every continuous function

f satisfying f = o(V -1 .φ • V.(ψ • V ) 1/s ), we have lim n→∞ ν η n (f ) = ν(f ).
Proof. Using [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF], there exists n 0 ∈ N such that for every n n 0 , we have

E ψ • V (X n+1 ) ψ • V (X n ) |F n 1 + γ n+1 β -αφ • V (X n ) V (X n ) .
Since the function dened on R * + by y → y 1/s is concave, we use the Jensen's inequality and obtain

E ψ • V (X n+1 ) 1/s ψ • V (X n ) 1/s |F n 1 + γ n+1 β -αφ • V (X n ) V (X n ) 1/s 1 + γ n+1 (β -αφ • V (X n )) sV (X n ) .
Now, we use [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF] and it follows that there exists λ ∈ [0, 1), C λ 0, such that

Q γ n+1 (ψ • V ) 1/s (X n ) (ψ • V ) 1/s (X n ) + γ n+1 s (ψ • V ) 1/s (X n )V -1 (X n )(β -αφ • V (X n )) (ψ • V ) 1/s (X n ) + γ n+1 ψ • V (X n ) 1/s-1 C λ s - α(1 -λ) s V -1 (X n )φ • V (X n ) .
or equivalently,

V -1 (X n )φ • V (X n )ψ • V (X n ) 1/s - s α(1 -λ) Ãγ n+1 (ψ • V ) 1/s (X n ) + C λ α(1 -λ) ( inf x v * |ψ(x)|) 1/s-1 .
Consequently, the result follows from ( 14), E[ψ • V (X n0 )] < ∞, and the fact that ψ is lower bounded on [v * , ∞).

Identication of the limit

In Theorem 3.1, we obtained tightness of (ν η n ) n∈N * . It remains to show that any limit point of this sequence is an invariant measure for the process with innitesimal generator A. This is the interest of the following Theorem which uses the innitesimal approximation. Theorem 3.2. Let n 0 ∈ N * . We assume that for every f ∈ D(A), we have

P-a.s. lim n→∞ ν η n ( Ãγ k f ) = 0. (16)
We also assume that E( Ãγ , A) (see ( 9)), holds. Then

P-a.s. lim n→∞ ν η n (Af ) = 0 (17)
It follows that, P-a.s., every (weak) limiting distribution ν η ∞ of the sequence (ν η n ) n∈N * is an invariant distribution for the semigroup (P t ) t 0 with innitesimal generator A.

A direct consequence of this result is that if (ν η n ) n∈N * is almost surely tight and the semigroup (P t ) t 0 with innitesimal generator A admits a unique invariant measure ν, then almost surely (ν η n ) n∈N * converges to ν.

Proof. First we write

ν η n ( Ãγ k f ) -ν η n (Af ) = 1 H n n k=1 η k Ãγ k f (X k-1 ) -η k Af (X k-1
). Now we use the short time approximation E( Ãγ , A) (see [START_REF] Lemaire | Estimation récursive de la mesure invariante d'un processus de diusion[END_REF]) and it follows that there exists n 0 ∈ N * such that

1 H n n k=n0 η k Ãγ k f (X k-1 ) -η k Af (X k-1 ) = C H n n k=n0 η k Λ f (X k-1 , γ k ).
Moreover, we have the following decomposition

∀x ∈ R d , ∀t ∈ [0, γ n0 ], Λ f (x, t) = q i=1 Gi E[ Λf,i (x, γ, U i (x, Θ i ) t , Θ i )]π i (dΘ i )g i (x)
with for every

Θ i ∈ G i , U i (x, Θ i ) = (U i (x, Θ i ) t ) t 0 a R N -valued random process and sup x∈E,t∈[0,γn 0 ],i∈{1,.,q} Gi E[ Λf,i (x, t, U i (x, Θ i ) t , Θ i )]π i (dΘ i ) < ∞.
Now we assume that E ergo ( Λf,i , g i ) (see [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF] and ( 13)) holds for Λ f,i . If E loc ( Λf,i , g i ) (see [START_REF] Mei | Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic dierential equations with markov switching[END_REF] and ( 11)) holds instead of E ergo ( Λf,i , g i ), the proof is similar by simpler so we leave it out. In order to obtain the desired convergence, we rst x Θ i ∈ G i and study

1 H n n k=n0 η k Λf,i (X k-1 , γ k , U i (X k-1 , Θ i ) γ k , Θ i )g i (X k-1 ).
We assume that (12) holds. If instead (13) is satised, the proof is similar but simper so we leave it to the reader. For R > 0, we denote B R = {x ∈ E, |x| R}. Using E ergo ( Λf,i , g i ) (see ( 12)), we have immediately lim n→∞ Λf,i (X n-1 , γ n , U i (X n-1 , Θ i ) γn , Θ)1 |Xn-1| R = 0 a.s. Then, since g i is a continuous function, as an immediate consequence of the Cesaro's lemma, we obtain

lim n→∞ 1 H n n k=n0 η k Λf,i (X k-1 , γ k , U i (X k-1 , Θ i ) γ k , Θ)g i (X k-1 )1 {|X k-1 | R} = 0 a.s.
Moreover, using [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF], for every n n 0 , we have lim |x|→∞ Λf,i (x, γ n , U i (x, Θ i ) γn , Θ i ) = 0 a.s.. Then, almsot surely, we obtain

1 H n n k=n0 η k Λf,i (X k-1 , γ k , U i (X k-1 , Θ i ) γ k , Θ i )g(X k-1 )1 {|X k-1 |>R} sup |x|>R,t∈[0,γn 0 ] | Λf,i (x, t, U i (x, Θ i ) t , Θ i )| sup n∈N * ν η n (g i ).
We let R tends to innity and since sup n∈N * ν η n (g i ) < ∞, the left hand side of the above equation converges almost surely to 0. It remains to obtain the hypothesis of the Dominated Convergence Theorem.

We have, for every n ∈ N * , n n 0 ,

E[ Gi 1 H n n k=n0 η k Λf,i (X k-1 ,γ k , U i (x, Θ i ) γ k , Θ i )g i (X k-1 ) π i (dΘ i )] sup x∈E,t∈[0,γn 0 ] E[ Gi | Λf,i (x, t, U i (x, Θ i ) t , Θ i )|π i (dΘ)] sup n∈N * ν η n (g i ). with sup x∈E,t∈[0,t0] E[ Gi | Λf,i (x, t, U i (x, Θ i ) t , Θ i )|π i (dΘ i )] < ∞. Consequently, it follows from the Domi- nated Convergence Theorem that lim n→∞ 1 H n n k=1 η k Ãγ k f (X k-1 ) -ν η n Af = 0.
It follows that almost surely lim n→∞ ν η n Af = 0 for every f ∈ D(A). Since Af beglongs to C 0 (E), we obtain that ν η ∞ (Af ) = 0 a.s. for any limit point ν η ∞ of the sequence ν η n and the conclusion follows from the Echeverria Weiss theorem (see Theorem 2.1).

3.2.3 General approach to prove ( 14) and ( 16)

Through Theorem 3.1 and Theorem 3.2, we showed how to obtain existence of limit points for (ν η n ) n∈N * and how to identify these limit points to invariant measures for A. This simply requires a recursive control assumption, an innitesimal approximation hypothesis and also [START_REF] Rabiet | A stochastic equation with censored jumps related to multi-scale Piecewise Deterministic Markov Processes[END_REF] and ( 16). This section is dedicated to the study of ( 14) and ( 16). This assumptions are not trivial to verify in most cases but a general approach can be used. We present it from here.

Let ρ ∈ (1, 2] and an increasing function I : R + → R + . For f, g : E → R and (X n ) n∈N a sequence of random variables with X n ∈ σ(X i , i ∈ {0, . . . , n}) for every n ∈ N, we assume that there exists n 0 ∈ N, and C > 0 such that, for every n n 0 ,

ĨX (f, g, X, ρ, I ) E[|f (X n+1 ) -X n | ρ |X n ] C I (γ n+1 )|g(X n )|. ( 18
)
and

SW I,γ,η (g, ρ, I ) ∞ n=1 η n H n γ n ρ I (γ n )E[|g(X n )|] < ∞. (19)
and we will also use the notation SW I,γ,η (g, ρ, I ) We will also use the hypothesis

SW II,γ,η (f ) ∞ n=0 (η n+1 /γ n+1 -η n /γ n ) + H n+1 E[|f (X n )|] < ∞. ( 20
)
with the convention η 0 /γ 0 = 1. One notices that this last assumption holds as soon as the sequence (η n /γ n ) n∈N * is non increasing We propose a rst result which enlightens the interest of these hypothesis and will be of particular interest when f = (ψ • V ) 1/s in the study of the tightness and when f ∈ D(A)

for the identication part.

Lemma 3.1. Let ρ ∈ (1, 2], g : E → R + , f : E → R, such that Ãγ n f exists for every n ∈ N * , I : R + → R + an increasing function and (X n ) n∈N a sequence of random variables with X n ∈ σ(X i , i ∈ {0, . . . , n}) for every n ∈ N We assume I X (f, g, X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) and SW I,γ,η (g, ρ, I ) (see (19)) hold. We have the following properties

A. If f : E → R + and SW II,γ,η (f ) (see (20)) holds, then P-a.s. sup n∈N * - 1 H n n k=1 η k Ãγ k f (X k-1 ) < ∞. (21)
B. If f is bounded and

lim n→∞ 1 H n n k=1 |η k+1 /γ k+1 -η k /γ k | = 0, (22)
Then P-a.s.

lim n→∞ 1 H n n k=1 η k Ãγ k f (X k-1 ) = 0 (23)
Proof. We write

- n k=1 η k Ãγ k f (X k-1 ) = - n k=1 η k γ k (f (X k ) -f (X k-1 )) + n k=1 η k γ k (f (X k ) -Q γ k f (X k-1 ))
We study the rst term of the right hand side.First we write

- 1 H n n k=1 η k γ k (f (X k ) -f (X k-1 )) = η 1 H n γ 1 f (X 0 ) - η n H n γ n f (X n ) + 1 H n n k=2 η k γ k - η k-1 γ k-1 f (X k-1 )
.

First, we assume that f :

E → R + and SW II,γ,η (f ) (see (20)) holds. From SW II,γ,η (f ) (see (20))
together with Kronecker's lemma, we obtain

lim n→∞ 1 H n n k=2 η k γ k - η k-1 γ k-1 + E[f (X k-1 )] = 0,
and since f is positive, we deduce that

sup n∈N - 1 H n n k=1 η k γ k (f (X k ) -f (X k-1 )) < ∞ a.s. Now, when f is bounded, we deduce from (22), that lim n→∞ η n /(H n γ n ) = 0 and lim n→∞ 1 H n n k=1 η k γ k (f (X k ) -f (X k-1 )) = 0 a.s.
. This concludes the study of the rst term and now we focus on the second one. From Kronecker lemma, it remains to prove the almost sure convergence towards zero of the martingale (M n ) n∈N * such that M 0 := 0 and for every n ∈ N * ,

M n := n k=1 η k γ k H k (f (X k ) -Q γ k f (X k-1 )).
From the Chow's theorem, this convergence will be a consequence of the niteness of the series

n k=1 η k γ k H k ρ E[|f (X k ) -Q γ k f (X k-1 )| ρ ]. Moreover E[|f (X k ) -Q γ k f (X k-1 )| ρ |X k-1 ] 1/ρ E[|f (X k ) -X k-1 | ρ |X k-1 ] 1/ρ +E[|X k-1 -Q γ k f (X k-1 )| ρ |X k-1 ] 1/ρ with E[|X k-1 -Q γ k f (X k-1 )| ρ |X k-1 ] E[|E[X k-1 -f (X k )|F k-1 ]| ρ |X k-1 ] E[|f (X k ) -X k-1 | ρ |X k-1 ]
We conclude using I X (f, g, X, s, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) with SW I,γ,η (g, s, I ) (see ( 19)).

The following Lemma presents a L 1 -niteness property that we can obtain under recursive control hypothesis and strong mean reverting assumption (φ = I d ). This result is thus useful to prove SW I,γ,η (g, ρ, I ) (see ( 19)) or SW II,γ,η (f ) (see (20)) for well chosen functions f and g in this particular situation.

Lemma 3.2. Let

v * > 0, V : E → [v * , ∞), ψ : [v * , ∞) → R + , such that Ãγ n ψ • V exists for every n ∈ N. We assume that I Q,V (ψ, I d ) (see (19)) hold and that E[ψ • V (X n0 )] < ∞ for every n 0 ∈ N * . Then sup n∈N E[ψ • V (X n )] < ∞ (24)
Proof. First, we deduce from (8) that there exists n 0 ∈ N such that for n n 0

, I Q,V (ψ, I d ) can be rewritten E[ψ • V (X n+1 )|X n ] ψ • V (X n )(1 -γ n+1 αV -1 (X n )V (X n )) + γ n+1 βψ • V (X n )V -1 (X n ) ψ • V (X n )(1 -γ n+1 (1 -λ)α) + γ n+1 C λ .
Applying a simple induction we deduce that

E[ψ • V (X n )] E[ψ • V (X n0 )] ∨ C λ .
(1-λ)α .

The same results holds if with assume only that 19)). Now, we provide a general way to obtain SW I,γ,η (g, ρ, I ) and SW II,γ,η (f ) for some specic g and f as soon as a recursive control hypothesis hold but without making strong mean reversion assumptions.

E[ψ • V (X n0 )] < ∞ for the same n 0 as in I Q,V (ψ, I d ) (see (
Lemma 3.3. Let v * > 0, V : E → [v * , ∞), ψ, φ : [v * , ∞) → R + , such that Ãγ n ψ • V exists for every n ∈ N.
We also introduce the non increasing sequence

(Θ n ) n∈N * such that n 1 Θ n γ n < ∞. We assume that I Q,V (ψ, φ) (see (8)) hold. Then ∞ n=1 Θ n γ n E[V -1 (X n-1 )φ • V (X n-1 )ψ • V (X n-1 )] < ∞ (25)
In particular, let ρ ∈ (1, 2] and an increasing function I : R + → R + . If, we also assume

SW I,γ,η (ρ, I ) γ -1 n I (γ n ) η n H n γ n ρ n∈N * is non increasing and ∞ n=1 η n H n γ n ρ I (γ n ) < ∞, (26)
then we have

SW I,γ,η (V -1 φ • V ψ • V, ρ, I ) (see (19)). Moreover, if SW II,γ,η γ -1 n (η n+1 /γ n+1 -η n /γ n ) + H n n∈N * is non increasing and ∞ n=1 (η n+1 /γ n+1 -η n /γ n ) + H n < ∞, (27)
then we have

SW II,γ,η (V -1 φ • V ψ • V ) (see (20)).
Proof. There exists n 0 ∈ N such that for n n 0 , I Q,V (ψ, φ) can be rewritten

γ n+1 V -1 (X n )ψ • V (X n )(φ • V (X n ) -β/α) ψ • V (X n ) -E[ψ • V (X n+1 )|X n ] α .
Using [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF], and since the sequence (Θ n ) n∈N * is non increasing, we obtain a telescopic decomposition as follows

Θ n+1 γ n+1 V -1 (X n )ψ • V (X n )φ • V (X n ) Θ n+1 ψ • V (X n ) -E[ψ • V (X n+1 )|X n ] α(1 -λ) + γ n+1 Θ n+1 C λ /α Θ n ψ • V (X n ) -Θ n+1 E[ψ • V (X n+1 )|X n ] α(1 -λ) + γ n+1 Θ n+1 C λ α(1 -λ) .

Applications

In this section, we apply the general approach presented above to practical cases. Before doing it, we give some standard notations and properties that will be used extensively in the sequel. First, for α ∈ (0, 1] and f a α-Hölder function we denote

[f ] α = sup x =y |f (y) -f (x)|/|y -x| α . Now, let d ∈ N.
For any R d×d -valued symmetric matrix S, we dene λ S := sup{λ S,1 , ..λ S,d , 0}, with λ S,i the i-th eigenvalue of S.

We follow with some useful polynomial inequalities. Let u, v ∈ R + , then ∀α ∈ (0, 1),

(u + v) α u α + v α . ( 28
) ∀α 1, (u + v) α u α + α2 α-1 (u α-1 v + v α ). (29)
Let l ∈ N * . We have also

∀α > 0, u i ∈ R d , i = 1, . . . , l, l i=1 u i α l (α-1)+ l i=1 |u i | α . ( 30
)
We also recall the Burkhoder Davies Gundy (BDG) inequality for discrete martingales. Let p 1 and ( Mn ) n∈N a R d -valued martingale and dene F n M = σ( Mk , k ∈ {0, . . . , n}). Then, there exists C p 0

such that E[| Mn | p ] C p n-1 k=0 E[| Mk+1 -Mk | 2 |F k M ] p/2 (31)
In the following, we propose some applications for three dierent congurations always under weak mean reverting assumptions. The rst one treats the case of the Euler scheme for Markov Switching diusions for test functions with polynomial growth. The second, we prove convergence for the Milstein scheme for test functions with polynomial or exponential growth. Finally, we consider the Euler scheme for general diusion processes with jump and test functions with polynomial growth. For each of the three applications, we give the proof of the recursive control assumption, of the innitesimal approximation hypothesis and also of ( 14) and [START_REF] Soize | The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions[END_REF]. We invite the reader to refer to the previous section to see how this assumptions interact together in order to obtain convergence and to identify the limit. The reader may notice that each of these three applications are treated independently from one another and then can be read in any desired order.

The Euler scheme for a Markov Switching diusion

In this part of the paper we study ergodic regimes for Markov switching Brownian diusions. This study is a complement to the study made in [START_REF] Mei | Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic dierential equations with markov switching[END_REF]. More particularly they treat the convergence (ν η n ) n∈N * under strong mean reverting assumption that is φ = I d . In this paper, we do not restrict to that case and consider weak mean reverting assumption that is φ(y) = y a , a ∈ (0, 1] for every y ∈ [v * , ∞). Similarly as in their study we consider polynomial test functions ψ such that ψ(y) = y p , p 1 for every y ∈ [v * , ∞). Nevertheless, a slight dierence with this paper is that they consider only p 4. Now, we present the Markov switching model, its decreasing step Euler approximation and the hypothesis necessary to obtain the convergence of (ν η n ) n∈N * built with this Euler scheme. We consider a d-dimension Brownian motion (W t ) t 0 and (ζ t ) t 0 a continuous time Markov chain taking values in the nite state space {1, .., M 0 }, M 0 ∈ N * with generator Q = (q z,w ) z,w∈{1,..,M0} and independent from W .

We are interested in the solution of the d dimensional stochastic equation

X t = x + t 0 b(X s , ζ s )ds + t 0 σ(X s , ζ s )dW s (32)
where for every z ∈ {1, .., M 0 }, b(., z) : R d → R d and σ(., z) → R d×d , l ∈ {1, . . . d}, are locally bounded and continuous functions. We recall that q z,w 0 for z = w and q z,1 + q(z, 1) for z, w ∈ {1, .., M 0 }. The innitesimal generator of this process is given by

Af (x, z) = b(x, z), ∇ x f (x, z) + d i,j=1 (σσ * ) i,j (x, z) ∂ 2 f ∂x i ∂x j (x, z) + M0 w=1 q z,w f (x, w) (33)
for every (x, z) ∈ R d × {1, .., M 0 }. We study the Euler scheme for this process such that for every n ∈ N and t ∈ [Γ n , Γ n+1 ], we have

X t =X Γn + (t -Γ n )b(X Γn , ζ t ) + σ(X Γn , ζ t )(W t -W Γn ) (34)
We will also denote ∆X n+1 = X Γn+1 -X Γn and

∆X 1 n+1 =γ n+1 b(X Γn , ζ Γn ), ∆X 2 n+1 =σ(X Γn , ζ Γn )(W Γn+1 -W Γn ). (35) and X i Γn+1 = X Γn + i j=1 ∆X i n+1 .
In the sequel we will use the notation

U n+1 = γ -1/2 n+1 (W Γn+1 -W Γn ).
Actually, we introduce a weaker assumption than Gaussian distribution for the sequence (U n ) n∈N * . Let q ∈ N * , p 0. We suppose that (U n ) n∈N * is a sequence of independent random variables such that

M N ,q (U ) ∀n ∈ N * , ∀q ∈ {1, . . . , q}, E[(U n ) ⊗q ] = E[(N (0, I d )) ⊗q ] (36) M p (U ) sup n∈N * E[|U n | p ] < ∞ (37)
Now, we assume that the Lyapunov function V :

R d × {1, .., M 0 } → [v * , ∞), v * > 0, satises L V (see (7)) with E = R d × {1, .., M 0 }, and |∇ x V | 2 C V V, sup x∈R d ,z∈{1,..,M0} |D 2 x V (x, z)| < ∞ (38) and ∀z ∈ {1, .., M 0 }, ∃c V,z 0, ∀x ∈ R d , V (x, z) c V,z inf w∈{1,..,M0} V (x, w) (39)
We also dene

∀x ∈ R d , z ∈ {1, .., M 0 }, λ ψ (x, z) := 1 2 λ D 2 x V (x,z)+∇xV (x,z) ⊗2 ψ •V (x,z)ψ •V (x) -1 . (40)
When ψ(y) = y p , we will also use the notation λ p instead of λ ψ . We suppose that there exists C > 0 such that, for every

x ∈ R d , z ∈ {1, .., M 0 }, B(φ) |b(x, z)| 2 + |σσ * (x, z)| Cφ • V (x, z) (41)
We now introduce the key hypothesis in order to obtain recursive control for the polynomial case, that is for p 1, we have ψ(y) = y p for every y ∈ [v * , ∞). We assume that there exists β ∈ R + , α > 0 and > 0, such that for every x ∈ R d , z ∈ {1, .., M 0 }, we have

R p ∇V (x, z), b(x, z) + χ p (x, z) β -αφ • V (x, z), (42)
with

χ p (x, z) = λ p ∞ 2 (2p-3)+ Tr[σσ * (x, z)] + V 1-p (x, z) M0 w=1 (q z,w + )V p (x, w) (43)
ψ : [v * , ∞) → R + such that ψ(y) = y p .
We assume that the sequence (U n ) n∈N * satises M N ,2 (U ) (see ( 36)) and M 2p (U ) (see (37)). We suppose that (38), B(φ) (see (41)), R p (see (42)), are satised. Then, there exists α > 0, β ∈ R + and n 0 ∈ N * , such that

∀n n 0 , x ∈ R d , ∀z ∈ {1, .., M 0 }, Ãγ n ψ • V (x, z) V -1 (x, z)ψ • V (x, z)(β -αφ • V (x)). (44)
Moreover, when φ = Id we have

sup n∈N E[ψ • V (X Γn , ζ Γn )] < ∞. (45)
Proof. First we write

V p (X Γn+1 , ζ Γn+1 ) -V p (X Γn , ζ Γn ) =V p (X Γn+1 , ζ Γn ) -V p (X Γn , ζ Γn ) +V p (X Γn+1 , ζ Γn+1 ) -V p (X Γn+1 , ζ Γn )
We study the rst term. From the Taylor's formula and the denition of λ ψ (see ( 40)), we have

ψ • V (X Γn+1 , ζ Γn ) =ψ • V (X Γn , ζ Γn ) + X Γn+1 -X Γn , ∇ x V (X Γn , ζ Γn ) ψ • V (X Γn , ζ Γn ) + 1 2 (D 2 V (ξ n+1 , ζ Γn )ψ • V (ξ n+1 , ζ Γn ) + ∇V (ξ n+1 , ζ Γn ) 2 ψ • V (ξ n+1 , ζ Γn ))(X Γn+1 -X Γn ) ⊗2 . ψ • V (X Γn , ζ Γn ) + X Γn+1 -X Γn , ∇ x V (X Γn , ζ Γn ) ψ • V (X Γn , ζ Γn ) +λ ψ (ξ n+1 )|ψ • V (ξ n+1 , ζ Γn )(X Γn+1 -X Γn )| 2 . with ξ n+1 ∈ (X Γn , X Γn+1 ). First, from (38), we have sup x∈R d λ p (x) < ∞. Now, since (U n ) n∈N * is a sequence of independent random variables satisfying M N ,1 (U ) (see (36)), we have E[X Γn+1 -X Γn |X Γn , , ζ Γn ] = γ n+1 b(X Γn , ζ Γn ) E[|X Γn+1 -X Γn | 2 |X Γn , , ζ Γn ] = γ n+1 Tr[σσ * (X Γn , ζ Γn )] + γ 2 n+1 |b(X Γn , ζ Γn )| 2
Assume rst that p = 1. Using (41), for every α ∈ (0, α), there exists n 0 ( α) such that for every n n 0 ( α),

λ 1 ∞ γ 2 n+1 |b(X Γn , ζ Γn )| 2 γ n+1 (α -α)φ • V (X Γn , , ζ Γn ). (46)
From assumption (43), we conclude that

γ -1 n+1 E[V (X Γn+1 , ζ Γn )-V (X Γn , ζ Γn )|X Γn ] + 2 z=1 (q ζΓ n ,w + )V (X Γn , z) β -αφ • V (X Γn , ζ Γn ) Assume now that p > 1.Since |∇V | C V V (see (38)), then √ V is Lipschitz. Using (30), it follows that V p-1 (ξ n+1 , ζ Γn ) √ V (X Γn , ζ Γn ) + [ √ V ] 1 |X Γn+1 -X Γn | 2p-2 2 (2p-3)+ (V p-1 (X Γn , ζ Γn ) + [ √ V ] 2p-2 1 |X Γn+1 -X Γn | 2p-2 )
We focus on the study of the second term of the remainder. First, using B(φ) (see (41)), for any p 1,

|X Γn+1 -X Γn | 2p c p γ p n+1 φ • V (X Γn , ζ Γn ) p (1 + |U n+1 | 2p ).
Let α ∈ (0, α). Therefore, we deduve from M 2p (U ) (see (37)) that there exists n 0 ( α) ∈ N such that for any n n 0 (α), we have

E[|X Γn+1 -X Γn | 2p |X Γn , ζ Γn ] γ n+1 φ • V (X Γn , ζ Γn ) p α -α φ/I d p-1 ∞ λ p ∞ 2 (2p-3)+ [ √ V ] 2p-2
To treat the other term we proceed as in (69

) with λ 1 ∞ replaced by λ p ∞ 2 2p-3 [ √ V ] 2p-2 1
, α replace by α and α ∈ (0, α). We gather all the terms together and using R p (see ( 42) and ( 43)), for every n n 0 (α) ∨ n 0 ( α), we obtain

E[V p (X Γn+1 , ζ Γn )-V p (X Γn , ζ Γn )|X Γn , ζ Γn ] + V 1-p (x, z) M0 w=1 (q z,w + )V p (x, w) γ n+1 pV p-1 (X Γn , ζ Γn )(β -αφ • V (X Γn , ζ Γn )) +γ n+1 pV p-1 (X Γn , ζ Γn ) φ • V (X Γn , ζ Γn )(α -α) + (α -α) V 1-p (X Γn , ζ Γn )φ • V (X Γn , ζ Γn ) p φ/I d p-1 ∞ γ n+1 V p-1 (X Γn , ζ Γn )(βp -αpφ • V (X Γn , ζ Γn )).
Now, we focus on the second term. First, since ζ and W are independent, it follows that

E[V p (X Γn+1 , ζ Γn+1 ) -V p (X Γn+1 , ζ Γn )|X Γn , ζ Γn , ∆X n+1 ] = γ n+1 2 z=1 (q ζΓ n ,z + o n→∞ (γ n+1 ))V p (X Γn+1 , z)
Now, we use the same reasoning as in the study of the rst term and for every z ∈ {1, .., M 0 }, we obtain

E[V p (X Γn+1 , z) -V p (X Γn , z)|X Γn , ζ Γn ] C(γ 1/2 n+1 V p-1 (X Γn , z)φ • V (X Γn , z) + γ p n+1 φ • V (X Γn , z) p ) Cγ 1/2 n+1 V p (X Γn , z)
where C > 0 is a constant which can change from line to line. It follows that there exists ε :

R + → R + satisfying lim t→0 ε(t) = 0, such that we have E[V p (X Γn+1 , ζ Γn+1 ) -V p (X Γn+1 , ζ Γn )|X Γn , ζ Γn ] =γ n+1 M0 z=1 (q ζΓ n ,z + o n→∞ (γ n+1 ))E[V p (X Γn+1 , z)|X Γn , ζ Γn ] γ n+1 M0 z=1 (q ζΓ n ,z + ε(γ n+1 ))V p (X Γn , z)
and ( 44) is a direct consequence of R p (see (42) and ( 43)). The proof of (45) is immediate application of Lemma 3.2 as soon as we notice that the increments of the Euler scheme (for Markov Switching diusions) have nite polynomial moments which imples (24).

Innitesimal control

Proposition 4.2. We suppose that the sequences

(U n ) n∈N * satisess M N ,2 (U ) (see (36)), M 2 (U ) (see ( 37
)). We also assume that b and σ are locally bounded functions, that φ has sublinear growth, that B(φ) (see (41)) holds and that sup n∈N * ν η n (|σ| 2 ) < ∞. Then, we have E( Ãγ , A) (see ( 9)). Proof. First we recall that D

(A) = C 2 K (R d ) and we write f (X Γn+1 , ζ Γn+1 ) -f (X Γn , ζ Γn ) =f (X Γn+1 , ζ Γn ) -f (X Γn , ζ Γn ) +f (X Γn+1 , ζ Γn+1 ) -f (X Γn+1 , ζ Γn ).
Since W and ζ are independent, we have

E[f (X Γn+1 , ζ Γn+1 ) -f (X Γn+1 , ζ Γn )|X Γn , ζ Γn , ∆X n+1 ] =γ n+1 M0 z=1 (q ζΓ n ,z + o n→∞ (γ n+1 ))f (X Γn+1 , z)
Using Taylor expansions of order one and two, for every z ∈ {1, .., M 0 } and the fact that U n+1 is centered, we obtain

E[f (X Γn+1 , z)-f (X Γn , z)|X Γn = x, ζ Γn ] =E[f (X Γn + ∆X 1 n+1 , z) -f (X Γn , z)|X Γn = x, ζ Γn ] +E[f (X Γn+1 , z) -f (X Γn + ∆X 1 n+1 , z)|X Γn = x, ζ Γn ] 1 0 |∇ x f (x + θb(x, ζ Γn )γ n+1 , z)||b(x, ζ Γn )γ n+1 |dθ + 1 0 |D 2 x f (x + b(x, ζ Γn )γ n+1 + θσ(x, ζ Γn ) √ γ n+1 v, z)||σ(x, ζ Γn ) √ γ n+1 v| 2 dθp U (dv).
where p N denotes the density of the centered Gaussian random variable taking values in R d of which covariance matrix is the identity matrix. Combining the two last inequalities,we obtain,

γ -1 n+1 E[f (X Γn+1 , ζ Γn+1 ) -f (X Γn+1 , ζ Γn )|X Γn , ζ Γn ] M0 z=1 q ζΓ n ,z f (X Γn , z) + o n→∞ (γ n+1 ) f ∞ + M0 z=1 (|q ζΓ n ,z | + o n→∞ (γ n+1 ))(Λ f,1 (X Γn , ζ Γn , γ n+1 )|b(X Γn , ζ Γn )| + Λ f,2 (X Γn , ζ Γn , γ n+1 )|σσ * (X Γn , ζ Γn )|). Now we dene E = R d × {1, .., M 0 }, G 1 = [0, 1], Θ 1 = θ, π 1 the measure dened on (G 1 , B(G 1 )) (with B(G 1
) the sigma elds endowed by the Borelians of G 1 ) by π(dΘ 1 ) = dΘ 1 (that is the Lebesgue measure), and for every

(x, z) ∈ R d × {1, .., M 0 } = E, we have Λ f,1 (x, z, γ) = G1 Λf,1 (x, z, γ, Θ 1 )π 1 (dΘ 1 ), with Λf,1 : R d × {1, .., M 0 } × R + × [0, 1] → R + (x, z, γ, θ) → M0 w=1 ∇ x f (x + θb(x, z)γ, w)|γ|,
and g 1 (x, z) = |b(x, z)|. We are going to proove that E loc ( Λf,1 , g 1 ) (see [START_REF] Mei | Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic dierential equations with markov switching[END_REF] and ( 11)) holds. Using (41) and the fact that φ(x) C|x|, the functions b have sublinear growth: there exists C b 0 such that |b(x, z)|+ C b (1 + |x|) for every x ∈ R d and z ∈ {1, .., M 0 }. Therefore, since f has compact support, it follows that there exists t 0 > 0 and R > 0 such that sup |x|>R,z∈{1,..,M0} sup γ t0 | Λf,1 (x, z, γ, θ)| = 0 for every θ ∈ [0, 1]. Moreover since ∇ x f is bounded and b is locally bounded, we conclude that Λf,1 , satises E loc ( Λf,1 , g 1 ) (see [START_REF] Mei | Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic dierential equations with markov switching[END_REF] and ( 11)).

We focus on the other term. We dene

G 2 = R d × [0, 1], Θ 2 = (v, θ), π 2 the measure dened on (G 2 , B(G 2 )) (with B(G 2 ) the sigma elds endowed by the Borelians of G 2 ) by π 2 (dΘ 2 ) = dθp U (dv), and for every (x, z) ∈ R d × {1, .., M 0 } = E, we have Λ f,2 (x, z, γ) = G2 Λf,2 (x, z, γ, Θ 2 )π 2 (dΘ 2 ), with Λf,2 : R d × {1, .., M 0 } × R + × R N × [0, 1] → R + (x, z, γ, v, θ) → M0 w=1 D 2 x f (x + b(x, z)γ + θσ(x, z) √ γv, w)| √ γv| 2 ,
and g 2 (x, z) = |σ(x, z)| 2 . We are going to proove that E ergo ( Λf,2 , g 2 ) (see [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF]) holds. We x v ∈ R N and θ ∈ [0, 1]. Now using (41) and the fact that φ(x)

C|x|, the functions b and σ, have sublinear growth: there exists C b,σ 0 such that |b(x, z)| + |σ(x, z)| C b,σ (1 + |x|) for every x ∈ R d and z ∈ {1, .., M 0 }. Therefore, since f has compact support, it follows that there exists t 0 > 0 and R > 0 such that sup |x|>R,z∈{1,..,M0} sup γ t0 | Λf,2 (x, z, γ, v, θ)| = 0. Moreover since D 2

x f is bounded and b and σ are locally bounded, we conclude that we have E ergo ( Λf,2 , g 2 ) (see [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF]).

Besides, it is immediate to show that E ergo ( o n→∞ (γ n+1 ) f ∞ , 1) (see (13)) holds. Finally, it remains to study E[f (X Γn+1 , ζ Γn ) -f (X Γn , ζ Γn )|X Γn , ζ Γn ].
Using once again Taylor expansions of order one and two, we have

γ -1 n+1 E[f (X Γn+1 , ζ Γn )-f (X Γn , ζ Γn )|X Γn = x, ζ Γn = z] -∇ x f (x, z), b(x, z) - d i,j=1 (σσ * ) i,j (x, z) ∂ 2 f ∂x i ∂x j (x, z) 1 0 |∇ x f (x + θb(x, z)γ n+1 , z) -∇ x f (x)||b(x, z)|dθ + 1 0 |D 2 x f (x + b(x, z)γ n+1 + θσ(x, z) √ γ n+1 v, z) -D 2 x f (x)||σ(x, z)v| 2 dθp U (dv).
Using the same reasonning as before, one can show that E loc ( Λf,3 , g 1 ) (see [START_REF] Mei | Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic dierential equations with markov switching[END_REF] and [START_REF] Milstein | Weak approximation of solutions of systems of stochastic dierential equations[END_REF]) and E ergo ( Λf,4 , g 2 ) (see [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF]) hold with Λf,3 :

R d × {1, .., M 0 } × R + × [0, 1] → R + (x, z, γ, θ) → |∇ x f (x + θb(x, z)γ, z) -∇ x f (x, z)|,
and Λf,4 :

R d × {1, .., M 0 } × R + × R N × [0, 1] → R + (x, z, γ, v, θ) → |D 2 x f (x + b(x, z)γ + θσ(x, z) √ γv, z) -D 2 x f (x)||v| 2
We gather all the terms together and the result follows.

4.1.3 Proof of ( 14) and ( 16)

Proposition 4.3. Let p 1, a ∈ (0, 1], ρ, s ∈ (1,
P-a.s. sup n∈N * - 1 H n n k=1 η k Ãγ k (ψ • V ) s (X Γ k-1 , ζ Γ k-1 ) < ∞. (47)
Moreover,

P-a.s. sup n∈N * ν η n (V p/s+a-1 ) < ∞, (48)
and that if L V (see [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF]) holds, then 22) also holds, then

(ν η n ) n∈N * is tight B. If f ∈ D(A) and (
P-a.s. lim n→∞ 1 H n n k=1 η k Ãγ k f (X Γ k-1 , ζ Γ k-1 ) = 0 (49)
Proof. ). In the same way, since ρ s(1 -(1 -a)/p), we deduce from SW II,γ,η (see (27)) and Lemma 3.3 that SW II,γ,η (V p/s ) (see (20)) holds. Now,we are going to prove ĨX (f, V a+p-1 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) for f ∈ D(A) and f = V p/s and the proof of (47) and (49) will be completed. Notice that (48) will follow from I Q,V (ψ, φ) (see [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF]) and Theorem 3.1. This is a consequence of Lemma 4.1 which is given below. We notice indeed that Lemma 4.1 and the fact that under B(φ) (see (41)) and p 1, we have |σσ * | CV p+a-1 , imply that for every f ∈ D(A) and f = V p/s , there exists a sequence X, such that ĨX (f, V a+p-1 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) holds and the proof is completed. Lemma 4.1. Let p 1, a ∈ (0, 1], ρ ∈ (1, 2] and, ψ(y) = y p and φ(y) = y a . We suppose that the sequence (U n ) n∈N * satises M 2pρ/s (U ) (see (37)). Then, for every n ∈ N, we have

∀f ∈ D(A), E[|f (X Γn+1 , ζ Γn+1 ) -f (X 1 Γn , ζ Γn )| ρ |X Γn , ζ Γn ] Cγ ρ/2 n+1 1 ∨ |σσ * (X Γn , ζ Γn )| ρ/2 . (50) with D(A) = {f : R d ×{1, .., M 0 }, ∀z ∈ {1, .., M 0 }, f (., z) ∈ C 2 K (R d )}.
In other words, for every f ∈ D(A), we have ĨX (f, |σσ * | ρ/2 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) with X n = f (X 1 Γn , ζ Γn ) for every n ∈ N and I (t) = t ρ/2 for every t ∈ R + .

Moreover, if (38), (39) and B(φ) (see (41)) hold and ρ s(1 -(1 -a)/p), then, for every n ∈ N, we have

E[|V p/s (X Γn+1 , ζ Γn+1 ) -V p/s (X Γn , ζ Γn )| ρ |X Γn , ζ Γn ] Cγ ρ/2 n+1 V p+a-1 (X Γn , ζ Γn ), (51)
In other words, we have ĨX (V p/s , V p+a-1 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) with X n = (ψ • V ) 1/s (X Γn , ζ Γn ) for every n ∈ N and I (t) = t ρ/2 for every t ∈ R + .

Proof. We begin by noticing that

|X Γn+1 -X 1 Γn | Cγ 1/2 n+1 |σσ * (X Γn , ζ Γn )| 1/2 |U n+1 | Let f ∈ D(A).
We employ this estimation and since for f ∈ D(A) then, for every z ∈ {1, .., M 0 }, f (., z) is Lipschitz, and it follows,

∀f ∈ D(A), E[|f (X Γn+1 , ζ Γn ) -f (X 1 Γn , ζ Γn )| ρ |X Γn , ζ Γn ] Cγ ρ/2 n+1 |σσ * (X Γn , ζ Γn )| ρ/2 Cγ ρ/2 n+1 V aρ/2 (X Γn , ζ Γn ) Cγ ρ/2 n+1 V a+p-1 (X Γn , ζ Γn ). Moreover, E[|f (X Γn+1 , ζ Γn+1 )-f (X Γn+1 , ζ Γn )| ρ |X Γn , ζ Γn ] =γ n+1 2 z=1 (q ζΓ n ,z + o n→∞ (γ n+1 ))E[|f (X Γn+1 , z) -f (X Γn+1 , ζ Γn )| ρ |X Γn , ζ Γn ] Cγ n+1 f ρ ∞ ,
which concludes the study for f ∈ D(A). We focus now on the case f = V p/s . We notice that (41) implies that for any n ∈ N,

|X Γn+1 -X Γn | Cγ 1/2 n+1 φ • V (X Γn , ζ Γn )(1 + |U n+1 |)
Once again we rewrite the term that we study as follows

V p/s (X Γn+1 , ζ Γn+1 ) -V p/s (X Γn , ζ Γn ) =V p/s (X Γn+1 , ζ Γn ) -V p/s (X Γn , ζ Γn ) +V p/s (X Γn+1 , ζ Γn+1 ) -V p/s (X Γn+1 , ζ Γn )
We study the rst term. Using (29) with α = 2p/s, it follows from (38) that for any z ∈ {1, .., M 0 }, V (., z) is Lipschitz and we have

V p/s (X Γn+1 , z) -V p/s (X Γn , z) 2 2p/s p/s(V p/s-1/2 (X Γn , z)| √ V (X Γn+1 , z) - √ V (X Γn , z)| +| √ V (X Γn+1 , z) - √ V (X Γn , z)| 2p/s ) 2 2p/s p/s([ √ V ] 1 V p/s-1/2 (X Γn , z)|X Γn+1 -X Γn | +[ √ V ] 2p/s 1 |X Γn+1 -X Γn | 2p/s ).
We use the assumption ρ s(1 -(1 -a)/p) and it follows from B(φ) (see (41)) that

E[|V p/s (X Γn+1 , z) -V p/s (X Γn , z)| ρ |X Γn , z] Cγ ρ/2 n+1 V p+a-1 (X Γn , z).
In order to treat the rst term, we put z = ζ Γn in this estimation. It remains to study the second term. We notice that since ρ s(1 -(1 -a)/p), it is immediate from the previous inequality that for every z ∈ {1, .., M 0 }, we have

E[V pρ/s (X Γn+1 , z)|X Γn , z] CV p+a-1 (X Γn , z).
. We focus on the term to estimate and using this inequality, we obtain

E[|V p/s (X Γn+1 ,ζ Γn+1 ) -V p/s (X Γn+1 , ζ Γn )| ρ |X Γn , ζ Γn ] =γ n+1 M0 z=1 (q ζΓ n ,z + o n→∞ (γ n+1 ))E[|V p/s (X Γn+1 , z) -V p/s (X Γn+1 , ζ Γn )| ρ |X Γn , ζ Γn ] Cγ n+1 M0 z=1 (|q ζΓ n ,z | + o n→∞ (γ n+1 )(V p+a-1 (X Γn , z) + V p+a-1 (X Γn , ζ Γn )) Cγ n+1 V p+a-1 (X Γn , ζ Γn ),
where the last inequality follows from (39). We rearrange the terms and the proof is completed.

The Milstein scheme

In this part we treat the case of a Milstein scheme (introduced in [START_REF] Milstein | Weak approximation of solutions of systems of stochastic dierential equations[END_REF]) with decreasing step for a Brownian diusion process. As far as we know, there is no study concerning that scheme for the algorithm we use as for high weak or strong order numerical scheme. We propose two approaches under weak mean reverting assumption. The rst one relies on polynomial and the second one relies on exponential test functions. More particularly we use an approach with test functions ψ such that ψ(y) = y p , p 0 for every y ∈ [v * , ∞). The other approach is based on test functions ψ(y) = exp(λy p ), p ∈ [0, 1/2], λ 0, for every y ∈ [v * , ∞).

We consider a d-dimension Brownian motion (W t ) t 0 . We are interested in the solution of the d dimensional stochastic equation

X t = x + t 0 b(X s )ds + t 0 σ(X s )dW s (52)
where b : R d → R d and σ, ∂ x l σ : R d → R d×d , l ∈ {1, . . . d}, are locally bounded and continuous functions.

The innitesimal generator of this process is given by

Af (x) = b(x), ∇f (x) + d i,j=1 (σσ * ) i,j (x) ∂ 2 f ∂x i ∂x j (x) (53)
Now, we introduce the Milstein scheme for (X t ) t 0 such that for every n ∈ N and t ∈ [Γ n , Γ n+1 ], we have

X t =X Γn + (t -Γ n )b(X Γn ) + σ(X Γn )(W t -W Γn ) + d i,j=1 d l=1 ∂ x l σ i (X Γn )σ l,j (X Γn ) t Γn s Γn dW j u dW i s (54) with σ l : R d → R d , x → σ l (x) = (σ 1,l (x), . . . , σ d,l (x)). ∆X 1 n+1 =γ n+1 b(X Γn ), ∆X 2 n+1 = d i,j=1 d l=1 ∂ x l σ i (X Γn )σ l,j (X Γn ) Γn+1 Γn s Γn dW j u dW i s , ∆X 3 n+1 =σ(X Γn )(W Γn+1 -W Γn ). ( 55
)
and

X i Γn+1 = X Γn + i j=1 ∆X i n+1 .
In the sequel we will use the notation

U n+1 = γ -1/2 n+1 (W Γn+1 -W Γn )
and W n+1 = (W i,j n+1 ) i,j∈{1,...,d} with W i,j n+1 = γ -1 n+1

Γn+1

Γn s Γn dW j u dW i s . Actually, for the polynomial case, we introduce a weaker assumption for the sequence (U n ) n∈N * and (W n ) n∈N * . Let q ∈ N * , p 0. We suppose that (U n ) n∈N * is a sequence of independent random variables such that U satises

M N ,q (U ) ∀n ∈ N * , ∀q ∈ {1, . . . , q}, E[(U n ) ⊗q ] = E[(N (0, I d )) ⊗q ] (56)
M p (U ) sup

n∈N * E[|U n | p ] < ∞ (57)
Moreover, we assume that (W n ) n∈N * is a sequence of independent and centered random variables such that

M p (W) sup n∈N * E[|W n | p ] < ∞ (58)
Now, we assume that the Lyapunov function V :

R d → [v * , ∞), v * > 0, satises L V (see (7)) and |∇V | 2 C V V, sup x∈R d |D 2 V (x)| < ∞ (59)
We also dene

∀x ∈ R d , λ ψ (x) := 1 2 λ D 2 V (x)+∇V (x) ⊗2 ψ •V (x)ψ •V (x) -1 . ( 60
)
When ψ(y) = y p , we will also use the notation λ p instead of λ ψ . We will suppose that, for every x ∈ R d ,

B(φ) |b(x)| 2 + |σσ * (x)| + d i,j,l=1 |∂ x l σ i (x)σ l,j (x)| 2 Cφ • V (x) (61)
We now introduce the key hypothesis in order to obtain recursive control for polynomial and exponential form for ψ.

Polynomial case . First for the polynomial case, let p 0. We assume that there exists β ∈ R + , α > 0, such that for every x ∈ R d , we have

R p ∇V (x), b(x) + χ p (x) β -αφ • V (x), (62)
with

χ p (x) = λ 1 ∞ Tr[σσ * (x)] if p 1 λ p ∞ 2 (2p-3)+ Tr[σσ * (x)] if p > 1. (63)
Exponential case . For the exponential case we modify this assumption in the following way. Let p 1/2. We assume that there exists β ∈ R + , α > 0, such that for every x ∈ R d , we have

R p,λ ∇V (x), b(x) + κ(x) + χ p (x) β -αφ • V (x), (64)
with

κ(x) = 1 2 d i=1 d l=1 ∂ x l σ i (x)σ l,i (x) + λpV p-1 (x) 1 φ • V (x) Tr[σσ * (x)]∇V (x) (65)
and

χ p (x) = V 1-p (x) φ • V (x) C σ (x) -1 ln(2 d / det(Σ(x))) (66)
for C σ (x) : R d → R * + a continuous function such that inf x∈R d C σ (x) > 0 and that for every x ∈ R d , the matrix Σ(x) ∈ R d×d dened by

             Σ(x) i,j = -2C σ (x)[ √ V ] 1 v p-1/2 * 1 2 d l=1 |∂ x l σ i (x)σ l,j (x)| ∀i, j ∈ {1, . . . , d}, i = j, Σ(x) i,i = 1 -2C σ (x)( D 2 V ∞ V p-1 (x) Tr[σσ * (x)] + [ √ V ] 1 v p-1/2 * 1 2 d l=1 |∂ x l σ i (x)σ l,i (x)|) ∀i ∈ {1, . . . , d}
is a positive denite matrix.

Recursive control Polynomial case

Proposition 4.4. Let v * > 0, and ψ, φ : [v * , ∞) → R * + a continuous function such that φ(y) Cy with C > 0. Now let p 0 and dene ψ(y) = y p . We suppose that (U n ) n∈N * is a sequence of independent random variables such that U satises M N ,2 (U ) (see ( 56)) and M 2p∨2 (U ) (see (57)). Moreover, we assume that (W n ) n∈N * is a sequence of independent and centered random variables such that M 2p∨2 (W) (see (58)) holds. We suppose that (59), B(φ) (see (61)), R p (see (62)), are satised. Then, there exists α > 0, β ∈ R + and n 0 ∈ N * , such that

∀n n 0 , x ∈ R d , Ãγ n ψ • V (x) V -1 (x)ψ • V (x)(β -αφ • V (x)). (67)
Moreover, when φ = Id we have

sup n∈N E[ψ • V (X Γn )] < ∞. (68)
Proof. First ,we focus on the case p 1. From the Taylor's formula and the denition of λ ψ (see (60)),

we have

ψ • V (X Γn+1 ) =ψ • V (X Γn ) + X Γn+1 -X Γn , ∇V (X Γn ) ψ • V (X Γn ) + 1 2 (D 2 V (ξ n+1 )ψ • V (ξ n+1 ) + ∇V (ξ n+1 ) 2 ψ • V (ξ n+1 ))(X Γn+1 -X Γn ) ⊗2 . ψ • V (X Γn ) + X Γn+1 -X Γn , ∇V (X Γn ) ψ • V (X Γn ) +λ ψ (ξ n+1 )|ψ • V (ξ n+1 )(X Γn+1 -X Γn )| 2 .
with ξ n+1 ∈ (X Γn , X Γn+1 ). First, from (59), we have sup

x∈R d λ p (x) < ∞. Since W is made of centered random variables, we deduce from M N ,2 (U ) (see (56)), M 2 (U ) (see (58)) and M 2 (W) (see (57)), that E[X Γn+1 -X Γn |X Γn ] = γ n+1 b(X Γn ) E[|X Γn+1 -X Γn | 2 |X Γn ] γ n+1 Tr[σσ * (X Γn )] + γ 2 n+1 |b(X Γn )| 2 + c d γ 2 n+1 d i,j,l=1 |∂ x l σ i (X Γn )σ l,j (X Γn )| 2 + c d γ 3/2 n+1 d i,j,l=1 |∂ x l σ i (X Γn )σ l,j (X Γn )||σ(X Γn )|
with c d a positive constant. Assume rst that p = 1. Using B(φ) (see (61)), for every α ∈ (0, α), there exists n 0 (α) such that for every n n 0 (α),

λ 1 ∞ γ 2 n+1 (|b(X Γn )| 2 +c d | d i,j,l=1 ∂ x l σ i (X Γn )σ l,j (X Γn )| 2 ) (69) + λ 1 c d γ 3/2 n+1 d i,j,l=1 |∂ x l σ i (X Γn )σ l,j (X Γn )||σ(X Γn )| γ n+1 (α -α)φ • V (X Γn ).
From assumption R p (see ( 62) and ( 63)), we conclude that

Ãγ n ψ • V (x) β -αφ • V (x) Assume now that p > 1.Since |∇V | C V V (see (59)), then √ V is Lipschitz. Using (30), it follows that V p-1 (ξ n+1 ) √ V (X Γn ) + [ √ V ] 1 |X Γn+1 -X Γn | 2p-2 2 (2p-3)+ (V p-1 (X Γn ) + [ √ V ] 2p-2 1 |X Γn+1 -X Γn | 2p-2 )
We focus on the study of the second term of the remainder. First, using B(φ) (see ( 61)), for any p 1,

|X Γn+1 -X Γn | 2p c p γ p n+1 φ • V (X Γn ) p (1 + |U n+1 | 2p + |W n+1 | 2p ).
Let α ∈ (0, α). Then, we deduce from M 2p (U ) (see ( 58)), M 2p (W) (see (57)), that there exists n 0 (α) ∈ N such that for any n n 0 (α), we have

E[|X Γn+1 -X Γn | 2p |X Γn ] γ n+1 φ • V (X Γn ) p α -α φ/I d p-1 ∞ λ p ∞ 2 (2p-3)+ [ √ V ] 2p-2 1
To treat the other term we proceed as in (69

) with λ 1 ∞ replaced by λ p ∞ 2 2p-3 [ √ V ] 2p-2 1
, α replace by α and α ∈ (0, α). We gather all the terms together and using (63), for every n n 0 (α) ∨ n 0 (α), we obtain

E[V p (X Γn+1 ) -V p (X Γn )|X Γn ] γ n+1 pV p-1 (X Γn )(β -αφ • V (X Γn )) +γ n+1 pV p-1 (X Γn ) φ • V (X Γn )( α -α) + (α -α) V 1-p (X Γn )φ • V (X Γn ) p φ/I d p-1 ∞ γ n+1 V p-1 (X Γn )(βp -αpφ • V (X Γn )).
which is exactly the recursive control for p > 1. Now,we treat the case p < 1. Since x → x p is concave, we have

V p (X Γn+1 ) -V p (X Γn ) pV p-1 (X Γn )(V (X Γn+1 ) -V (X Γn ))
We have just proved that we have the recursive control I Q,V (ψ, φ) holds for ψ = I d (with some constants β ∈ R + and α > 0), and since V takes positive values, we obtain

E[V p (X Γn+1 ) -V p (X Γn )|X Γn ] pV p-1 (X Γn )E[V (X Γn+1 ) -V (X Γn )|X Γn ] V p-1 (X Γn )(pβ -pαφ • V (X Γn )),
which completes the proof of (67). The proof of 68) is immediate application of Lemma 3.2 as soon as we notice that the increments of the Milstein scheme have nite polynomial moments which imples (24).

Exponential case

In this section we will not relax the assumption on the Gaussian structure of the increment as we do in the polynomial case with hypothesis (see ( 56), ( 57) and ( 58)). In order to obtain our result, we introduce a supplementary assumption in order to express the iterated stochastic integrals in terms of products of the increments of the Brownian motion. The so called commutative noise assumption is the following ∀x ∈ R d , ∀i, j ∈ {1, . . . , d},

d l=1 ∂ x l σ i (x)σ l,j (x) = d l=1 ∂ x l σ j (x)σ l,i (x). (70)
In this case, with the notation from (55), we have

∆X 2 n+1 = 1 2 d i,j=1 d l=1 ∂ x l σ i (X Γn )σ l,j (X Γn )(W j Γn+1 -W j Γn )(W i Γn+1 -W i Γn ) (71) - 1 2 γ n+1 d i=1 d l=1 ∂ x l σ i (X Γn )σ l,i (X Γn ).
In the sequel we will adopt the following notation

∆X 1 n+1 =γ n+1 b(X Γn ) - 1 2 γ n+1 d i=1 d l=1 ∂ x l σ i (X Γn )σ l,i (X Γn )., ∆X 2 n+1 = 1 2 d i,j=1 d l=1 ∂ x l σ i (X Γn )σ l,j (X Γn )(W j Γn+1 -W j Γn )(W i Γn+1 -W i Γn ) (72)
Lemma 4.2. Let Λ i,j ∈ R, i, j ∈ {1, . . . , d} and U a R d -valued random variable made with d independent and indentically distributed standard normal random variables U = (U i ) i∈{1,..,d} , U i ∼ N (0, 1). We dene Σ ∈ R d×d such that Σ i,i = 1-2Λ i,i , i ∈ {1, . . . , d} and Σ i,j = -2Λ i,j , i, j ∈ {1, . . . , d}, i = j. We assume that Σ is a positive denite matrix. Then

E[exp( d i,j=1 Λ i,j |U i U j |)] 2 d det(Σ) -1/2 . ( 73
)
Proof. A direct computation yields

E[exp( d i,j=1 Λ i,j |U i U j |)] = R d (2π) -d/2 exp( d i,j=1 Λ i,j |u i u j | -1/2 d i=1 |u i | 2 )du ζ∈{-1,1} d R d (2π) -d/2 exp( d i,j=1 Λ i,j ζ i u i ζ j u j -1/2 d i=1 |ζ i u i | 2 )du =2 d R d (2π) -d/2 exp( d i,j=1 Λ i,j u i u j -1/2 d i=1 |ζ i u i | 2 )du = 2 d det(Σ) -1/2 .
Lemma 4.3. Let U a R d -valued random variable made with d independent and indentically distributed standard normal random variables U = (U i ) i∈{1,..,d} , U i ∼ N (0, 1). for every h ∈ (0, 1), we have

∀v ∈ R d , E[exp( √ h v, U + h i,j Λ i,j |U i U j |)] exp( h 2(1 -h) |v| 2 )2 hd det(Σ) -h/2 (74)
Proof. Using the Hölder inequality we have

E[exp( √ h v, U + h d i,j=1 Λ i,j |U i U j |)] E[exp( √ h 1 -h v, U )] 1-h E[exp(h d i,j=1 Λ i,j |U i U j |)]
The results follows from Lemma 4.2.

Using those results, we deduce the recursive control for exponential test functions.

Proposition 4.5. Let v * > 0, and φ : [v * , ∞) → R + a continuous function such that φ(y) Cy with C > 0 and lim y→∞ φ(y) = ∞. Now let p ∈ [0, 1/2], λ 0 and dene ψ : [v * , ∞) → R + such that ψ(y) = exp(λy p ). We suppose that (59), B(φ) (see (61)), R p,λ (see (64)), are satised.We also assume that

∀x ∈ R d , Tr[σσ * (x)]|b(x)|( √ V (x) + |b(x)|) CV 1-p (x)φ • V (x) (75)
Then, there exists α > 0, β ∈ R + and n 0 ∈ N * , such that

∀n n 0 , x ∈ R d , Ãγ n ψ • V (x) V -1 (x)ψ • V (x)(β -αφ • V (x)). (76)
Moreover, when φ = Id we have

sup n∈N E[ψ • V (X Γn )] < ∞. (77)
Proof. First, with notations (72), we rewrite

V p (X Γn+1 ) -V p (X Γn ) =V p (X Γn + ∆X 1 n+1 + ∆X 3 n+1 ) -V p (X Γn ) +V p (X Γn + ∆X 1 n+1 + ∆X 3 n+1 ) -V p (X Γn )
an we study each term separately. Since p 1, the function dened on [v * , ∞) by y → y p is concave. Using then the Tayor expansion of order 2 of the function V , for every x, y ∈ R d , there exists θ ∈ [0, 1]

such that V p (y) -V p (x) pV p-1 (x)(V (y) -V (x)) =pV p-1 (x)( ∇V (x), y -x + 1 2 Tr[D 2 V (θx + (1 -θ)y)(y -x) ⊗2 ])
and then,

V p (y) -V p (x) pV p-1 (x)( ∇V (x), y -x + 1 2 D 2 V ∞ |y -x| 2 . ( 78
)
Using this inequality with x = X Γn and y = X Γn + ∆X

1 n+1 + ∆X 3 n+1 , it follows that V p (X Γn + ∆X 1 n+1 + ∆X 3 n+1 ) -V p (X Γn ) pV p-1 (X Γn )( ∇V (X Γn ), ∆X 1 n+1 + ∆X 3 n+1 + 1 2 pV p-1 (X Γn ) D 2 V ∞ | ∆X 1 n+1 + ∆X 3 n+1 | 2 ).
Now, we study the other term. Since p 1/2, then the function dened on [v * , ∞) by y → y 2p is concave and we obtain

V p (X Γn+1 )-V p (X Γn + ∆X 1 n+1 + ∆X 3 n+1 ) pV p-1/2 ( ∆X 1 n+1 + ∆X 3 n+1 )( √ V (X Γn+1 ) - √ V (X Γn + ∆X 1 n+1 + ∆X 3 n+1 )) p[ √ V ] 1 v p-1/2 * | ∆X 2 n+1 |
In the sequel, we will use the notation

∀x ∈ R d , b(x) = b(x) + 1 2 d i=1 d l=1 ∂ x l σ i (x)σ l,i (x). It follows that E[exp(λV p (X Γn+1 )) -exp(λV p (X Γn ))|X Γn ] H γn+1 (X Γn )L γn+1 (X Γn ) with, for every x ∈ R d , t ∈ R * + , H t (x) = exp(λV p (x) + tλpV p-1 (x) ∇V (x), b(x) + t 2 1 2 λp D 2 V ∞ V p-1 (x)| b(x)| 2 ) and L t (x) =E[exp( √ tλpV p-1 (x) ∇V (x), σ(x)U + tλp D 2 V ∞ V p-1 (x) Tr[σσ * (x)]|U | 2 +tλp[ √ V ] 1 v p-1/2 * 1 2 d i,j=1 d l=1 |∂ x l σ i (x)σ l,j (x)||U i U j | + t 3/2 λpV p-1 (x) D 2 V ∞ 2 b(x), σ(x)U )]
where U = (U 1 , . . . U d ), with U i , i ∈ {1, . . . , d}, some independent and identically distributed standard normal random variables. In order to compute L t (x), we use Lemma 4.3 (see (74

)) with h = C σ (x) -1 tλp, v = C σ (x)λpV p-1 (x)σ * (x)(∇V (x) + t D 2 V ∞ 2b(x)
) and Σ(x) the matrix such that for every i, j ∈ {1, . . . , d},

             Σ(x) i,j = -2C σ (x)[ √ V ] 1 v p-1/2 * 1 2 d l=1 |∂ x l σ i (x)σ l,j (x)| ∀i, j ∈ {1, . . . , d}, i = j, Σ(x) i,i = 1 -2C σ (x)( D 2 V ∞ V p-1 (x) Tr[σσ * (x)] + [ √ V ] 1 v p-1/2 * 1 2 d l=1 |∂ x l σ i (x)σ l,i (x)|) ∀i ∈ {1, . . . , d}
where C σ : R d → R * + is such that inf x∈R d C σ (x) > 0 and for every x ∈ R d then Σ(x) is a positive denite matrix. We apply Lemma 4.3 and it follows that for t inf x∈R d C σ (x)/(2λp)

L t (x) exp( tλpC σ (x) -1 2(1 -λpC σ (x) -1 t) |v| 2 + tλpC σ (x) -1 (ln(2 d / det(Σ(x)))) exp(tλpC σ (x) -1 |v| 2 + tλpC σ (x) -1 (ln(2 d / det(Σ(x))))
At this point, we focus on the rst term of the exponential. We have

|v| 2 C σ (x)λp Tr[σσ * (x)]V 2p-2 (x)(|∇V (x)| 2 + t D 2 V ∞ 4 ∇V (x), b(x) + t 2 4 D 2 V 2 ∞ | b(x)| 2 )
Using B(φ), (75) and R p,λ (see (64)), it follows there exists C > 0 such that

H t (x)L t (x) exp(λV p (x) + tλpV p-1 (x)(β -αφ • V (x)) + Ct 2 V p-1 (x)φ • V (x))
Then, we have

H t (x)L t (x) exp((1 -tpαV -1 (x)φ • V (x))λV p (x) +tpαV -1 (x)φ • V (x)V p (x)( β αφ • V (x)
+ tC/(αp))).

Using the convexity of the exponential function, we have for tpαV

-1 (x)φ • V (x)) < 1, H t (x)L t (x) exp(λV p (x)) -tpαV -1 (x)φ • V (x))) exp(λV p (x)) +tpαV -1 (x)φ • V (x) exp(V p (x)( β αφ • V (x)
+ tC/(αp))).

At this point we notice that (24) holds with this ψ which will be useful in order to obtain (77). Moreover and independently from that, the function dened on R d by x → exp(V p (x)( β αφ•V (x) + tC/(αp))) is continuous and bounded on any compact set. Moreover φ tends to innity at the innity and then we have

φ • V (x) exp(V p (x)( β αφ • V (x) + tC/(αp))) = O x→∞ exp(λV p (x))
for every t < λαp/C, and the proof of the recursive control (76) is completed. Combining it with (24) (which is obtained above) and applying Lemma 3.2 gives (77).

Proof of the innitesimal estimation

Proposition 4.6. We suppose that the sequence (U n ) n∈N * satises M N ,2 (U ) (see ( 36)) and M 2 (U ) (see (57)) and that the sequence (W n ) n∈N * is centered and satises M 2 (W) (see ( 58)). We also assume that b and σ are locally bounded functions, that φ has sublinear growth, that B(φ) (see ( 61)) holds and that we have

sup n∈N * ν η n (|σ| 2 ) < ∞ and sup n∈N * ν η n ( d i,j,l=1 |∂ x l σ i σ l,j |) < ∞.
Then, we have E( Ãγ , A) (see ( 9)).

Proof. First, we recall that D(A) ⊂ C 2 K (E). Using the Taylor expansion, we have

f (X 1 Γn ) -f (X Γn-1 ) = ∇f (x), ∆X 1 n + R 1 (X Γn-1 , X 1 Γn-1 ) with R 1 (x, y) = f (y) -f (x) -∇f (x), y -x . First we notice that E[∆X 1 n ] = γ n b(X Γn-1
). Now, we focus on the expectation of R 1 (X Γn-1 , X 1 Γn-1 ). First, we dene the function from

R d × R d to R + as follows r 1 (x, y) = sup θ∈[0,1] |∇f (x + θ(y -x)) -∇f (x)|. (79)
Then r 1 is a bounded continuous function such that r 1 (x, x) = 0. Moreover, it folows immediately that R 1 (x, y) r 1 (x, y)|y -x|.

Therefore, we deduce that

E[R 1 (X Γn-1 , X 1 Γn-1 )|X Γn-1 ] E[|∆X 1 n |r 1 (X Γn-1 , X 1 Γn-1 )|X Γn-1 ] Cγ n |b(X Γn-1 )|r 1 (X Γn-1 , X 1 Γn-1 )
Now, we notice that hypothesis (61) and the fact that φ(x) C|x|, imply that b has a sublinear growth:

there exists C b 0 such that b(x) C b (1 + |x|). Now since f has a compact support, there exists R > 0 such that f (x) = 0 for every x ∈ R d such that |x| > R. As a consequence if |x| > 2R and γ t 0 = R/(C b (1 + 2R)) then for y = x + γb(x), |y| > |x| -γC b (1 + |x|) > R,
and then r 1 (x, y) = 0. It follows that the function Λf,1 : (x, γ) → γr 1 (x, x + γx) is uniformly continuous on R d × [0, t 0 ] and then we obtain E loc ( Λf,1 , |b|) (see [START_REF] Mei | Convergence and convergence rates for approximating ergodic means of functions of solutions to stochastic dierential equations with markov switching[END_REF] and ( 11)) with t 0 = R/(C b (1 + 2R)). In the same way we have

f (X 2 Γn ) -f (X 1 Γn ) = ∇f (X 1 Γn ), ∆X 2 n + R 1 (X 1 Γn-1 , X 2 Γn-1 ).
The rst term of the right hand side of the above equation is a centered random variable and we obtain

E[R 1 (X 1 Γn-1 ,X 2 Γn-1 )|X Γn-1 ] d i,j,l=1 |∂ x l σ i (X Γn-1 )σ l,j (X Γn-1 )|E[| γn γn-1 s γn-1 dW j u dW i s |r 1 (X 1 Γn , X 2 Γn-1 )|X Γn-1 ]. Now we dene G 2 = R d×d , Θ 2 = v, π 2 the measure dened on (G 2 , B(G 2 )) (with B(G 2 )
dW j u dW i s |r 1 (X 1 Γn , X 2 Γn-1 )|X Γn-1 = x] = R d×d Λi,j f,2 (x, γ n , v)W(dv) = G2 Λi,j f,2 (x, γ n , Θ 2 )π 2 (dΘ 2 ) with Λi,j f,2 : R d × R + × R d×d → R + (x, γ, v) → γ|v i,j |r 1 (x + γb(x), x + γb(x) + γ d i,j,l=1 ∂ x l σ i (x)σ l,j (x)v i,j ).
We are going to prove that E ergo (

d i,j=1
Λi,j f,2 , d i,j,l=1 |∂ x l σ i σ l,j |) (see [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF]) holds.First, we notice that B(φ) (61) and the fact that φ has sublinear growth, the functions b and ∂ x l σ i σ l,j , i, j, l ∈ {1, . . . , d}, have sublinear growth: there exists C b,σ 0 such that |b(x)| + d i,j,l=1 |∂ x l σ i (x)σ l,j (x)| C b,σ (1 + |x|) for every x ∈ R d . Therefore, in the same way as above, we obtain E ergo (

d i,j=1 Λi,j f,2 , d i,j,l=1 |∂ x l σ i σ l,j |) from sup n∈N * ν η n ( d i,j,l=1 |∂ x l σ i σ l,j |) < ∞.
In order to treat the last term, we write

f (X 3 Γn ) -f (X 2 Γn ) = ∇f (X 2
Γn ), ∆X

3 n + 1 2 Tr[D 2 f (X 2 Γn )(∆X 3
n ) ⊗2 ] + R 2 (X 2 Γn-1 , X 3
Γn ) with R 2 (x, y) = f (y) -f (x) -∇f (x), y -x -1 2 Tr[D 2 f (x)(y -x) ⊗2 ]. First we study E[Tr[(D 2 f (X 2 Γn ) -D 2 f (X Γn-1 ))(∆X 3
n ) ⊗2 ]|X Γn-1 ].
We dene

Λ1 f,3 : R d × R + × R d×d → R + (x, γ, v) → γ(2 Tr[v] + d)|D 2 f (x + γb(x) + γ d i,j,l=1 ∂ x l σ i (x)σ l,j (x)v i,j ) -D 2 f (x)|,
we have

E[Tr[(D 2 f (X 2 Γn )-D 2 f (X Γn-1 ))(∆X 3 n ) ⊗2 ]|X Γn-1 = x] C|σ| 2 (x) Λ1 f,3 (x, γ n , v)W(dv)
Using once again the fact that f has a compact support and the functions b and ∂ x l σ i σ l,j , i, j, l ∈ {1, . . . , d}, have sublinear growth, in the same way as before from, it follows from

sup n∈N * ν η n (|σ| 2 ) < ∞ that E ergo ( Λ1 f,3 , |σ| 2 ) (with π 1 3 = W) holds.
Now, we consider the other term. Similarly as before, we dene the function from

R d × R d to R + as follows r 2 (x, y) = sup θ∈[0,1] |D 2 f (x + θ(y -x)) -D 2 f (x)|. ( 80
)
Then r 2 is a bounded continuous function such that r 2 (x, x) = 0. Moreover, we have

R 2 (x, y) r 2 (x, y)|y -x| 2 .
We dene now

Λ2 f,3 : R d × R + × R d×d → R + (x, γ, v) → γ(2 Tr[v] + d) ζ∈{-1,1} d r 2 (x + γb(x) + γ d l=1 ∂ x l σ i (x)σ l,j (x)v i,j , x + γb(x) + γ d i,j,l=1 ∂ x l σ i (x)σ l,j (x)v i,j + √ γ d i=1 σ i (x)ζ i 2v i,i + 1) It follows that E[|∆X 3 n | 2 r 2 (X 2 Γn , X 3 Γn-1 )|X Γn-1 = x] C|σ| 2 (x) Λ2 f,3 (x, γ n , v)dW(dv)
Once again, since b, σ and ∂ x l σ i σ l,j , i, j, l ∈ {1, . . . , d}, have sublinear growth, it follows from

sup n∈N * ν η n (|σ| 2 ) < ∞ that E ergo ( Λ2 f,3 , |σ| 2 ) (with π 2 3 = W) holds.
We gather all the terms together and the proof is completed.

4.2.3 Proof of ( 14) and ( 16) Polynomial case Proposition 4.7. Let p 0, a ∈ (0, 1], ρ, s ∈ (1, 2] and, ψ(y) = y p , φ(y) = y a and I (t) = t ρ/2 . We suppose that (U n ) n∈N * is a sequence of independent random variables such that U satises M N ,2 (U ) (see ( 56)) and M 2p∨2pρ/s∨2 (U ) (see ( 57)). Moreover, we assume that (W n ) n∈N * is a sequence of independent and centered random variables such that M 2p∨2pρ/s∨2 (W) (see ( 58)) holds. We also assume that (59), B(φ) (see ( 61)) and R p (see ( 62)), with this p, hold. We also suppose that SW I,γ,η (ρ, I ) (see ( 26)) hold.

Then SW I,γ,η (V p∨1+a-1 , ρ, I ) (see ( 19)) holds and we have the following properties A. If SW II,γ,η (V p/s ) (see ( 20)) and SW pol (p, a, s, ρ) (see (85)) hold, then

P-a.s. sup n∈N * - 1 H n n k=1 η k Ãγ k (ψ • V ) s (X Γ k-1 ) < ∞, (81)
and we also have,

P-a.s. sup n∈N * ν η n (V p/s+a-1 ) < ∞. ( 82
)
Moreover, when p/s p ∨ 1 + a -1, the assumption SW II,γ,η (V p/s ) (see ( 20)) can be replaced by SW II,γ,η (see ( 27)). Besides, if we also suppose that L V (see ( 7)) holds and that p/s + a - 22) is satised, then

1 > 0, then (ν η n ) n∈N * is tight B. If f ∈ D(A) and (
P-a.s. lim n→∞ 1 H n n k=1 η k Ãγ k f (X Γ k-1 ) = 0 (83)
Proof. The result is an immediate consequence of Lemma 3.1. It remains to check the assumption of this Lemma.

First, we show SW I,γ,η (V p∨1+a-1 , ρ, I ) (see ( 19)). First we notice that for any p 1 then R p (see (62)) implies R 1 . Since (59), B(φ) (see (61)) and R p (see ( 62)) hold, it follows from Proposition 4.5 that I Q,V ( ψ, φ) (see [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF]) is satised with the function ψ : [v * , ∞) → R + dened by ψ(y) = y p∨1 . Then, using SW I,γ,η (ρ, I ) (see (26)) with Lemma 3.3, gives SW I,γ,η (V p∨1+a-1 , ρ, I ) (see ( 19)). In the same way, for p/s a + p -1, we deduce from SW II,γ,η (see ( 27)) and Lemma 3.3 that SW II,γ,η (V p/s ) (see (20)) holds. Now,we are going to prove ĨX (f, V a+p∨1-1 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) for f ∈ D(A) and f = V p/s and the proof of ( 81) and (83) will be completed. Notice that (82) will follow from I Q,V (ψ, φ) (see [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF]) and Theorem 3.1. The proof is a consequence of Lemma 4.4 which is given below. We notice indeed that B(φ

) (see (61)) gives |σσ * | ρ/2 + d i=1 d l=1 |∂ x l σ i (x)σ l,i | ρ CV ρa/2
. This observation combined with (86) implies that for every f ∈ D(A) and f = V p/s , there exists a sequence X, such that ĨX (f, V a+p∨1-1 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) holds and the proof is completed. Lemma 4.4. Let p 0, a ∈ (0, 1], ρ ∈ (1, 2] and, ψ(y) = y p and φ(y) = y a . We suppose that the sequence 57)) and that the sequence (W n ) n∈N * satises M ρ∨2pρ/s (W) (see (58)). Then, for every n ∈ N, we have: for every f ∈ D(A),

(U n ) n∈N * satises M ρ∨2pρ/s (U ) (see (
E[|f (X Γn+1 ) -f (X 1 Γn )| ρ |X Γn ] Cγ ρ/2 n+1 |σσ * (X Γn )| ρ/2 + Cγ ρ n+1 d i=1 d l=1 |∂ x l σ i (x)σ l,i (x)| ρ . (84) with D(A) = C 2 K (R d ).
In other words, for every f ∈ D(A), we have ĨX (f, g σ , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) with

g σ = |σσ * | ρ/2 + d i=1 d l=1 |∂ x l σ i (x)σ l,i | ρ , X n = f (X 1
Γn ) for every n ∈ N and I (t) = t ρ/2 for every t ∈ R + .

Moreover, if (59) and B(φ) (see (61)) hold and

SW pol (p, a, s, ρ)      s(2/ρ -1)(a + p -1) + s -2 0, if 2p/s < 1, (2 -s)/(2 -ρ) a s/ρ, if 2p/s 1 and p < 1, ρ s(1 -(1 -a)/p), if p 1. (85)
Then, for every n ∈ N, we have

E[|V p/s (X Γn+1 ) -V p/s (X Γn )| ρ |X Γn ] Cγ ρ/2 n+1 V p+a-1 (X Γn ). (86)
In other words, we have ĨX (V p/s , V p+a-1 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) with X n = V p/s (X Γn ) for every n ∈ N and I (t) = t ρ/2 for every t ∈ R + .

Proof. We begin by noticing that

|X Γn+1 -X 1 Γn | Cγ 1/2 n+1 |σσ * (X Γn )| 1/2 |U n+1 | + Cγ n+1 | d i=1 d l=1 |∂ x l σ i (X Γn )σ l,i (X Γn )| 2 | 1/2 |W n+1 | Let f ∈ D(A).
Then f is Lipschitz and the previous inequality gives (84).

We focus now on the case f = V p/s . We notice that B(φ) (see ( 61))implies that for any n ∈ N,

|X Γn+1 -X Γn | Cγ 1/2 n+1 φ • V (X Γn )(1 + |U n+1 | + |W n+1 ||)
First, we assume that 2p/s 1. Let x, y ∈ R d . Then, the function from R + to R + such that y → y 2p/s is concave and since √ V is Lipschitz (from (59)), we deduce that

V p/s (y) -V p/s (x) 2p/s √ V 2p/s-1 (x)( √ V (y) - √ V (x)) 2p/s[ √ V ] 1 V p/s-1/2 (x)|y -x|. Now, since 2(s/ρ -1)(a + p -1) + s -2 0 and V takes values in [v * , ∞)
, we deduce that there exists C > 0 such that for every x ∈ R d , we have V ρp/s-ρ/2 (x) CV a(1-ρ)+p-1 (x). Now, we assume that 2p/s 1. Using (29) with α = 2p/s and since

√

V is Lipschitz, we have

V p/s (X Γn+1 ) -V p/s (X Γn ) 2 2p/s p/s(V p/s-1/2 (X Γn )| √ V (X Γn+1 , z) - √ V (X Γn )| +| √ V (X Γn+1 ) - √ V (X Γn , z)| 2p/s ) 2 2p/s p/s([ √ V ] 1 V p/s-1/2 (X Γn )|X Γn+1 -X Γn | +[ √ V ] 2p/s 1 |X Γn+1 -X Γn | 2p/s ).
In order to obtain (86), it remains to use the assumptions B(φ) (see (61)) and then ρ s(1

-(1 -a)/p) if p 1 and (2 -s)/(2 -ρ) a s/ρ together with 2p/s 1 if p < 1. Exponential case Proposition 4.8. Let p ∈ [0, 1/2], λ 0, ρ ∈ (1, 2] and,ψ, φ : [v * , ∞) → R + with ψ(y) = exp(λy p
) and φ a continuous function such that φ(y) Cy with C > 0 and I (t) = t ρ/2 . We assume that (59), B(φ) (see (61)) and R p,λ (see ( 64)) hold and that ρ < s. We also suppose that SW I,γ,η (ρ, I ) (see (26)) and (75) hold. Then SW I,γ,η (V -1 φ • V exp(λV p ), ρ, I ) (see ( 19)) hold and we have the following properties A. If SW II,γ,η (see ( 27)) holds, then we have SW II,γ,η (exp(λ/sV p )) (see (20)) and

P-a.s. sup n∈N * - 1 H n n k=1 η k Ãγ k (ψ • V ) s (X Γ k-1 ) < ∞, (87)
and we also have,

P-a.s. sup n∈N * ν η n (V -1 φ • V exp(λ/sV p )) < ∞. (88)
Besides, when L V (see [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF]) holds, then

(ν η n ) n∈N * is tight. B. If f ∈ D(A)
and ( 22) is satised, then

P-a.s. lim n→∞ 1 H n n k=1 η k Ãγ k f (X Γ k-1 ) = 0 (89)
Proof. The result is an immediate consequence of Lemma 3.1. It remains to check the assumption of this Lemma.

First, we show SW I,γ,η (V -1 φ • V exp(λ/sV p ), ρ, I ) (see ( 19)). We begin by noticing that R p,λ (see (64)) implies R p, λ for every λ λ. Since (59), B(φ) (see (61)), R p,λ (see (64)) and (75) hold, it follows from Proposition 4.5 that I Q,V ( ψ, φ) (see [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF]) is satised for every function ψ : [v * , ∞) → R + such that ψ(y) = exp( λV p ) with λ λ. At this point, we notice that this property and the fact that φ has sublinear growth imply (90). Then, using SW I,γ,η (ρ, I ) (see ( 26)) with Lemma 3.3, gives SW I,γ,η (V -1 φ • V exp(λV p ), ρ, I ) (see ( 19)). In the same way, we deduce from SW II,γ,η (see ( 27)) and Lemma 3.

3 that SW II,γ,η (V -1 φ • V exp(λ/sV p )) (see (20)) holds.
Now,we are going to prove ĨX (f, V -1 φ • V exp(λV p ), X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) for f ∈ D(A) and f = V p/s and the proof of ( 87) and (89) will be completed. Notice that (82) will follow from I Q,V (ψ, φ) (see [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF]) and Theorem 3.1. The proof is a consequence of Lemma 4.4 (see (83)) and Lemma 4.5 which is given below. We notice indeed that B(φ

) (see (61)) gives |σσ * | ρ/2 + d i=1 d l=1 |∂ x l σ i (x)σ l,i | ρ (φ • V ) ρ
. Moreover, we have already shown that (90) is satised. These observations combined with (91) imply that for every f ∈ D(A) and f = exp(λ/sV p ), there exists a sequence X, such that ĨX (f, V -1 φ • V exp(λV p ), X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) holds and the proof is completed. Lemma

4.5. Let p ∈ [0, 1/2], λ 0, ρ, s ∈ (1, 2] and,ψ, φ : [v * , ∞) → R + with ψ(x) = exp(λx p )
and φ a continuous function such that φ(x) Cx with C > 0. We assume that (59) and B(φ) (see (61)) hold, that ρ < s, and there exists n 0 ∈ N * , such that

∀ λ λ, ∃C 0, ∀n n 0 , E[exp( λV p (X Γn+1 ))|X Γn ] C exp( λV p (X Γn )). (90)
Then, for every n n 0 , we have

E[| exp(λ/sV p (X Γn+1 )) -exp(λ/sV p (X Γn ))| ρ |X Γn ] Cγ ρ/2 n+1 φ • V (X Γn ) V (X Γn ) exp(λV p (X Γn )), (91)
In other words, we have ĨX (exp(λ/sV p ), V -1 φ•V exp(λV p ), X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) with X n = exp(λ/sV p (X Γn )) for every n ∈ N and I (t) = t ρ/2 for every t ∈ R + .

Proof. Before we prove the result, we notice that B(φ) (see (61)) implies that for any n ∈ N

|X Γn+1 -X Γn | Cγ 1/2 n φ • V (X Γn )(1 + |U n+1 | 2 + |W n+1 | 2 ).
First, we assume that p 1/2. Let x, y ∈ R d . First, since the function x → x 2p is concave, we have

V p (y) -V p (x) 2p √ V 2p-1 (x)( √ V (y) - √ V (x)) 2p[ √ V ] 1 V p-1/2 (x)|y -x|. Moreover, exp(λ/sV p (y)) -exp(λ/sV p (x)) λ s (exp(λ/sV p (y)) + exp(λ/sV p (x)))|V p (y) -V p (x)|.
We combine those two inequalities and use Hölder inequality in order to obtain

E[| exp(λ/sV p (X Γn+1 )) -exp(λ/sV p (X Γn ))| ρ |X Γn ] C exp(λρ/sV p (X Γn ))V pρ-ρ/2 (X Γn )E[|X Γn+1 -X Γn | ρ |X Γn ] +CV pρ-ρ/2 (X Γn )E[exp(λρ/sV p (X Γn+1 ))|X Γn+1 -X Γn | ρ |X Γn ] C exp(λρ/sV p (X Γn ))V pρ-ρ/2 (X Γn )E[|X Γn+1 -X Γn | ρ |X Γn ] +CV pρ-ρ/2 (X Γn )E[exp(λρθ/sV p (X Γn+1 ))|X Γn ] 1/θ E[|X Γn+1 -X Γn | ρθ/(θ-1)
|X Γn ] (θ-1)/θ , for every θ > 1. Now, we use (90) and since ρ < s, for every θ ∈ (1, s/ρ], we obtain E[exp(λρθ/sV p (X Γn+1 ))|X Γn ] C exp(λρθ/sV p (X Γn )).

Rearranging the terms and since ρ < s, we conclude from B(φ) (see (61)) that

E[| exp(λ/sV p (X Γn+1 )) -exp(λ/sV p (X Γn+1 ))| ρ |X Γn+1 ] Cγ ρ/2 n V pρ-ρ/2 (X Γn )|φ • V (X Γn )| ρ/2 exp(λρ/sV p (X Γn )) Cγ ρ/2 n V -1 (X Γn )φ • V (X Γn ) exp(λV p (X Γn )).

Application to processes with jump

The purpose of this section is to build an invariant measure using a decreasing step Euler scheme for a Feller diusion process with jump which is not necessarily a Levy process. This study extends the one in [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic dierential equation driven by a lévy process[END_REF] where the author treat the convergence of (ν η n ) n∈N * for miscellaneous decreasing step Euler scheme for Levy processes. The interest of our approach is that we consider process with some general jump components which involve Levy processes but also diusion process with censored jump or piecewise deterministic Markov processes. We consider weak mean reverting assumption that is φ(y) = y a , a ∈ (0, 1] for every y ∈ [v * , ∞). Similarly as in its study we consider polynomial test functions ψ such that ψ(y) = y p , p 0 for every y ∈ [v * , ∞).

We consider a Poisson point process p with state space (F ; B(F )) where F = F × R + . We refer to [START_REF] Ikeda | Stochastic dierential equations and diusion processes[END_REF] for more details. We denote by N the counting measure associated to p. We have N ([0, t) × A) = #{0 s < t; p s ∈ A} for t 0 and A ∈ B(F ). We assume that the associated intensity measure is given by N (dt, dz, dv

) = dt × λ(dz) × 1 [0,∞) (v)dv where (z, v) ∈ F = F × R + .
We will use the notation Ñ = N -N . We also consider a d-dimension Brownian motion (W t ) t 0 independent from N . We are interested in the solution of the d dimensional stochastic equation In this paper, we do not discuss existence of such processes. This processes can be seen as extension of Levy process (put c(z, x) = c(x)z and ζ = 1). Especially, if we want decomposition (92) to make sense, we must at least assume that for every x ∈ R d , we have

X t = x + t 0 b(X s -)ds + t 0 σ(X s -)dW s + t 0 F c(z, X s -)1 v ζ(z,X s -) 1 [0,h] (|z|) Ñ (ds, dz, dv) + t 0 F c(z, X s -)1 v ζ(z,X s -) 1 (h,∞) (|z|)N (ds, dz, dv).
F |c(z, x)| 2 ζ(z, x)1 [0,h] (|z|)λ(dz) < ∞, (93)
and

F |c(z, x)|ζ(z, x)1 (h,∞) (|z|)λ(dz) < ∞. (94)
The main dierence with Levy processes is that the intensity of jump ξ(x, z)λ(dz) may depend on the position of the process. Actually, this type of process can also be seen as an extension of SDE with censored jump component. Indeed, if for every x ∈ R d , we have

F |c(z, x)|ζ(z, x)λ(dz)dt < ∞, (95)
it comes down to study the solution of following SDE with censored jump part:

X t = x + t 0 b(X s -)ds + t 0 σ(X s -)dW s + t 0 F c(z, X s -)1 v ζ(z,X s -) 1 [0,∞) (|z|)N (ds, dz, dv). (96)
with, for every

x ∈ R d , b(x) = b(x) + F c(z, x)ζ(z, x)1 [0,h] (|z|)λ(dz).
The study of this family of processes in the literature is expensing. In [START_REF] Fournier | Jumping sdes: absolute continuity using monotonicity[END_REF], the author focus on the case σ = 0 and prove the existence of an absolutely continuous (with respect to the Lebesgue measure) density. In the PhD thesis [START_REF] Rabiet | A stochastic equation with censored jumps related to multi-scale Piecewise Deterministic Markov Processes[END_REF] , the author extends existence and uniqueness results for SDE with non null Brownian part and censored jump part and also show that they can be considered as limit processes of some general piecewise deterministic Markov processes. Besides, he studies ergodicity of those processes using a regenerated procedure.

This procedure provides a Doeblin (locally lower Lebesgue bounded) condition which enables to prove recurrence in Harris sense and then ergodicity. Finally, one may notice that the study for SDE with censored jump part with form (96), is equivalent to the study of (92) with h = 0. Consequently, we study the approximation of invariant measures for solutions of (92) and the results we provide in this part apply to SDE with censored jump as soon as we put h = 0.

The innitesimal generator of this process is given by

Af (x) = b(x), ∇f (x) + d i,j=1 (σσ * ) i,j (x) ∂ 2 f ∂x i ∂x j (x) (97)
+ F (f (x + c(z, x)) -f (x) -c(z, x), ∇f (x) 1 [0,h] (|z|))ζ(z, x)λ(dz).
The rst step is to consider an `truncated' approximation for the process (X t ) t 0 with nite big jump intensity. In this case we can introduce an Euler scheme for this `truncated' process and then prove convergence of the measure dened in ( 5) toward an invariant measure of the' process (X t ) t 0 . In order to show that the limit of the measure dened in ( 5) and built with the Euler scheme of the truncated process is an invariant measure for the process (X t ) t 0 , it is necessary to introduce a supplementary hypothesis. This hypothesis enables to control of the distance between the generator of (X t ) t 0 and the generator of the `truncated' process with jump size at most M . When λ({z, h < |z|}) = ∞, we assume that lim

M →∞ lim n→∞ ν η n (λ M ) = 0 with ∀M h, ∀x ∈ R d , λ M (x) := F |c(z, x)||ζ(z, x)|1 (M,∞) (|z|)λ(dz) = 0.
(98)

Approach and preliminary results

Let M ∈ R + ∪{+∞} such that λ({z, h < |z| < M }) < ∞. We dene N M (ds, dz, dv) := 1 |z|<M 1 v ζ ∞ N (ds, dz, dv). Now, we introduce the process (X M t ) t 0 which satises the following equation

X M t = x + t 0 b(X M s -)ds + t 0 σ(X M s -)dW s + t 0 F c(z, X M s -)1 v ζ(z,X M s -) 1 [0,h] (|z|) ÑM (ds, dz, dv) + t 0 F c(z, X M s -)1 v ζ(z,X M s -) 1 (h,∞) (|z|)N M (ds, dz, dv) (99)
Since NM is nite, we represent the random measure N M using a compound Poisson process. We introduce the Poisson processes (J M t ) t 0 , independent from ÑM , with intensity ζ ∞ λ({z, h < |z| < M }) and jump times (T M k ) k∈N * . We introduce the sequences of independent random variables (and independent from J M and ÑM )

Z M k ∼ λ({z, h < |z| < M }) -1 1 h<|z|<M dz,
and

V k ∼ ζ -1 ∞ (v)1 v ζ ∞ dv.
Therefore, (99) can be rewritten

X M t = x + t 0 b(X M s -)ds + t 0 F c(z, X M s -)1 v ζ(z,X M s -) 1 [0,h] (|z|) ÑM (ds, dz, dv) + t 0 σ(X M s -)dW s + J M t k=1 c(Z M k , X M T - k )1 V k ζ(Z M k ,X M T - k ) (100)
The innitesimal generator of this process is given by

A M f (x) = b(x), ∇f (x) + d i,j=1 (σσ * ) i,j (x) ∂ 2 f ∂x i ∂x j (x) (101)
+ F (f (x + c(z, x)) -f (x) -c(z, x), ∇f (x) 1 [0,h] (|z|))1 (h,M ) (|z|)ζ(z, x)λ(dz).
Now, we introduce an approximation for X M . We will use a Euler type scheme such that for every n ∈ N and t ∈ [Γ n , Γ n+1 ], we have

X M t = X M Γn + (t -Γ n )b(X M Γ - n ) + t Γn c(z, X M Γ - n )1 v ζ(z,X M Γ - n ) 1 [0,h] (|z|) ÑM (ds, dz, dv) (102) + σ(X M Γ - n )(W t -W Γn ) + J M t k=1+J M Γn c(Z M k , X M Γ - n )1 V k ζ(Z M k ,X M Γ - n
) .

which is well dened since λ({z, h < |z| < M }) < ∞. Now, h > 0, x ∈ R d and t 0, we dene

M h 1,t (x) := t 0 F c(z, x)1 v ζ(z,x) 1 [0, h] (|z|)N M (ds, dz, dv), (103)
M h 1,t (x) := t 0 F c(z, x)1 v ζ(z,x) 1 [0, h] (|z|) ÑM (ds, dz, dv), M h 2,t (x) := t 0 F c(z, x)1 v ζ(z,x) 1 ( h,M ) (|z|)N M (ds, dz, dv), M h 2,t (x) := t 0 F c(z, x)1 v ζ(z,x) 1 ( h,M ) (|z|) ÑM (ds, dz, dv).
In order to simplify the writing, we will use the notations: can be simulated at time Γ n . This assumption prevails in this paper. When it is not possible a solution is given in [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic dierential equation driven by a lévy process[END_REF]. It consists in localizing the small jumps of X M Γn -X M Γn-1 on a strict subset [h n , h] (h n > 0) of [0, h] with lim n→∞ h n = 0 and in assuming that the small jumps with size contained in [h n , h] can be simulated. This specic study in our case is very similar to the one for the Levy process made in [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic dierential equation driven by a lévy process[END_REF],

∆X M,1 n+1 =γ n+1 b(X M Γn ), ∆X M,2 n+1 = σ(X M Γn )(W Γn+1 -W Γn ), (104)
∆X M,3 n+1 = M h 1,γn+1 (X M Γn ), ∆X M,4 n+1 = M h 2,γn+1 (X M Γn ),
and also to the one we do when we suppose that ∆X M,3 n+1 can be simulated. Consequently, we propose a study in which we assume ∆X M,3 n+1 we invite the reader to refer [13] in order to generalize it to the case where it can not be simulated.

For every n ∈ N * and t ∈ [Γ n , Γ n+1 ], the innitesimal generator of (X M t ) T 0 is given by A

M x0 f (x) = b(x 0 ), ∇f (x) + Tr[σσ * (x 0 )D 2 f (x)] (105) + F (f (x + c(z, x 0 )) -f (x) -c(z, x 0 ), ∇f (x) 1 [0,h] (|z|)ζ(z, x 0 )1 [0,M ) (|z|)λ(dz), on the set {X M Γn = x 0 }, with the notation D 2 f (x) = ∂ 2 f ∂xi∂xj (x)
. At this point we notice that Af (x) = A x f (x) which will be a key property in order to prove the innitesimal estimation E( Ãγ , A) (see [START_REF] Lemaire | Estimation récursive de la mesure invariante d'un processus de diusion[END_REF]) in the sequel (in particular for the jump part).

the sequel we will use the notation U n+1 = γ -1/2 n+1 (W Γn+1 -W Γn ). Actually, we introduce a weaker assumption than Gaussian distribution for the sequence (U n ) n∈N * . Let q ∈ N * , p 0. We suppose that (U n ) n∈N * is a sequence of independent random variables such that

M N ,q (U ) ∀n ∈ N * , ∀q ∈ {1, . . . , q}, E[(U n ) ⊗q ] = E[(N (0, I d )) ⊗q ] (106)
M p (U ) sup

n∈N * E[|U n | p ] < ∞ (107)
Now we introduce some hypothesis concerning the parameters. First, we introduce the hypothesis concerning the jump components. In the sequel, we will denote

τ q,h (x) = F c(z, x) 2q ζ(z, x)1 [0,h] (|z|)λ(dz), τ p,h (x) = F c(z, x) 2p ζ(z, x)1 (h,M ) (|z|)λ(dz), (108)
and

τ p (x) = F c(z, x) 2p ζ(z, x)1 [0,M ) (|z|)λ(dz),
for p, q 0. We assume the following nitness hypothesis: Let p, q 0. for every x ∈ R d , we have

H q h τ q,h (x) < ∞, H p h τ p,h (x) < ∞, H p τ p (x) < ∞. (109)
It is immediate to notice that H p implies both H p h and H p h for any h > 0. Moreover, we introduce the following classical hypothesis: H q h implies H q h for q q and H p h implies H p h for p p . Now, we assume that the Lyapunov function V : R d → [v * , ∞), v * > 0, satises L V (see [START_REF] Lamberton | Recursive computation of the invariant distribution of a diusion[END_REF]) and

|∇V | 2 C V V, sup x∈R d |D 2 V (x)| < ∞ (110)
We also dene

∀x ∈ R d , λ ψ (x) := 1 2 λ D 2 V (x)+∇V (x) ⊗2 ψ •V (x)ψ •V (x) -1 . (111)
When ψ(x) = |x| p , we will also use the notation λ p instead of λ ψ . We will suppose that, for every x ∈ R d ,

B p,q (φ) |b(x) + κ p,q (x)| 2 + |σσ * (x)| Cφ • V (x) (112)
where

κ p,q (x) = R l c(z, x)ζ(z, x)(1 p>1/2 1 (h,M ) (|z|) -1 p,q 1/2 1 [0,h] (|z|))λ(dz). (113)
The reader may notice that κ p,q is well dened when H q h and H p h hold. When p > 1/2 we will also use the notation κ p instead of κ p,q . For p, q 0 and φ a positive function, we introduce the following hypothesis

H q h (φ, V ) τ q,h (x) Cφ • V (x) q , (114)
H p h (φ, V ) τ p,h (x) Cφ • V (x) p , H p (φ, V ) τ p (x) Cφ • V (x) p , for every x ∈ R d .
Remark 4.1. We notice that H p (φ, V ) implies both H p h (φ, V ) and H p h (φ, V ) for any h 0. Moreover, H q h (φ, V ) implies H q h(φ, V ) for any h ∈ (0, h].

We now introduce the key hypothesis in order to obtain recursive control that we will use for polynomial functions ψ. We assume that there exists β ∈ R + , α > 0, such that for every x ∈ R d , we have R p,q ∇V (x), b(x) + κ p,q (x) + χ p,q (x)

β -αφ • V (x), (115)
with

χ p,q (x) = λ 1 ∞ (Tr[σσ * (x)] + τ 1 (x)) + p -1 V 1-p (x) χp,q,h (x) if p 1 λ p ∞ 2 (2p-3)+ (Tr[σσ * ] + τ 1 (x) + [ √ V ] 2p-2 1 V 1-p (x)τ p (x)) if p 1. ( 116
) with χp,q,h (x) =(1 p,q 1/2 [ √ V ] 2(p∨q) 1 v p-p∨q * + 1 p 1/2 1 q>1/2 C q [V p-1 ∇V ] 2q-1 +1 p>1/2 C p∨q [V p∨q-1 ∇V ] 2(p∨q)-1 )τ p∨q,h (x) +(1 p 1/2 [ √ V ] 2p 1 + 1 p>1/2 C p [V p-1 ∇V ] 2p-1 )τ p,h (x)
and C q and C p∨q the constant from the BDG inequality dened in (31).

Proof of the recursive control

In order to obtain the recursive mean reverting control, we require a rst result concerning the evolution of the jump components.

Lemma 4.6. We have the following properties

B. Let q ∈ [0, 1/2] and assume that H q h (see (109)) hold. Then, ∀n ∈ N,

E[|M h 1,t (x)| 2q ] tτ q,h (x). (118)
C. Let q ∈ [1/2, 1] and assume that H q h (see (109)) hold. Then, ∀n ∈ N,

E[| M h 1,t (x)| 2q ] C q tτ q,h (x). (119)
with C q the constant which appears in the BDG inequality (see (31)). D. Let p > 1. We assume that H p (see (109)) hold. Then, there exists ξ > 1, which does not depend on h, such that for we have

E[| M h 1,t (x) + M h 2,t (x)| 2p ] t(τ p (x) + c p τ p,h (x)) + C p,h t ξ φ • V (x) p . ( 120
)
where c p 0, and C p,h 0 is nite if

H 1 h (φ, V ), H p h (φ, V ), H 1/2 h (φ, V ) and H p h (φ, V ) (see (114)) hold. Now, let p ∈ [1/2, 1). Assume that H p (see (109)) holds. Then E[| M h 1,t (x) + M h 2,t (x)| 2p ] C p tτ p (x) (121)
with C q the constant which appears in the BDG inequality (see (31)).

If we assmue instead that H 1 and H p (see (109)) hold, there exists C 0 such that

E[| M h 1,t (x) + M h 2,t (x)| 2p ] C(t p τ p 1 (x) + tτ p (x)) (122)
Finally, if p = 1, and only H 1 (see (109)) holdq then

E[| M h 1,t (x) + M h 2,t (x)| 2 ] = tτ 1 (x) (123)
Proof. We prove point A.. Let (J M t ) t 0 , a Poisson process with intensity ζ ∞ µ({z, h < |z| < M }) and jump times (T M k ) k∈N * . We introduce the sequences of independent random variables (and independent from J M )

Z M k ∼ λ({z, h < |z| < M }) -1 1 h<|z|<M dz, and V k ∼ ζ -1 ∞ (v)1 v ζ ∞ dv. We rewrite, M h 2,t (x) = J M t k=1 c(Z M k , x)1 V k ζ(Z M k ,x) . Now we denote λ = ζ ∞ λ({z, h < |z| < M }). Therefore, E J M t k=1 c(Z M k , x)1 V k ζ(Z M k ,x) 2p =E k 1 1 k=J M t k l=1 c(Z M l , x)1 V l ζ(Z M l ,x) 2p = k 1 E k l=1 c(Z M l , x)1 V l ζ(Z M l ,x) 2p e -λt ( λt) k k! .
Now, using the inequality (30), it follows that

E k l=1 c(Z M l , x)1 U l ζ(Z M l ,x) 2p k (2p-1)+ k l=1 E |c(Z M l , x)1 V l ζ(Z M l ,x) | 2p =k 1+(2p-1)+ λ-1 R l c(z, X M Γn ) 2p ζ(z, X M Γn )1 (h,M ) (|z|)λ(dz).
Moreover,

e -λt k 1 k 1+(2p-1)+ ( λt) k k! = e -λt λt k 0 (k + 1) (2p-1)+ ( λt) k k!
Now, we are going to use the inequalities (28) and (29). If p < 1 then e -λt λt k 0

(k + 1) (2p-1)+ ( λt) k k! λt + λt k 0 k (2p-1)+ ( λt) k k! = λt + λte -λt k 1 k (2p-1)+ ( λt) k k! .
Using this reasoning recursively, we obtain e -λt λt k 0

(k + 1) (2p-1)+ ( λt) k k! λt (2p-1)+ i=0 ( λt) i ,
and the proof is completed. Now, we assume that p 1. Then e -λt λt k 0

(k + 1) (2p-1)+ ( λt) k k! λt + λt(2p -1) + 2 (2p-1)+-1 k 0 (k + k (2p-1)+ ) ( λt) k k! ,
and similarly as before, a recursive approach yields (117).

We focus on the proof of point B.. We apply inequality (30) and compensation formula, and (118)

follows from

E[|M h 1,t (x)| 2q ] s t |M h 1,s (x) -M h 1,s -(x)| 2q = tτ q,h (x)
.

Point C. (see ( 119)) is a direct consequence of the BDG inequality (see (31)).

Finally, we consider the proof of point D.. First we treat the case p = 1. In this case, the process

(M t ) t 0 such that M t := ( M h 1,t + M h 2,t ) 2 -τ 1 (x), is a martingale and then E[| M h 1,t + M h 2,t | 2 ] = tτ 1 (x). Now, let p > 1. Let h ∈ [0, h].
Using the BDG inequality (see (31)), ws obtain

E[| M h 1,t | 2p ] C p E | s t | M h 1,s -M h 1,s -| 2 | p = C p E | s t |∆ M h 1,s | 2 | p .
In order to obtain our result, we are going to use a recursive approach. For any k ∈

N * , M h,k 1,t := s t |∆ M h 1,s | 2 k - tτ 2 k-1 ,h (x) is a martingale. Using (30) for the martingale ( M h,k 1,t ) t 0 ) E | s t |∆ M h 1,s | 2 k | p/2 k-1 =E | M h,k 1,t + tτ 2 k-1 ,h (x)| p/2 k-1 2 (p/2 k-1 -1)+ E | s t ∆ M h,k 1,s | p/2 k-1 + 2 (p/2 k-1 -1)+ |tτ 2 k-1 ,h (x)| p/2 k-1 2 (p/2 k-1 -1)+ C p E | s t |∆ M h 1,s | 2 k+1 | p/2 k + 2 (p/2 k-1 -1)+ |tτ 2 k-1 ,h (x)| p/2 k-1
Now, let k 0 = inf{k ∈ N * ; 2 k p}. Using (28), we have

E | s t |∆ M h 1,s | 2 k 0 +1 | p/2 k 0 E s t |∆ M h 1,s | 2p = tτ p,h (x) Since 2 k < p for any k < k 0 , it follows that E[| M h 1,t | 2p ] c p tτ p,h (x) + c p k0 k=1 |tτ 2 k-1 ,h (x)| p/2 k-1 c p tτ p,h (x) + c p |tτ 1,h (x)| p ∨ |tτ 2 k 0 -1 ,h (x)| p/2 k 0 -1
with c p 0 a constant which can change from line to line. Since we have H 1 h (φ, V ) and H p h (φ, V ), it follows that there exists ξ > 1 such that

E[| M h 1,t | 2p ] c p tτ p,h (x) + c p t ξ φ • V (x) p
Now, using (117) , we have

E[|M h 2,t | 2p ] t(1 + (t))τ p,h (x)
.

From (29), it follows that

E | M h 2,t | 2p ] t(1 + (t))τ p,h (x) + p2 2p |t 1 + (t))τ p,h (x)| 1-1/(2p) |tτ 1/2,h (x) + t 2p |τ 1/2,h (x)| 2p Since we have H 1/2 h (φ, V ) and H p h (φ, V ), it follows that there exists ξ > 1 such that E | M h 2,t | 2p ] tτ p,h (x) + c p t ξ φ • V (x) p
Now since M h 1 and M h 2 are independent, using (29), we obtain

E | M h 1,t + M h 2,t | 2p ] E M h 2,t | 2p ] + p2 2p (E | M h 1,t |]E | M h 2,t | 2p-1 ] + E | M h 1,t | 2p ]) t(τ p,h (x) + c p τ p,h (x)) + c p t ξ φ • V (x) p ,
Finally, let p ∈ [1/2, 1). Using the BDG inequality (see (31)), (28) and the compensation formula, we have

E[| M h 1,t + M h 2,t | 2p ] C p E | s t |∆M h 1,s + ∆M h 2,s | 2 | p . C p E s t |∆M h 1,s + ∆M h 2,s | 2p . =C p tτ p (x)
Moreover, (122) follows from Jensen's inequality and the proof is completed.

Lemma 4.7. Let x ∈ R d p, q ∈ [0, 1]. We assume that H p h and H p∨q h (see (109)) hold. Then, there exists : R + → R a locally bounded function which satises (t)/t C such that for every x 0 ∈ R d , we have

E[V p (x 0 + M h 1,t (x) + M h 2,t (x) -κ p,q (x)) -V p (x 0 )] t χp,q,h (x) + 1 p 1/2 [ √ V ] 2p 1 t (t)τ p,h (x)
where κ p,q dened in (113) and χp,q,h is dened in (116) and is given by

χp,q,h (x) =(1 p,q 1/2 [ √ V ] 2(p∨q) 1 v p-p∨q * + 1 p 1/2 1 q>1/2 C q [V p-1 ∇V ] 2q-1 +1 p>1/2 C p∨q [V p∨q-1 ∇V ] 2(p∨q)-1 )τ p∨q,h (x) +(1 p 1/2 [ √ V ] 2p 1 + 1 p>1/2 C p [V p-1 ∇V ] 2p-1 )τ p,h (x)
Let p, q 1/2. Assume that H p h and H p∨q h (see (109)) hold. Then, there exist : R + → R a locally bounded function, which satises (t)/t C such that for every x 0 ∈ R d , we have

E[V p (x 0 + M h 1,t (x) + M h 2,t (x)) -V p (x 0 )] [ √ V ] 2(p∨q) 1 v p-p∨q * tτ p∨q, h(x) (124)
+[ √ V ] 2p 1 t(1 + (t))τ p, h (x)
Let p 1/2 and q > 1/2. Assume that H p h and H q h (see (109)) hold. Then, there exist : R + → R a locally bounded function, which satises (t)/t C such that for every x 0 ∈ R d , we have

E[V p (x 0 + M h 1,t (x) + M h 2,t (x)) -V p (x 0 )] C q [V p-1 ∇V ] 2q-1 tτ q, h(x) (125) +[ √ V ] 2p 1 t(1 + (t))τ p, h(x)
Let p > 1/2 and assume that H p h and H p∨q h (see (109)) hold. Then, for every x 0 ∈ R d , we have

E[V p (x 0 + M h 1,t (x) + M h 2,t (x)) -V p (x 0 )] C p∨q [V p∨q-1 ∇V ] 2(p∨q)-1 tτ p∨q, h(x) (126)
+C

p [V p-1 ∇V ] 2p-1 tτ p, h(x)
Proof. Assume rst that p 1/2. Using (28) with α = p/2, and since √ V is Lipschitz, it follows from the same approach as in the proof of Lemma 4.6, point A., that

E[V p (x 0 + M h 2,t (x)) -V p (x 0 )] E[| √ V (x 0 + M h 2,t (x)) - √ V (x 0 )| 2p ] [ √ V ] 2p 1 E[|M h 2,t (x)| 2p ] [ √ V ] 2p 1 t(1 + (t)) F c(z, x) 2p ζ(z, x)1 (h,M ] (|z|)λ(dz). with : R + → R a locally bounded function, which satises (t)/t C. Let x 0 ∈ R d . We study E[V p (x 0 + M h 1,t (x)) -V p (x 0 )], if q 1/2 and E[V p (x 0 + M h 1,t (x)) -V p (x 0 )] if q > 1/2.
First, we consider the case q p. In this case H q h implies H p h . Using once again (28) with α = p/2, and since √

V is Lipschitz,it follows from the same approach as in the proof of Lemma 4.6, point B., that

E[V p (x 0 + M h 1,t (x)) -V p (x 0 )] E[| √ V (x 0 + M h 1,t (x)) - √ V (x 0 )| 2p ] [ √ V ] 2p 1 E[|M h 1,t (x)| 2p ] [ √ V ] 2p 1 t F c(z, x) 2p ζ(z, x)1 [0,h] (|z|)λ(dz).
Now let p < q. First, let q 1/2. Using (28) with α = 2q, the concavity of the function y → y p/q , and since

√ V is Lipschitz, we deduce that E[V p (x 0 + M h 1,t (x)) -V p (x 0 )] E[| √ V p/q (x 0 + M h 1,t (x)) - √ V p/q (x 0 )| 2q ] V p-q (x 0 )E[| √ V (x 0 + M h 1,t (x)) - √ V (x 0 )| 2q ] [ √ V ] 2q 1 v p-q * E[|M h 1,t (x)| 2q ] [ √ V ] 2q 1 v p-q * t F c(z, x) 2q ζ(z, x)1 [0,h] (|z|)λ(dz).
We assume that q > 1/2. Using Taylor expansion of order one, we obtain

E[V p (x 0 + M h 1,t (x)) -V p (x 0 )] = E[V p-1 (ξ) ∇V (ξ), M h 1,t (x) ], with ξ ∈ [x 0 , x 0 + M h 1,t (x)]. Now since √ V is Lipschitz, we can proove that x → V p-1 (x)∇V (x) is 2q -1
holder in this case (see [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic dierential equation driven by a lévy process[END_REF], Lemma 3) and since M h 1,t (x) is centered, it follows from the same approach as in the proof of Lemma 4.6, point C. that

E[V p (x 0 + M h 1,t (x)) -V p (x 0 )] E[V p-1 (x 0 )∇V (x 0 ) M h 1,t (x) + [V p-1 ∇V ] 2q-1 | M h 1,t (x)| 2q ] =[V p-1 ∇V ] 2q-1 E[| M h 1,t (x)| 2q ] C q [V p-1 ∇V ] 2q-1 t F c(z, x) 2q ζ(z, x)1 [0,h] (|z|)λ(dz)
Now, we assume that p > 1/2. Let p q. Using once again the fact x → V p-1 (x)∇V (x) is 2p -1

Hölder, similarly as in Lemma 4.6, point D., we deduce that

E[V p (x 0 + M h 1,t (x) + M h 2,t (x)) -V p (x 0 )] E[V p-1 (x 0 )∇V (x 0 )( M h 1,t (x) + M h 2,t (x)) +[V p-1 ∇V ] 2p-1 | M h 1,t (x) + M h 2,t (x)| 2p ] C p [V p-1 ∇V ] 2p-1 t F c(z, x) 2p ζ(z, x)1 [0,M ) (|z|)λ(dz)
Now, let p < q. In the same way

E[V p (x 0 + M h 2,t (x)) -V p (x 0 )] C p [V p-1 ∇V ] 2p-1 t F c(z, x) 2p ζ(z, x)1 [h,M ] (|z|)λ(dz)
Finally, as in the proof for q > 1/2 p, we obtain 114)) when p 1/2, that there exists α ∈ (0, α) and n 0 ( α) N * such that for every n n 0 (α), we have

E[V p (x 0 + M h 1,t (x)) -V p (x 0 )] C q [V p-1 ∇V ] 2q-1 t F c(z, x) 2q ζ(z, x)1 [0,h] (|z|)λ(dz)
E[V p (X M Γn+1 )) -V p (X M,2 Γn+1 + γ n+1 κ p,q (X M Γn ))|X M Γn ] γ n+1 χp,q,h (X M Γn ) +γ n+1 ( α -α)pV p-1 (X M Γn )φ • V (X M Γn ).
Gathering all the terms together and using R p,q (see (115)) yields the recursive contol (129). The proof of ( 128) is an immediate application of Lemma 3.2 as soon as we notice that the increments of the Euler scheme (102) have nite polynomial moments (under the hypothesis from A.) which imples (24).

Proof of the innitesimal estimation

In order to btain the result, it is necessary to introduce some structural assumption concerning the jump process. For x ∈ R d , let us dene the process (M t (x)) t 0 such that M t (x) = M h 1,t (x) + M h 2,t (x) (see (103) for notations) for every t 0. We assume that ∀z ∈ F, lim

Now, we give the result that provides the innitesimal estimation.

Lemma 4.8. Let p 0 and let q ∈ [0, 1]. We consider the sequence (U n ) n∈N * which satises M N ,2 (U ) (see ( 106)) and M 2 (U ) (see ( 107)). Moreover, we assume that H p h and H q h (see (109)) hold. We also suppose that b and σ are locally bounded functions with sublinear growth, that (132) holds and that we have sup n∈N * ν η n (|σ| 2 ) < ∞, and also sup n∈N * ν η n (τ q,h ) < ∞ when q ∈ (1/2, 1]. Then, we have E( Ãγ , A M ) (see ( 9)), with A M dened in (101).

Proof. In this proof we will use the function ω b,σ,t : R d × R d → R d such that ω b,σ,t (x, v) = x + tb(x) + √ tσ(x)v. We also introduce the random process (M h t ) t 0 independent from (U n ) n∈N ,such that for every t 0, we have M h t (x) = M h 1,t (x) + M h 2,t (x) (see (103) for notations). First we write t -1 (E[f (ω b,σ,t (x, v)+M h t (x)] -f (ω b,σ,t (x, v)) = A M 3 f (x) + R A3 (x, t, v) Case q ∈ [1/2, 1]. We focus on R A3 . At this point, we assume that τ q,h takes strictly positive values (otherwise R A3 = 0). We denote π(dΘ) = p U (dv)1 [0,h] (|z|)λ(dz)dθd for Θ = (v, z, θ)

∈ G = R d × F × [0, 1]. It follows that (for U ∼ U 1 ), E[R A3 (x, t, U )] = τ q,h (x)E[ G τ -1 q,h (x)R A3 (x, t, M h tθ (x), v, z, θ)π(dΘ)]
First we show [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF]. We recall that b and σ have sublinear growth. Therefore, as a direct consequence of (132) and since f has a compact support, there exists t 0 0 such that ∀Θ = (v, z, θ) ∈ G, lim |x|→∞ sup t∈[0,t0] τ -1 q,h (x)R A3 (x, t, M h tθ (x), Θ) = 0 a.s.

Finally, since f is continuous with compact support, then it is uniformly continuous and since (M tθ (x)) t 0 ) is a left limited right continuous process, we deduce that for any compact subset K of R d , we have ∀Θ = (v, z, θ) ∈ G, lim t→0 sup x∈K τ -1 q,h (x)R A3 (x, t, M h tθ (x), Θ) = 0 a.s.

Consequently [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF] holds. Now, we show that E ergo (τ -1 q,h R A3 , τ q,h ) holds.

Using Taylor expansion of order one, we obtain R A3 (x, t, u, v, z, θ) From Taylor expansion of order two, it also follows that R A3 (x, t, u, v, z, θ) 1 2 and E ergo (τ -1 q,h R A3 , τ q,h ) follows from sup n∈N * ν η n (τ q,h ) < ∞. First we show [START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF]. We recall that b and σ have sublinear growth. Therefore, as a direct consequence of (132) and since f has a compact support, there exists t 0 0 such that ∀Θ = (v, z, θ) ∈ G, lim |x|→∞ sup t∈[0,t0] R A3 (x, t, M h tθ (x), Θ) = 0 a.s.

Finally, since f is continuous with compact support, then it is uniformly continuous and since (M tθ (x)) t 0 ) is a left limited right continuous process, we deduce that for any compact subset K of R

(x), Θ)|]π(dΘ) C f ∞ ζ ∞ λ({z, h < |z| < M }) < ∞,
and E ergo (R A3 , 1) follows. To complete the proof, it remains to study (for U ∼ U 1 )

t -1 E[f (ω b,σ,t (x, U ) -f (x)].
This is already done in the proof of Proposition 4.6, so we invite the reader to refer to this part of the paper for more details.

Case q ∈ [0, 1/2]. In this case the study is the same as for q ∈ [1/2, 1]. We notice that κ p,q (see (113) for notations) is well dened and that (132) implies that it has sublinear growth. Consequenty, since λ({0 < |z| < M }) < ∞, if we replace b by b -κ p,q and then take h = 0, we obtain the result with a similar proof.

The following result show how to obtain lim n→∞ ν η n (Af ) = 0 from Lemma 4.8. This is a key result which allow us to work with truncated jumps and nevertheless obtain convergence towards the invariant measure of the process with unbounded jumps.

Proposition 4.10. We assume that λ({z, h < |z|}) = ∞. For M h, we dene We assume that ( 16) holds for every process that belongs to the family of processes ((X M t ) t 0 ) M M0 , for some M 0 h, that is: ∀M M 0 , ∀f ∈ D(A) P-a.s.

lim n→∞ 1 H n n k=1 η k γ k E[f (X M Γ k ) -f (X M Γ k-1 )|X M Γ k-1 ] = 0, (133)
with (X M t ) t 0 dened in (102). We also suppose that (98) is satised, that is lim M →∞ lim n→∞ ν η n (λ M ) = 0

. Finally we suppose that the hypothesis from Lemma 4.8 are satised with M replaced by M (and

η k A M f (X M Γ k-1 ) = 0.
Finally, we let M tends to innity and since lim M →∞ lim n→∞ ν η n (λ M ) = 0, the proof is completed.

4.3.4 Proof of ( 14) and ( 16)

Proposition 4.11. Let p 0, q ∈ [0, 1], a ∈ (0, 1], ρ, s ∈ (1, 2] and, ψ(y) = y p , φ(y) = y a and I (t) = t ρ/2 . We assume that the sequence (U n ) n∈N * satises M N ,2 (U ) (see ( 106)) and M 2p∨2 (U ) (see (107)) and that (110) holds. Then, we have the following properties,

A. We assume that B p,q (φ) (see ( 112)) and R p,q (115) hold. We also suppose that , ρ, I ) by SW I,γ,η (V aρ/2 , ρ, I ). A solution to obtain SW I,γ,η (V aρ/2 , ρ, I ) when aρ/2 a + p -1 is provided by point A. that is SW I,γ,η (V p+a-1 , ρ, I ). When aρ/2 > a + p -1 a possible solution consists if replacing p by p 0 in A. with p 0 satisfying aρ/2 a + p 0 -1

Lemma 2 . 1 .

 21 (Kronecker). Let (a n ) n∈N * and (b n ) n∈N * be two sequences of real numbers. If (b n ) n∈N * is non-decreasing, positive, with lim n→∞ b n = ∞ and n 1 a n /b n converges in R, then lim

  the sigma elds endowed by the Borelians of G 2 ) by π 2 (dΘ 2 ) = W(dv) where W denotes the law of the R d×d -valued random variable with components s for i, j ∈ {1, . . . , d}. Then for every x ∈ R d = E,

  b : R d → R d , σ : R d → R d×d and c(z, .) : R d → R d , z ∈ R l are locally bounded and continuous functions.

  this point, we precise that we implicitely suppose that ∆X M,3 n+1

  |x|→∞ |c(z, x)|/|x| = 0, and ∃t 0 ∈ R + , ∀t t 0 , lim |x|→∞ |M t (x)|/|x| = 0 a.s.

FF 0 FR 0 FR

 00 (f (x + c(z, x)) -f (x) -c(z, x), ∇f (x) )1 [0,h] (|z|))ζ(z, x)λ(dz) + (f (x + c(z, x)) -f (x))1 (h,M ) (|z|)ζ(z, x)λ(dz) It follows that we can decompose R A3 (x, t, v) in the following way: R A3 (x, t, v) = R A3 (x, t, v) + R A3 (x, t, v) with R A3 (x, t, v) =E[ 1 A3 (x, t, v, M tθ (x), z, θ)1 [0,h] (|z|)λ(dz)dθ] with R A3 : R d × R + × R d × R d × F × [0, 1] → R + (x, t, u, v, z, θ) → f (ω b,σ,t (x, v) + u + c(z, x)) -f (ω b,σ,t (x, v) + u) -c(z, x), ∇f (ω b,σ,t (x, v) + u) -(f (x + c(z, x)) -f (x) -c(z, x), ∇f (x) )ζ(z, x), and R A3 (x, t, v) =E[ 1 A3 (x, t, v, M tθ (x), z, θ)1 (h,M ) (|z|)λ(dz)dθ] with R A3 : R d × R + × R d × R d × F × 1] → R + (x, t, u, v, z, θ) → (f (ω b,σ,t (x, v) + u + c(z, x)) -f (ω b,σ,t (x, v) + u))ζ(z, x) -(f (x + c(z, x)) -f (x))ζ(z, x), R A3 (x, t, v) =E[ ω b,σ,t (x, v) + M h tθ (x) + c(z, x)) -f (ω b,σ,t (x, v) + M h tθ (x))ζ(z,x)1 (h,M ) (|z|)λ(dz)dθ] x + c(z, x)) -f (x))ζ(z, x)1 (h,M ) (|z|)λ(dz)dθ].

1 0

 1 |c(z, x)||∇f (ω b,σ,γ (x, v) + u + ϑc(z, x)) -∇f (ω b,σ,γ (x, v) + u) +∇f (x + ϑc(z, x)) -∇f (x))ζ(z, x)|dϑ.

1 0

 1 |c(z, x)| 2 |D 2 f (ω b,σ,γ (x, v) + u + ϑc(z, x)) -D 2 f (x + ϑc(z, x))|ζ(z, x)dϑ Therefore, for any r ∈ [1, 2], R A3 (x, t, u, v, z, θ) C D 2 f ∞ ∨ ∇f ∞ |c(z, x)| r ζ(z, x) Taking r = q, the hypothesis H q h (see (109) brings sup x∈R d sup t∈R+ τ -1 q,h (x) G E[|R A3 (x, t, M h tθ (x), Θ)|]π(dΘ) < ∞,

Now, we focus

  on R A3 . We denote π(dΘ) = p U (dv)1 [0,h] (|z|)λ(dz)dθd for Θ = (v, z, θ) ∈ G = R × F × [0, 1]. It follows that (for U ∼ U 1 ), E[R A3 (x, t, U )] = E[ G R A3 (x, t, M h tθ (x), v, z, θ)π(dΘ)]

λ

  M (x) = F |c(z, x)||ζ(z, x)|1 [M ,∞) (|z|)λ(dz).

t 3 f

 3 ) t 0 replaced by (X M t ) t 0 ) for every M M 0 . Then, we have∀f ∈ D(A) P-a.s. lim Af (X M Γ k-1 ) = 0.(134)Proof. We notice that Af= A M f + R A3,A M x + c(z, x)) -f (x))1 [M ,∞) (|z|)ζ(z, x)λ(dz) ∇f ∞ F c(z, x)ζ(z, x)1 [M ,∞) (|z|)λ(dz) ∇f ∞ λ M (x) Now, we write ν η n (A M f ) = ν η n (Af ) + ν η n (R A3,A M) and then we obtain:|ν η n (Af )| |ν η n (A M f )| + ∇f ∞ |ν η n (λ M )|.Since (133) holds, then Lemma 4.8 with Theorem 3.2 give, ∀f ∈ D(A) P-a.s.

  + a continuous function such that φ(y) Cy with C > 0. Now let p 1 and dene

4.1.1 Recursive control Proposition 4.1. Let v * > 0, and φ : [v * , ∞) → R *

  The result is an immediate consequence of Lemma 3.1. It remains to check the assumption of this Lemma. First, we show SW I,γ,η (V p+a-1 , ρ, I ) (see (19)). Since (38), B(φ) (see (41)) and R p (see (42)) hold, it follows from Proposition 4.1 that I Q,V (ψ, φ) (see[START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF]) holds. Then, using SW I,γ,η (ρ, I ) (see (26)) with Lemma 3.3 gives SW I,γ,η (V p+a-1 , ρ, I ) (see(19)

  A. Let p 0. Assume that H p h (see (109)) hold. There exists a locally bounded function : R + → R which satises (t)/t C, and such that ∀n ∈ N,

	E[|M h 2,t (x)	2p ] t(1 + (t))τ p,h (x).

  + → R a locally bounded function which satises (t)/t C. It follows from H

	p h (φ, V ) (see
	(

Now, we are able to present the recursive control under weak mean reverting assumption for test functions with polynomial growth. with : R

  d , we have∀Θ = (v, z, θ) ∈ G, limConsequently[START_REF] Pagès | Sur quelques algorithmes récursifs pour les probabilités numériques[END_REF] holds. Now, we show that E ergo (R A3 , 1) holds. As a direct consequence of λ({z, h < |z| < M }) < ∞,

			t→0	sup x∈K	R A3 (x, t, M h tθ (x), Θ) = 0 a.s.
	we obtain
	sup x∈R d	sup t∈R+	E[|R A3 (x, t, M h tθ
		G

  SW I,γ,η (ρ, I ) (see (26)) hold and that: i) If p 1, we assume that H p , H ) If p ∈ (0, 1) and let q ∈ [0, 1], we assume that H V ) (see (114)) if p 1/2. Then SW I,γ,η (V p+a-1 , ρ, I ) (see (19)) holds and we have the following property: If in addition SW II,γ,η (V p/s ) (see (20)), SW pol (p, a, s, ρ) (see (139)), H p∨q h (φ, V ) and HMoreover, when p/s p + a -1, the assumption SW II,γ,η (V p/s ) (see (20)) can be replaced by SW II,γ,η (see (27)). Besides, if we also suppose that L V (see (7)) holds and that p/s + a -1 > 0, then (ν η n ) n∈N * is tight B. If f ∈ D(A), (22), H q h and H p h (see (109) are satised and SW I,γ,η (|σσ * | ρ/2 + 1 q =0 τ 1∧ρ/(2q) V ) and if we also replace SW I,γ,η (|σσ * | ρ/2 + τ 1∧ρ/(2q)

			1/2
	p h and H p∨q h h (φ, p (see (109)) hold and that we have p h (φ, V ) are satised, then H P-a.s. sup n∈N * -n 1 k=1 H n η k Ãγ (135)
	and we also have,
									(136)
									q	+
	1 p =0 τ	1∧ρ/(2p) p	, ρ, I ) (see (19)) holds, then
			P-a.s.	lim n→∞	1 H n	k=1 n	η k	Ãγ k f (X	M Γ k-1 ) = 0	(137)
	Remark 4.2. The reader may notice that (137) remains true if we replace H q h and H tively H q h (φ, V ) and H p h (φ, q 1∧ρ/(2p) p h by respec-+ τ p

p h (φ, V ) and H 1 h (φ, V ) (see (

114

)) are satised and that, if

p > 1, H 1 h(φ, V ),H p h(φ, V ), H h (φ, V ) and H p h(φ, V ) hold for any h ∈ (0, h]. iik (ψ • V ) s (X M Γ k-1 ) < ∞, P-a.s. sup n∈N * ν η n (V p/s+a-1 ) < ∞.

APPLICATIONS

Proposition 4.9. Let v * > 0, p 0, q ∈ [0, 1], and φ : [v * , ∞) → R * + a continuous function such that φ(y) Cy with C > 0 and dene also ψ : [v * , ∞) → R + such that ψ(y) = y p . We assume that the sequence (U n ) n∈N * satises M N ,2 (U ) (see ( 106)) and M 2p∨2 (U ) (see (107)) and that (110), B p,q (φ) (see (112)) and R p,q (115) hold. We have the following properties:

A. Assume that p 1. We also assume that H p , H p h (φ, V ) and H 1 h (φ, V ) (see (114)) are satised and that, if p > 1, H 1 h(φ, V ),H p h(φ, V ), H

1/2 h (φ, V ) and H p h(φ, V ) hold for any h ∈ (0, h]. Then, there exists α > 0, β ∈ R + and n 0 ∈ N * , such that

Moreover, when φ = Id we have

B. Assume that p ∈ [0, 1) and let q ∈ [0, 1]. Moreover, we assume that H p h and H p∨q h (see ( 109)) hold and that we have H p h (φ, V ) (see (114)) if p 1/2. Then, there exists α > 0, β ∈ R + and n 0 ∈ N * , such that

Moreover, when φ = Id we have

Proof. We focus on the proof of A.. From the Taylor's formula and the denition of λ ψ (see ( 111)), we have

with,

and

Using the Cauchy Schwarz inequality it follows that

Assume rst that p = 1. Using B p,q (φ) (see (112)) and H 1 h (φ, V ) (see (114)), for every α ∈ (0, α), there exists n 0 (α) such that for every n n 0 (α),

From assumption (116) and since τ 1,h + τ 1,h = τ 1 , we conclude that

We focus on the study of the second term of the remainder. First, using B p,q (see (112)) and

Moreover, using Lemma 4.6 (see (120)), there exists ξ > 1, such that for every h ∈ [0, h], then ∀n ∈ N,

Using the Jensen inequality and (28), we have

Now, since H p h (φ, V ) holds, then lim h→0 τ p, h/(φ • V ) p = 0 and then for any ε > 0 there exists

hold for every h ∈ (0, h]. Let α ∈ (0, α). Since p > 1/2, there exists 0 (α) ∈ N such that for any n n 0 ( α), we have

To treat the other term we proceed as in (131

, α replace by α and α ∈ (0, α). We gather all the terms together and using R p,q (see ( 115) and ( 116)), for every n n 0 (α) ∨ n 0 ( α), we obtain

which is exactly the recursive control for p > 1, that is (127). The proof of ( 128) is an immediate application of Lemma 3.2 as soon as we notice that the increments of the Euler scheme (102) have nite polynomial moments (under the hypothesis from A.) which imples (24). Now, we prove point B.. Since p 1, the function dened on (v * , ∞) by y → y p is concave. Using then the Tayor dexpansion of order 2 of the function V , for every x, y ∈ R d , there exists λ ∈

and then,

We apply this inequality, and with the notation (113), it follows that

As in the proof of the case p 1, it follows from B p,q (φ) (see (112)) that there exists α ∈ (0, α) and n 0 ( α) ∈ N * such that for every n n 0 (α), we have

Proof. The result is an immediate consequence of Lemma 3.1. It remains to check the assumption of this Lemma.

We focus on the proof of ( 135) and (136). First, we show SW I,γ,η (V p+a-1 , ρ, I ) (see ( 19)). Since (110), B p,q (φ) (see (112)) and R p,q (see ( 115)) hold, it follows (using hypothesis from point i) and point ii)) from Proposition 4.9 that I Q,V (ψ, φ) (see ( 8)) is satised. Then, using SW I,γ,η (ρ, I ) (see ( 26)) with Lemma 3.3 gives SW I,γ,η (V p+a-1 , ρ, I ) (see ( 19)). In the same way, for p/s a + p -1, we deduce from SW II,γ,η (see ( 27)) and Lemma 3.3 that SW II,γ,η (V p/s ) (see (20)) holds. Now,we are going to prove ĨX (V p/s , V a+p-1 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) and the proof of (135) will be completed. Notice that (136) will follow from I Q,V (ψ, φ) (see [START_REF] Lamberton | Recursive computation of the invariant distrbution of a diusion: The case of a weakly mean reverting drift[END_REF]) and Theorem 3.1. The proof is a consequence of Lemma 4.9 which is given below. We notice indeed that Lemma 4.9 (see (140)) implies that, there exists a sequence X, such that ĨX (V p/s , V a+p-1 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) holds and the proof of ( 135) and ( 136) is completed.

We complete the proof o the Proposition by noticing that (137) follows directly from Lemma 4.9 (see (138)). Lemma 4.9. Let p 0, q ∈ [0, 1], a ∈ (0, 1], ρ ∈ (1, 2] and, ψ(y) = y p and φ(y) = y a . We suppose that the sequence (U n ) n∈N * satises M ρ∨2pρ/s (U ) (see ( 107)) and that (110) holds. We also assume that H p h and H q h (see ( 109)) are satised. Then, for every n ∈ N we have: for every f ∈ D(A),

In other words, for every f ∈ D(A), we have

Γn ) for every n ∈ N and I (t) = t ρ/2 for every t ∈ R + . Now, we assume that (110), B p,q (φ) (see ( 112)), H p∨q h (φ, V ) and H p h (φ, V ) hold. Finally, we suppose that the following holds:

Then, for every n ∈ N, we have

In other words, we have ĨX (V p/s , V p+a-1 , X, ρ, I ) (see [START_REF] Talay | Stochastic hamiltonian dissipative systems with non globally lipschitz coecients: exponential convergence to the invariant measure and discretization by the implicit euler scheme[END_REF]) with X n = V p/s (X Γn ) for every n ∈ N and I (t) = t ρ/2 for every t ∈ R + .

Proof. Before we start the proof, we assume that q = 0 and p = 0. Otherwise, the proof is simpler so we leave it out.

Let f ∈ D(A). We introduce decomposition (with notations (104))

Now we study ∆X M,3 n+1 . We distinguish two cases: ρ/2 q and q < ρ/2. First, let ρ/2 q. Then from Cauchy Schwartz inequality,we obtain

4 APPLICATIONS 48 Now if q < ρ/2, then since f is Lipschitz and dened on a compact set, it is also 2q/ρ-Hölder, and then for every x 0 ∈ R d , we have

Moreover, from Lemma 4.6 point B. and C., it follows that

and we conclude that

Now if p < ρ/2, then since f is Lipschitz and dened on a compact set, it is also 2p/ρ-Hölder, and then for every x 0 ∈ R d , we have

Now, using Lemma 4.6 point A. since we have H p , it follows that

and we conclude that

and gathering all the terms as in the the initial decomposition gives (138).

We focus now on the case f = V p/s . First, we assume that 2p/s 1. Let x, y ∈ R d . Then, the function from R + to R + such that y → y 2p/s is concave and since

√

V is Lipschitz, we deduce that

Now, since s(2/ρ -1)(a + p -1) + s -2 0 from SW pol (p, a, s, ρ) (see (139), and V takes values in [v * , ∞), we deduce that there exists C > 0 such that for every x ∈ R d , we have V ρp/s-ρ/2 (x) CV a(1-ρ/2)+p-1 (x). Using B p,q (φ) (see ( 112)), we obtain

Now, we study ∆X M,3 n+1 . First we consider the case 2q > s. Since ρ s (from SW pol (p, a, s, ρ) (see (139))), we have 2q > ρ and it follows from Cauchy Schwartz inequality and Lemma 4.6 point C., that

Now, we notice that since p/s 1/2 then V p/s is α-Hölder for any α ∈ [2p/s, 1] (see Lemma 3. in [START_REF] Panloup | Recursive computation of the invariant measure of a stochastic dierential equation driven by a lévy process[END_REF]). It follows that for 2q s, V p/s is 2(p ∨ q)/s-Hölder. Then, using Cauchy-Schwartz inequality and since ρ s from SW pol (p, a, s, ρ), we have

where the last inequality is a consequence of Lemma

We gather all the terms together and the proof is competed for the case 2p s. Now, we consider the case 2p > s. Using (29) with α = 2p/s, it follows that

We study ∆X M,3 n+1 . We recall that ρ s from SW pol (p, a, s, ρ) (see ( 139)) and then 2p ρ in this case. We distinguish the case q p and q > p.