# Total variation distance between a jump-equation and its Gaussian
approximation

Vlad Bally  Université Gustave Eiffel, LAMA (UMR CNRS, UPEMLV, UPEC), MathRisk
INRIA, F-77454 Marne-la-Vallée, France. Email address<EMAIL_ADDRESS>Yifeng Qin  Université Gustave Eiffel, LAMA (UMR CNRS, UPEMLV, UPEC), MathRisk
INRIA, F-77454 Marne-la-Vallée, France. Email address<EMAIL_ADDRESS>
Abstract

We deal with stochastic differential equations with jumps. In order to obtain
an accurate approximation scheme, it is usual to replace the "small jumps" by
a Brownian motion. In this paper, we prove that for every fixed time $t$, the
approximate random variable $X^{\varepsilon}_{t}$ converges to the original
random variable $X_{t}$ in total variation distance and we estimate the error.
We also give an estimate of the distance between the densities of the laws of
the two random variables. These are done by using some integration by parts
techniques in Malliavin calculus.

Key words: Stochastic differential equations with jumps, Malliavin calculus,
Total variation distance, Gaussian approximation

###### Contents

  1. 1 Introduction
  2. 2 Main results
    1. 2.1 The basic equation and the hypotheses
    2. 2.2 Approximation
    3. 2.3 The main theorem
    4. 2.4 A typical example
  3. 3 Abstract integration by parts framework
    1. 3.1 Main consequences: Convergence in total variation distance
  4. 4 Malliavin calculus and stochastic differential equations with jumps
    1. 4.1 The splitting method
    2. 4.2 Malliavin calculus for Poisson point measures and space-time Brownian motions
    3. 4.3 Malliavin calculus applied to stochastic differential equations with jumps
    4. 4.4 Proof of the main result (Theorem 2.2)
  5. 5 Appendix
    1. 5.1 Proof of Lemma 4.1
      1. 5.1.1 Construction of the Euler scheme
      2. 5.1.2 Preliminary estimates
      3. 5.1.3 Estimations of $\|X_{t}^{n,M}\|_{L,l,p}$
    2. 5.2 Proof of Lemma 4.2
    3. 5.3 Some proofs concerning Section 4.2

##  1 Introduction

In this paper we consider the stochastic differential equation with jumps

$X_{t}=x+\int_{0}^{t}\int_{(0,1]}c(s,z,X_{s-})N_{\mu}(ds,dz)$

where $N_{\mu}$ is a Poisson random measure on $(0,1]$ with compensator
$\mu(dz)ds$ and $c$ is a coefficient which verifies strong regularity
hypotheses (see Hypotheses 2.1-2.4 in Section 2.1). The typical example that
we have in mind is $\mu(dz)=\frac{dz}{z^{1+\rho}}1_{\\{z\in(0,1]\\}}$, with
$\rho\in[0,1),$ so this is a truncated stable process - however, throughout
the paper, we keep the general framework in which $\mu$ is a measure which has
infinite mass around zero. Our aim is to replace the "small jumps" by a space-
time Brownian motion:

$\displaystyle X_{t}^{\varepsilon}$ $\displaystyle=$ $\displaystyle
x+\int_{0}^{t}\int_{\\{z>\varepsilon\\}}c(s,z,X_{s-}^{\varepsilon})N_{\mu}(ds,dz)$
$\displaystyle+\int_{0}^{t}b_{\varepsilon}(s,X_{s}^{\varepsilon})ds+\int_{0}^{t}\int_{(0,\varepsilon]}c(s,z,X_{s}^{\varepsilon})W_{\mu}(ds,dz),$

where $W_{\mu}(ds,dz)$ is a space-time Brownian motion (in the sense of Walsh
[36]) with covariance $\mu(dz)ds$, $x\in\mathbb{R}$, and the coefficient
$b_{\varepsilon}$ is defined by

$b_{\varepsilon}(s,x)=\int_{(0,\varepsilon]}c(s,z,x)\mu(dz).$

The interest of such approximations appears in various frameworks.

Our main motivation comes from numerical computations. If $\mu(E)<\infty$ then
there are a finite number of jumps in any compact interval of time, so $X_{t}$
may be represented by means of a compound Poisson process which may be
explicitly simulated. But if $\mu(E)=\infty$ this is not possible anymore
(except in very particular situations - see Talay and Protter [34] for
example), and the "small jumps" should be truncated to revert to the case of a
finite measure. This procedure is rather rough and gives large errors. In
order to improve the approximation scheme, one may replace the "small jumps",
namely those smaller than $\varepsilon$, by a stochastic integral with respect
to $W_{\mu}(ds,dz)$. Note that the Poisson measure $dN_{\mu}$ is not
compensated, which is why the drift corresponding to $b_{\varepsilon}$
appears. This idea goes back to Asmussen and Rosinski [3]. In the case of
$SDE^{\prime}s$ driven by a Lévy process, Fournier [16] gives a precise
estimate of the error and compares the approximation obtained just by
truncating the small jumps to the one obtained by adding a Gaussian noise as
in (1). An enlightening discussion on the complexity of the two methods is
also given. However, in that paper, the strong error is considered, while in
our paper we discuss the weak error.

A second motivation comes from modelization problems in chemistry and biology:
we are concerned by reactions which are naturally modelled by means of jump
processes containing two regimes: one is very rapid but the jumps are small,
and another is much slower and the jumps are larger – see for example [1],
[2], [4], [13], [28], [29]. In this case the regime corresponding to the rapid
scale may be modelled by a stochastic integral with respect to a Gaussian
process and the slow regime by a compound Poisson process. It may also be
reasonable to consider an intermediary regime and this would be modelled by a
drift term.

A third motivation is given by a class of statistical problems (see [11], [15]
and references therein), where a stochastic process is observed at various
times and it should be decided whether its increments are due to small jumps
or to a Gaussian component. In this framework it is important to estimate the
error in total variation sense. The authors explain that, if the error in
total variation between the laws of $X_{t}$ and of $X_{t}^{\varepsilon}$ goes
to zero, then there is no way to construct a test which decides if the noise
comes from small jumps or from the Brownian motion. So, asymptotically, the
two models contain the same information.

Let us now discuss briefly our results and the relation to previously
available estimates. If $L_{t}$ (respectively $L^{\varepsilon}_{t}$)
represents the infinitesimal operator of $X_{t}$ (respectively of
$X^{\varepsilon}_{t})$ then a development in Taylor series of order two gives

$\left\|(L_{t}-L^{\varepsilon}_{t})f\right\|_{\infty}\leq
C\left\|f\right\|_{3,\infty}\int_{(0,\varepsilon]}\left|\hat{c}(z)\right|^{3}\mu(dz),$

where $\hat{c}(z):=\sup\limits_{s\leq
T}\sup\limits_{x\in\mathbb{R}}\left|c(s,z,x)\right|$ and
$\left\|f\right\|_{3,\infty}:=\sum\limits_{0\leq i\leq
3}\|f^{(i)}\|_{\infty}$. Then a Trotter-Kato type argument yields

$\sup_{s\leq T}\left\|(P_{t}-P_{t}^{\varepsilon})f\right\|_{\infty}\leq
C\left\|f\right\|_{3,\infty}\int_{(0,\varepsilon]}\left|\hat{c}(z)\right|^{3}\mu(dz)$

where $P_{t}$ (respectively of $P_{t}^{\varepsilon})$ represents the semigroup
of $X_{t}$ (respectively of $X^{\varepsilon}_{t}).$

The drawback of the above estimate is that the bound on the error involves
$\left\|f\right\|_{3,\infty}$, so it only applies to smooth test functions.
The main contribution of our paper is to replace $\left\|f\right\|_{3,\infty}$
by $\left\|f\right\|_{\infty}$, so as to prove convergence in total variation
distance. This is done under non-degeneracy and regularity assumptions on the
coefficient $c.$ Moreover, under these hypotheses, we prove that
$\mathbb{P}(X_{t}(x)\in dy)=p_{t}(x,y)dy$ and
$\mathbb{P}(X_{t}^{\varepsilon}(x)\in dy)=p_{t}^{\varepsilon}(x,y)dy$ with
smooth densities $y\mapsto p_{t}(x,y)$ and $y\mapsto
p_{t}^{\varepsilon}(x,y).$ And, for every $k$ and every $\delta>0$, we obtain

$\left\|\partial_{y}^{k}p_{t}-\partial_{y}^{k}p_{t}^{\varepsilon}\right\|_{\infty}\leq
C_{k,\delta}\Big{(}\int_{(0,\varepsilon]}\left|\hat{c}(z)\right|^{3}\mu(dz)\Big{)}^{1-\delta}.$

This proves that $p_{t}^{\varepsilon}$ converges to $p_{t}$ in distribution
norms as $\epsilon\to 0$.

Our approach uses a strategy based on integration by parts (an abstract
Malliavin calculus) developed in [6].

The paper is organized as follows. In Section 2, we give the main results and
in Section 3, we recall the integration by parts technique introduced in [6]
and used here. In Section 4, we use these results in the framework of
stochastic equations with jumps and we prove the main result (Theorem 2.2).
The Appendix contains technical estimates concerning Sobolev norms in
Malliavin sense.

##  2 Main results

###  2.1 The basic equation and the hypotheses

A time horizon $T>0$ will be fixed throughout the paper. As already mentioned,
we deal with the one-dimensional jump equation

$X_{t}=x+\int_{0}^{t}\int_{(0,1]}c(s,z,X_{s-})N_{\mu}(ds,dz),$ (2)

where $N_{\mu}$ is a Poisson point measure with intensity
$\widehat{N_{\mu}}(ds,dz)=\mu(dz)ds$, and $\mu$ is a positive $\sigma$-finite
measure on $(0,1]$, $t\in[0,T]$.

For technical reasons which will be discussed in Section 4, we introduce the
following change of variables. Let $\theta:(0,1]\rightarrow[1,\infty)$ be the
function defined by $\theta(z)=\frac{1}{z}$, and let
$\nu(dz):=\mu\circ\theta^{-1}(dz)$. Then $\nu$ is a positive $\sigma$-finite
measure on $[1,\infty)$. Consider a Poisson point measure $N_{\nu}(ds,dz)$
with intensity $\widehat{N_{\nu}}(ds,dz)=\nu(dz)ds$. One may then check that
for every $t\in[0,T]$, $X_{t}$ has the same law as $\widehat{X}_{t}$, with
$(\widehat{X}_{t})_{t\in[0,T]}$ the solution of

$\displaystyle\widehat{X}_{t}=x+\int_{0}^{t}\int_{[1,\infty)}\widetilde{c}(s,z,\widehat{X}_{s-})N_{\nu}(ds,dz),$
(3)

where $\widetilde{c}(s,z,x):=c(s,\frac{1}{z},x)$.

Since this paper deals with the laws of the solution to (2),it is equivalent
to consider the equation (3). We formulate our hypotheses in terms of
$\widetilde{c}$ and $\nu$ (instead of $c$ and $\mu$).

Hypothesis 2.1 (Regularity with parameter $q^{\ast}$) The map
$(s,z)\mapsto\widetilde{c}(s,z,x)$ is continuous, and there exists a non-
negative and decreasing function $\bar{c}:[1,\infty)\rightarrow\mathbb{R}_{+}$
and a constant $q^{\ast}\in\mathbb{N}$ such that for every indices
$\beta_{1},\beta_{2}$, with $\beta_{1}\leq q^{\ast}$ and $\beta_{2}\leq
q^{\ast}$, we have

$\sup_{s\in[0,T]}\sup_{x\in\mathbb{R}}(|\widetilde{c}(s,z,x)|+|\partial_{z}^{\beta_{2}}\partial_{x}^{\beta_{1}}\widetilde{c}(s,z,x)|)\leq\bar{c}(z),\quad\forall
z\in[1,\infty),$

with

$\displaystyle\int_{[1,\infty)}|\bar{c}(z)|^{p}\nu(dz)=:\bar{c}_{p}<\infty,\quad\forall
p\geq 1.$ (4)

######  Remark.

We will use several times the following consequence of (4) and of Burkholder’s
inequality (see for example the Theorem 2.11 in [26]): We assume that
$\Phi(s,z,\omega)$ and $\varphi(s,\omega)$ are two non-negative functions such
that

$|\Phi(s,z,\omega)|\leq\bar{c}(z)\varphi(s,\omega).$

Then for any $p\geq 2$,

$\displaystyle\mathbb{E}\Big{|}\int_{0}^{T}\int_{[1,\infty)}\Phi(s,z,\omega)N_{\nu}(ds,dz)\Big{|}^{p}\leq
C\times C_{p}\,\mathbb{E}\int_{0}^{T}|\varphi(s,\omega)|^{p}ds,$ (5)

where
$C_{p}=max\\{(\bar{c}_{2})^{\frac{p}{2}},\bar{c}_{p},(\bar{c}_{1})^{p}\\}$ and
$C$ is a constant depending on $p$ and $T$.

###### Proof.

By compensating $N_{\nu}$, using Burkholder’s inequality and (4), we have

$\displaystyle\mathbb{E}|\int_{0}^{T}\int_{[1,\infty)}\Phi(s,z,\omega)N_{\nu}(ds,dz)|^{p}\leq
C(\mathbb{E}|\int_{0}^{T}\int_{[1,\infty)}\Phi(s,z,\omega)\widetilde{N_{\nu}}(ds,dz)|^{p}+\mathbb{E}|\int_{0}^{T}\int_{[1,\infty)}\Phi(s,z,\omega)\nu(dz)ds|^{p})$
$\displaystyle\leq
C(\mathbb{E}\int_{0}^{T}(\int_{[1,\infty)}|\Phi(s,z,\omega)|^{2}\nu(dz))^{\frac{p}{2}}ds+\mathbb{E}\int_{0}^{T}\int_{[1,\infty)}|\Phi(s,z,\omega)|^{p}\nu(dz)ds+\mathbb{E}\int_{0}^{T}|\int_{[1,\infty)}\Phi(s,z,\omega)\nu(dz)|^{p}ds)$
$\displaystyle\leq C\times
C_{p}\mathbb{E}\int_{0}^{T}|\varphi(s,\omega)|^{p}ds.$

∎

Hypothesis 2.2 There exists a non-negative function
$\breve{c}:[1,\infty)\rightarrow\mathbb{R}_{+}$ such that
$\int_{[1,\infty)}|\breve{c}(z)|^{p}\nu(dz)=:\breve{c}_{p}<\infty,\quad\forall
p\geq 1$, and

$\Big{|}\frac{\partial_{x}\widetilde{c}(s,z,x)}{1+\partial_{x}\widetilde{c}(s,z,x)}\Big{|}\leq\breve{c}(z),\quad\forall
s\in[0,T],x\in\mathbb{R},z\in[1,\infty).$

To avoid overburdening notation, since both hypotheses 2.1 and 2.2 apply, we
will take $\breve{c}(z)=\bar{c}(z)$.

Hypothesis 2.3 (Ellipticity) There exists a non-negative function
$\underline{c}:[1,\infty)\rightarrow\mathbb{R}_{+}$ such that for every
$s\in[0,T],x\in\mathbb{R},z\in[1,\infty)$,

$\left|\partial_{z}\widetilde{c}(s,z,x)\right|^{2}\geq\underline{c}(z)\quad
and\quad\left|\widetilde{c}(s,z,x)\right|^{2}\geq\underline{c}(z).$

Hypothesis 2.4 (Sector condition) This is a supplementary hypothesis
concerning the measure $\nu$. Two version of this hypothesis will be used; we
state them separately below. Let $I_{k}=[k,k+1),k\in\mathbb{N}$ and
$m_{k}=\nu(I_{k})$.

$(a)$ Strong sector condition: We say that the strong sector condition is
satisfied if there exist constants $\varepsilon_{\ast}>0$ and
$\alpha_{1}\geq\alpha_{0}>\alpha_{2}>0$, such that

$\displaystyle\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}$ $\displaystyle\geq$
$\displaystyle\mathbbm{1}_{I_{k}}(z)\frac{\varepsilon_{\ast}}{z^{1-\alpha_{1}}}dz\quad\
\ \text{for all $k\in\mathbb{N}$},$ (6) $\displaystyle\underline{c}(z)$
$\displaystyle\geq$ $\displaystyle e^{-z^{\alpha_{2}}}\qquad\quad\
\qquad\text{ for all $z\geq 1$ and,}$
$\displaystyle\int_{1}^{\infty}\frac{|\bar{c}(z)|^{p}}{z^{1-\alpha_{0}}}dz$
$\displaystyle<$ $\displaystyle\infty\qquad\qquad\qquad\quad\text{for all
$p\geq 1$}.$ (7)

Notice that if (6) is true for some $\alpha_{1}$, then it is also true for any
${\alpha}\leq\alpha_{1}$. So (6) also implies

$\displaystyle\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}\geq\mathbbm{1}_{I_{k}}(z)\varepsilon_{k}dz,\quad
with\quad\varepsilon_{k}=\frac{\varepsilon_{\ast}}{(k+1)^{1-{\alpha}}},$

for any ${\alpha}\leq\alpha_{1}$.

$(b)$ Weak sector condition: We say that the weak sector condition holds if
there exist constants $\varepsilon_{\ast}>0$ and $\alpha>0$, such that for
every $k\in\mathbb{N}$, we have

$\displaystyle\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}$ $\displaystyle\geq$
$\displaystyle\mathbbm{1}_{I_{k}}(z)\frac{\varepsilon_{\ast}}{z}dz\qquad\quad\text{for
all $k\in\mathbb{N}$},$ (8) $\displaystyle\underline{c}(z)$
$\displaystyle\geq$ $\displaystyle\frac{1}{z^{\alpha}}\qquad\qquad\
\qquad\text{ for all $z\geq 1$ and},$
$\displaystyle\int_{1}^{\infty}\frac{|\bar{c}(z)|^{p}}{z}dz$ $\displaystyle<$
$\displaystyle\infty\qquad\qquad\qquad\ \ \text{ for all $p\geq 1$}.$ (9)

We notice that (8) also implies
$\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}\geq\mathbbm{1}_{I_{k}}(z)\varepsilon_{k}dz$
with $\varepsilon_{k}=\frac{\varepsilon_{\ast}}{k+1}.$

######  Remark.

Notice that hypotheses 2.1, 2.2 and 2.3 are analogous to those in [10]
$(2-7,2-26,2-24)$.

Henceforth, we will suppose that hypotheses 2.1-2.3 hold, as well as either
2.4$(a)$ or 2.4$(b)$.

###  2.2 Approximation

We come back now to equation (2). The goal of this paper is to replace the
small jumps in (2) by a drift and a Brownian motion. In equation (2), the
Poisson point measure $N_{\mu}$ is not compensated, so the first step is to
introduce a drift (see $b_{\varepsilon}$ below) which represents the
compensator. Afterwards, we introduce a space-time Brownian motion $W_{\mu}$
in order to replace the "compensated small jumps":

$\displaystyle X_{t}^{\varepsilon}$ $\displaystyle=$ $\displaystyle
x+\int_{0}^{t}\int_{\\{z>\varepsilon\\}}c(s,z,X_{s-}^{\varepsilon})N_{\mu}(ds,dz)$
(10)
$\displaystyle+\int_{0}^{t}b_{\varepsilon}(s,X_{s}^{\varepsilon})ds+\int_{0}^{t}\int_{(0,\varepsilon]}c(s,z,X_{s}^{\varepsilon})W_{\mu}(dz,ds),$

where

$\displaystyle b_{\varepsilon}(s,x)=\int_{(0,\varepsilon]}c(s,z,x)\mu(dz)$

and $W_{\mu}$ is a space-time Brownian motion with covariance measure
$\mu(dz)ds$, which is independent of $N_{\mu}$.

Let us discuss this equation. We notice that we keep the "big jumps" with
$z>\varepsilon$ but we eliminate the "small jumps" with $z\leq\varepsilon.$ We
replace the "small jumps" by the drift with coefficient $b_{\varepsilon}$ and
by the stochastic integral with coefficient $c.$ This stochastic integral is
driven by the so called space-time Brownian motion $W_{\mu}$, as introduced by
Walsh in [36]. The existence and uniqueness of the solution to this equation
(10) are also given by Kunita (see
$\cite[cite]{[\@@bibref{}{ref20}{}{}]},\cite[cite]{[\@@bibref{}{ref13}{}{}]}$).

We recall that we work on a fixed interval of time $[0,T]$. We now precise the
filtration that we consider. Let

$\displaystyle\mathcal{F}_{t}^{W}=\sigma(W_{\mu}(\varphi\mathbbm{1}_{[0,t]}):\
\varphi\in L^{2}((0,1]\times[0,T],\mu\times Leb)),$
$\displaystyle\mathcal{F}_{t}^{N}=\sigma(N_{\mu}(\varphi\mathbbm{1}_{[0,t]}):\
\varphi\in L^{1}((0,1]\times[0,T],\mu\times Leb)),$
$\displaystyle\mathcal{F}_{t}=\mathcal{F}_{t}^{W}\bigvee\mathcal{F}_{t}^{N},$
(11)

where $Leb$ denotes the Lebesgue measure and

$W_{\mu}(\varphi)=\int_{0}^{T}\int_{(0,1]}\varphi(s,z)W_{\mu}(ds,dz),\quad
N_{\mu}(\varphi)=\int_{0}^{T}\int_{(0,1]}\varphi(s,z)N_{\mu}(ds,dz).$

So, $X_{t}^{\varepsilon}$ is $\mathcal{F}_{t}-$measurable and $X_{t}$ is
$\mathcal{F}_{t}^{N}-$measurable.

We denote

$\displaystyle
L^{2}(W)=\\{F\in\mathcal{F}_{T}^{W}:\mathbb{E}|F|^{2}<\infty\\},\quad
L^{2}(N)=\\{G\in\mathcal{F}_{T}^{N}:\mathbb{E}|G|^{2}<\infty\\}.$ (12)

######  Remark.

Let $\Phi$ be an adapted and piecewise constant process, that is

$\Phi(s,z,\omega)=\sum_{i=1}^{n}\sum_{j=1}^{m}\Phi_{i,j}(\omega)\mathbbm{1}_{[s_{i},s_{i+1})}(s)\mathbbm{1}_{B_{j}}(z),$

where $0\leq s_{1}<\cdots<s_{n},i=1,\cdots,n,$
$B_{j}\in\mathcal{B}((0,1]),j=1,\cdots,m,$ are disjoint sets. Suppose that
$\Phi_{i,j}$ are $\mathcal{F}^{W}_{s_{i}}-$measurable for all $j=1,\cdots,m$,
and $\sup\limits_{i,j}\mathbb{E}|\Phi_{i,j}|^{2}<\infty.$ Then for every $G\in
L^{2}(N)$, we have

$\displaystyle\mathbb{E}\Big{[}G\times\int_{0}^{T}\int_{(0,1]}\Phi(s,z,\omega)W_{\mu}(ds,dz)\Big{]}=0.$
(13)

###### Proof.

Since $W_{\mu}([s_{i},s_{i+1})\times B_{j})$ is centered and independent of
$\Phi_{i,j}$ and of $G$, for all $i=1,\cdots,n,\ j=1,\cdots,m$, it follows
that $\mathbb{E}[G\Phi_{i,j}W_{\mu}([s_{i},s_{i+1})\times B_{j})]=0$. Then it
extends by linearity, and so (13) is true. ∎

Now we write the infinitesimal operator of $X_{s}$ and $X_{s}^{\varepsilon}$,
respectively: For $\phi\in C_{b}^{3}(\mathbb{R})$ (the space of functions with
continuous and bounded derivatives up to order 3),

$\displaystyle L_{s}\phi(x)$ $\displaystyle=$
$\displaystyle\int_{(0,1]}(\phi(x+c(s,z,x))-\phi(x))\mu(dz)\qquad\text{ and}$
$\displaystyle L_{s}^{\varepsilon}\phi(x)$ $\displaystyle=$
$\displaystyle\int_{\\{z>\varepsilon\\}}(\phi(x+c(s,z,x))-\phi(x))\mu(dz)+\phi^{\prime}(x)b_{\varepsilon}(s,x)+\tfrac{1}{2}\phi^{\prime\prime}(x)a_{\varepsilon}(s,x),$
(14)

where

$a_{\varepsilon}(s,x)=\int_{(0,\varepsilon]}|c(s,z,x)|^{2}\mu(dz).$

Using Taylor’s formula of order 2, we find

$\displaystyle
L_{s}\phi(x)=\int_{\\{z>\varepsilon\\}}(\phi(x+c(s,z,x))-\phi(x))\mu(dz)+\phi^{\prime}(x)b_{\varepsilon}(s,x)+\tfrac{1}{2}\phi^{\prime\prime}(x)a_{\varepsilon}(s,x)+R_{s}(x),$

where

$\displaystyle|R_{s}(x)|$ $\displaystyle\leq$
$\displaystyle\tfrac{1}{6}\|\phi\|_{3,\infty}\int_{(0,\varepsilon]}|c(s,z,x)|^{3}\mu(dz),$

with $\left\|\phi\right\|_{l,\infty}:=\sum\limits_{0\leq i\leq
l}\|\phi^{(i)}\|_{\infty}$, the sum of all the uniform norms of the
derivatives of function $\phi$ up to order $l$. In conclusion, we find

$\displaystyle\left\|(L_{s}-L_{s}^{\varepsilon})\phi\right\|_{\infty}=\left\|R_{s}\right\|_{\infty}\leq\tfrac{1}{6}\left\|\phi\right\|_{3,\infty}\eta_{3}(\varepsilon),$
(15)

with

$\eta_{p}(\varepsilon)=\int_{(0,\varepsilon]}\left|\bar{c}(1/z)\right|^{p}\mu(dz)=\int_{[\varepsilon^{-1},\infty)}\left|\bar{c}(z)\right|^{p}\nu(dz),\quad
p\geq 1.$ (16)

Then, we can give an estimate of the distance between the semigroups. We use
the standard semigroup notation, which we remind below. Let
$[X_{t}(s,x)]_{t\geq s}$ and $[X_{t}^{\varepsilon}(s,x)]_{t\geq s}$ be the
solutions to (2) and (10), respectively, starting at time $s$ from point $x$.
Denote by $P_{s,t}\phi(x)=\mathbb{E}\phi(X_{t}(s,x))$ and
$P_{s,t}^{\varepsilon}\phi(x)=\mathbb{E}\phi(X_{t}^{\varepsilon}(s,x))$. Also,
set $P_{t}:=P_{0,t}$ and $P_{t}^{\varepsilon}:=P_{0,t}^{\varepsilon}$.

######  Lemma 2.1.

There exists a constant $C$ depending on $T$ such that for $\phi\in
C_{b}^{3}(\mathbb{R})$ and $0\leq t\leq T$, we have

$\left\|P_{t}\phi-P_{t}^{\varepsilon}\phi\right\|_{\infty}\leq
C\left\|\phi\right\|_{3,\infty}\eta_{3}(\varepsilon).$ (17)

###### Proof.

Step 1 Trotter-Kato method: We know from Kunita [27] (Theorem 4.5.1) that we
have the Kolmogorov forward and backward equations:

$\displaystyle\partial_{t}P_{s,t}\phi(x)=P_{s,t}L_{t}\phi(x),\quad\quad\partial_{t}P_{s,t}^{\varepsilon}\phi(x)=P_{s,t}^{\varepsilon}L_{t}^{\varepsilon}\phi(x);$
(18)
$\displaystyle\partial_{s}P_{s,t}\phi(x)=-L_{s}P_{s,t}\phi(x),\quad\partial_{s}P_{s,t}^{\varepsilon}\phi(x)=-L_{s}^{\varepsilon}P_{s,t}^{\varepsilon}\phi(x).$
(19)

Then using Newton-Leibniz’s formula and (18), (19),

$P_{t}^{\varepsilon}\phi(x)-P_{t}\phi(x)=\int_{0}^{t}\partial_{s}(P_{0,s}^{\varepsilon}P_{s,t})\phi(x)ds=\int_{0}^{t}(P_{0,s}^{\varepsilon}(L_{s}^{\varepsilon}-L_{s})P_{s,t})\phi(x)ds.$

It follows that

$\displaystyle\left\|P_{t}\phi-P_{t}^{\varepsilon}\phi\right\|_{\infty}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}\left\|P_{0,s}^{\varepsilon}(L_{s}^{\varepsilon}-L_{s})P_{s,t}\phi\right\|_{\infty}ds$
(20) $\displaystyle\leq$
$\displaystyle\int_{0}^{t}\left\|(L_{s}^{\varepsilon}-L_{s})P_{s,t}\phi\right\|_{\infty}ds$
$\displaystyle\leq$
$\displaystyle\tfrac{1}{6}\,\eta_{3}(\varepsilon)\int_{0}^{t}\left\|P_{s,t}\phi\right\|_{3,\infty}ds.$

Step 2 (propagation of regularity)  In [27], Kunita has shown in Theorem 3.4.1
and Theorem 3.4.2 the regularity of the flow associated with the jump-
diffusion. So in our case, we have

$\displaystyle\left\|P_{s,t}\phi\right\|_{3,\infty}$ $\displaystyle\leq$
$\displaystyle\sup\limits_{x\in\mathbb{R}}(\mathbb{E}|\phi(X_{t}(s,x))|+\mathbb{E}|\partial_{x}\phi(X_{t}(s,x))|+\mathbb{E}|\partial_{x}^{2}\phi(X_{t}(s,x))|+\mathbb{E}|\partial_{x}^{3}\phi(X_{t}(s,x))|)$
(21) $\displaystyle\leq$
$\displaystyle\left\|\phi\right\|_{3,\infty}\sup\limits_{x\in\mathbb{R}}\mathbb{E}[1+3|\partial_{x}X_{t}(s,x)|+3|\partial_{x}^{2}X_{t}(s,x)|+|\partial_{x}^{3}X_{t}(s,x)|]\leq{C}\left\|\phi\right\|_{3,\infty}.$

Substituting (21) into (20), we obtain (17). ∎

######  Remark.

A similar result has been obtained in [21] (Theorem 4.7). Besides, one may
also consider an approximate equation obtained just by discarding the small
jumps:

$\widetilde{X_{t}^{\varepsilon}}=x+\int_{0}^{t}\int_{\\{z>\varepsilon\\}}c(s,z,\widetilde{X_{s-}^{\varepsilon}})N_{\mu}(ds,dz).$

Then, if $\widetilde{L_{s}^{\varepsilon}}$ is the infinitesimal operator of
$\widetilde{X_{s}^{\varepsilon}}$, we have
$\|(L_{s}-\widetilde{L_{s}^{\varepsilon}})\phi\|_{\infty}\leq\|\phi\|_{1,\infty}\eta_{1}(\varepsilon)$.
So the same reasoning as above gives

$\big{\|}P_{t}\phi-\widetilde{P_{t}^{\varepsilon}}\phi\big{\|}_{\infty}\leq
C\left\|\phi\right\|_{1,\infty}\times\eta_{1}(\varepsilon)\rightarrow 0.$ (22)

The gain in (17) is that we have $\eta_{3}(\varepsilon)$ instead of
$\eta_{1}(\varepsilon)$ in (22), which means that we have a faster speed of
convergence.

###  2.3 The main theorem

We are finally ready to state the main results of this paper. Denote by
$d_{TV}(F,G)$ the total variation distance between the laws of two random
variables $F$ and $G$.

######  Theorem 2.2.

Assume that Hypotheses 2.1, 2.2. and 2.3 hold with
$q^{\ast}\geq\frac{3}{\delta}+1$ for some $\delta>0$.

(a) If in addition we assume Hypothesis 2.4 (a), then there exists a constant
$C$ depending on $\delta$ and $T$ such that

$\displaystyle d_{TV}(X_{t},X_{t}^{\varepsilon})\leq
C\eta_{3}(\varepsilon)^{1-\delta}.$ (23)

Under the above hypotheses, the laws of $X_{t}$ and $X_{t}^{\varepsilon}$ are
absolutely continuous with respect to the Lebesgue measure, with smooth
densities $p_{X_{t}}(x)$ and $p_{X_{t}^{\varepsilon}}(x)$. Moreover, if $l$ is
an index such that $q^{*}\geq\frac{3+l}{\delta}+1$, then there exists a
constant ${C}$ depending on $\delta$, $T$ and $l$ such that

$\displaystyle\|p_{X_{t}}-p_{X_{t}^{\varepsilon}}\|_{l,\infty}\leq{C}\eta_{3}(\varepsilon)^{1-\delta}.$
(24)

(b) If in addition we assume Hypothesis 2.4 (b), then there exists a constant
$C$ depending on $\delta$ and $T$ such that for every $t\in[0,T]$ with
$t>\frac{8\alpha(\frac{3}{\delta}-1)}{\varepsilon_{\ast}}$ (with
$\varepsilon_{\ast}$ and $\alpha$ given in Hypothesis 2.4 (b)), we have

$\displaystyle d_{TV}(X_{t},X_{t}^{\varepsilon})\leq
C\eta_{3}(\varepsilon)^{1-\delta}.$ (25)

For any index $l$ and for $t>\frac{8\alpha(3l+2)}{\varepsilon_{\ast}}$, both
the laws of $X_{t}$ and $X_{t}^{\varepsilon}$ have $l$-times differentiable
densities $p_{X_{t}}(x)$ and $p_{X_{t}^{\varepsilon}}(x)$. Assume moreover
that $q^{\ast}\geq\frac{3+l}{\delta}+1$. Then there exists a constant ${C}$
depending on $\delta$ ,$T$ and $l$ such that for
$t>\max\\{\frac{8\alpha}{\varepsilon_{\ast}}(\frac{3+l}{\delta}-1),\frac{8\alpha(3l+2)}{\varepsilon_{\ast}}\\}$,
we have

$\displaystyle\|p_{X_{t}}-p_{X_{t}^{\varepsilon}}\|_{l,\infty}\leq{C}\eta_{3}(\varepsilon)^{1-\delta}.$
(26)

The proof of this theorem is left to Section 4.4.

######  Remark.

Some recent results concerning the weak approximation of the SDE with jumps
are also given in
$\cite[cite]{[\@@bibref{}{ref39}{}{}]},\cite[cite]{[\@@bibref{}{ref40}{}{}]},\cite[cite]{[\@@bibref{}{ref41}{}{}]}$
for example. But they do not concern the convergence in total variation
distance.

###  2.4 A typical example

For $t\in[0,T]$, we consider the following SDE driven by a Lévy process:

$\displaystyle X_{t}=x+\int_{0}^{t}\sigma(X_{s-})dZ_{s},$ (27)

where $(Z_{t})_{t\in[0,T]}$ is a Lévy process of Lévy triplet $(0,0,\mu)$,
with $\mu(dz)=\mathbbm{1}_{(0,1]}(z)\frac{dz}{z^{1+\rho}}$, $0\leq\rho<1$.

We approximate (27) by

$\displaystyle
X_{t}^{\varepsilon}=x+\int_{0}^{t}\sigma(X_{s-}^{\varepsilon})dZ_{s}^{\varepsilon}+b(\varepsilon)\int_{0}^{t}\sigma(X_{s}^{\varepsilon})ds+c(\varepsilon)\int_{0}^{t}\sigma(X_{s}^{\varepsilon})dB_{s},$
(28)

where $(Z_{t}^{\varepsilon})_{t\in[0,T]}$ is a Lévy process of Lévy triplet
$(0,0,\mathbbm{1}_{\\{z>\varepsilon\\}}\mu(dz))$, $(B_{t})_{t\in[0,T]}$ is a
standard Brownian motion independent of $(Z_{t}^{\varepsilon})_{t\in[0,T]}$,
and

$b(\varepsilon)=\int_{(0,\varepsilon]}z\mu(dz),\quad
c(\varepsilon)=\sqrt{\int_{(0,\varepsilon]}z^{2}\mu(dz)}.$

Then we have the following theorem.

######  Theorem 2.3.

We assume that $\sigma\in C_{b}^{\infty}(\mathbb{R})$,
$0<\underline{\sigma}\leq\sigma(x)\leq\bar{\sigma}$ and
$-1<a\leq\sigma^{\prime}(x)\leq\bar{\sigma},\ \forall x\in\mathbb{R}$, for
some universal constants $\bar{\sigma},\underline{\sigma},a$, where
$\sigma^{\prime}$ is the differential of $\sigma$ in $x$. Then for any
$\delta>0$, there is a constant $C>0$ such that for any $t\in[0,T]$,

$d_{TV}(X_{t},X_{t}^{\varepsilon})\leq C\varepsilon^{3-\rho-\delta}.$

Moreover, the laws of $X_{t}$ and $X_{t}^{\varepsilon}$ have smooth densities
$p_{X_{t}}(x)$ and $p_{X_{t}^{\varepsilon}}(x)$ respectively. And for any
index $l$ and any $\delta>0$, there exists a constant $C>0$ such that

$\|p_{X_{t}}-p_{X_{t}^{\varepsilon}}\|_{l,\infty}\leq
C\varepsilon^{3-\rho-\delta}.$

###### Proof.

We notice that

$Z_{t}=\int_{0}^{t}\int_{(0,1]}zN_{\mu}(ds,dz),$

where $N_{\mu}$ is a Poisson point measure with intensity $\mu(dz)ds$. Then
(27) coincides with (2) with $c(s,z,x)=\sigma(x)z$, and (28) coincides with
(10) with $c(s,z,x)=\sigma(x)z$,
$b_{\varepsilon}(s,x)=b(\varepsilon)\sigma(x)$, and
$\int_{\\{z\leq\varepsilon\\}}zW_{\mu}(ds,dz)=c(\varepsilon)dB_{s}$.

Let $\theta:(0,1]\rightarrow[1,\infty)$ be a function defined by
$\theta(z)=\frac{1}{z}$. By a change of variables,

$\widetilde{c}(s,z,x)=c(s,\frac{1}{z},x)=\sigma(x)\times\frac{1}{z},\quad\nu(dz)=\mu\circ\theta^{-1}(dz)=\mathbbm{1}_{[1,\infty)}(z)\frac{dz}{z^{1-\rho}}.$

One can easily check that Hypothesis 2.1 is verified (for every
$q^{\ast}\in\mathbb{N}$) with $\bar{c}(z)=\bar{\sigma}\times\frac{1}{z}$ and

$\int_{1}^{\infty}|\bar{c}(z)|^{p}\nu(dz)=\int_{1}^{\infty}\frac{\bar{\sigma}^{p}}{z^{p+1-\rho}}dz<\infty,\
\forall p\geq 1.$

We recall that $I_{k}=[k,k+1),k\in\mathbb{N}$ and $m_{k}=\nu(I_{k})$. Then for
sufficiently large $z$, we have

$\min\\{|\partial_{z}\widetilde{c}(s,z,x)|^{2},|\widetilde{c}(s,z,x)|^{2}\\}\geq\underline{\sigma}^{2}\times\frac{1}{z^{4}}\geq
e^{-z^{\alpha_{2}}},$

with some $0<\alpha_{2}<1$. We also have

$\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}\geq\mathbbm{1}_{I_{k}}(z)\frac{1}{2}dz\geq\mathbbm{1}_{I_{k}}(z)\frac{1}{2}\frac{dz}{z^{1-\alpha_{0}}},$

with some $\alpha_{0}\in(\alpha_{2},1)$. Moreover, since for any $p\geq 1$,
$p+1-\alpha_{0}>1$, we have

$\int_{1}^{\infty}\frac{|\bar{c}(z)|^{p}}{z^{1-\alpha_{0}}}dz=\bar{\sigma}\int_{1}^{\infty}\frac{1}{z^{p+1-\alpha_{0}}}dz<\infty.$

So Hypothesis 2.3 and Hypothesis 2.4 $(a)$ are satisfied. Finally,

$\big{|}\frac{\partial_{x}\widetilde{c}(s,z,x)}{1+\partial_{x}\widetilde{c}(s,z,x)}\big{|}\leq\frac{\bar{\sigma}\times\frac{1}{z}}{1+a\times\frac{1}{z}}\leq\max\\{\frac{1}{1+a},1\\}\times\bar{\sigma}\times\frac{1}{z},$

so Hypothesis 2.2 is satisfied as well. Then we can apply Theorem 2.2$(a)$ for
the equation (27) and (28). Since

$\eta_{3}(\varepsilon)=\int_{(0,\varepsilon]}\bar{\sigma}^{3}\times
z^{3}\mu(dz)=\frac{\bar{\sigma}^{3}}{3-\rho}\varepsilon^{3-\rho},$

we obtain the estimates from Theorem 2.3. ∎

##  3 Abstract integration by parts framework

In order to obtain the main theorem (Theorem 2.2), we will apply some
techniques of Malliavin calculus. So firstly, we give the abstract integration
by parts framework introduced in [6]. This is a variant of the integration by
parts framework given in [10].

We consider a probability space ($\Omega$,$\mathcal{F}$,$\mathbb{P}$), and a
subset
$\mathcal{S}\subset\mathop{\bigcap}\limits_{p=1}^{\infty}L^{p}(\Omega;\mathbb{R})$
such that for every $\phi\in C_{p}^{\infty}(\mathbb{R}^{d})$ and every
$F\in\mathcal{S}^{d}$, we have $\phi(F)\in\mathcal{S}$ (with $C_{p}^{\infty}$
the space of smooth functions which, together with all the derivatives, have
polynomial growth). A typical example of $\mathcal{S}$ is the space of simple
functionals, as in the standard Malliavin calculus. Another example is the
space of "Malliavin smooth functionals".

Given a separable Hilbert space $\mathcal{H}$, we assume that we have a
derivative operator
$D:\mathcal{S}\rightarrow\mathop{\bigcap}\limits_{p=1}^{\infty}L^{p}(\Omega;\mathcal{H})$
which is a linear application which satisfies

$a)$

$\displaystyle D_{h}F:=\langle DF,h\rangle_{\mathcal{H}}\in\mathcal{S},\ for\
any\ h\in\mathcal{H},$ (29)

$b)$ $\underline{Chain\ Rule}$: For every $\phi\in
C_{p}^{\infty}(\mathbb{R}^{d})$ and
$F=(F_{1},\cdots,F_{d})\in\mathcal{S}^{d}$, we have

$\displaystyle D\phi(F)=\sum_{i=1}^{d}\partial_{i}\phi(F)DF_{i},$ (30)

Since $D_{h}F\in\mathcal{S}$, we may define by iteration the derivative
operator of higher order
$D^{q}:\mathcal{S}\rightarrow\mathop{\bigcap}\limits_{p=1}^{\infty}L^{p}(\Omega;\mathcal{H}^{\otimes
q})$ which verifies $\langle
D^{q}F,\otimes_{i=1}^{q}h_{i}\rangle_{\mathcal{H}^{\otimes
q}}=D_{h_{q}}D_{h_{q-1}}\cdots D_{h_{1}}F$. We also denote
$D_{h_{1},\cdots,h_{q}}^{q}F:=D_{h_{q}}D_{h_{q-1}}\cdots D_{h_{1}}F$. Then,
$D_{h_{1},\cdots,h_{q}}^{q}F=$ $D_{h_{q}}D_{h_{1},\cdots,h_{q-1}}^{q-1}F$
($q\geq 2$).

For $F=(F_{1},\cdots,F_{\tilde{d}})\in\mathcal{S}^{\tilde{d}}$, we define
$\sigma_{F}=(\sigma_{F}^{i,j})_{i,j=1,\cdots,{\tilde{d}}}$ to be the Malliavin
covariance matrix with $\sigma_{F}^{i,j}=\langle
DF_{i},DF_{j}\rangle_{\mathcal{H}}$ and we denote

$\displaystyle\Sigma_{p}(F)=\mathbb{E}(1/\det\sigma_{F})^{p}.$ (31)

For ${\tilde{d}}=1$, which is the case that we discuss in this paper,
$\det\sigma_{F}=\sigma_{F}=\langle DF,DF\rangle_{\mathcal{H}}$. We say that
$F$ is non-degenerated if $\Sigma_{p}(F)<\infty$, $\forall p\geq 1$.

We also assume that we have an Ornstein-Uhlenbeck operator
$L:\mathcal{S}\rightarrow\mathcal{S}$ which is a linear operator satisfying
the following duality formula:
$\underline{Duality}$: For every $F,G\in\mathcal{S}$,

$\displaystyle\mathbb{E}\langle
DF,DG\rangle_{\mathcal{H}}=\mathbb{E}(FLG)=\mathbb{E}(GLF).$ (32)

As an immediate consequence of the duality formula, we know that
$L:\mathcal{S}\subset L^{2}(\Omega)\rightarrow L^{2}(\Omega)$ is closable.

######  Definition 3.1.

If $D^{q}:\mathcal{S}\subset L^{2}(\Omega)\rightarrow
L^{2}(\Omega;\mathcal{H}^{\otimes q})$, $\forall q\geq 1$, are closable, then
the triplet $(\mathcal{S},D,L)$ will be called an IbP (Integration by Parts)
framework.

Now, we introduce the Sobolev norms. For any $l\geq 1$, $F\in\mathcal{S}$,

$\displaystyle\left|F\right|_{1,l}$ $\displaystyle=$
$\displaystyle\sum_{q=1}^{l}\left|D^{q}F\right|_{\mathcal{H}^{\otimes
q}},\quad\left|F\right|_{l}=\left|F\right|+\left|F\right|_{1,l}.$ (33)

We remark that $|F|_{0}=|F|$ and $|F|_{1,l}=0$ for $l=0$. For
$F=(F_{1},\cdots,F_{d})\in\mathcal{S}^{d}$, we set

$\displaystyle\left|F\right|_{1,l}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{d}\left|F_{i}\right|_{1,l},\quad\left|F\right|_{l}=\sum_{i=1}^{d}\left|F_{i}\right|_{l}.$

Moreover, we associate the following norms. For any $l,p\geq 1$,

$\displaystyle\left\|F\right\|_{l,p}$ $\displaystyle=$
$\displaystyle(\mathbb{E}\left|F\right|_{l}^{p})^{1/p},\quad\left\|F\right\|_{p}=(\mathbb{E}\left|F\right|^{p})^{1/p},$
$\displaystyle\left\|F\right\|_{L,l,p}$ $\displaystyle=$
$\displaystyle\left\|F\right\|_{l,p}+\left\|LF\right\|_{l-2,p}.$ (34)

We denote by $\mathcal{D}_{l,p}$ the closure of $\mathcal{S}$ with respect to
the norm $\left\|\circ\right\|_{L,l,p}:$

$\mathcal{D}_{l,p}=\overline{\mathcal{S}}^{\left\|\circ\right\|_{L,l,p}},$
(35)

and

$\mathcal{D}_{\infty}=\mathop{\bigcap}\limits_{l=1}^{\infty}\mathop{\bigcap}\limits_{p=1}^{\infty}\mathcal{D}_{l,p},\quad\mathcal{H}_{l}=\mathcal{D}_{l,2}.$

For an IbP framework $(\mathcal{S},D,L)$, we now extend the operators from
$\mathcal{S}$ to $\mathcal{D}_{\infty}$. For $F\in\mathcal{D}_{\infty}$,
$p\geq 2$, there exists a sequence $F_{n}\in\mathcal{S}$ such that
$\left\|F-F_{n}\right\|_{p}\rightarrow 0$,
$\left\|F_{m}-F_{n}\right\|_{q,p}\rightarrow 0$ and
$\left\|LF_{m}-LF_{n}\right\|_{q-2,p}\rightarrow 0$. Since $D^{q}$ and $L$ are
closable, we can define

$D^{q}F=\lim_{n\rightarrow\infty}D^{q}F_{n}\quad in\quad
L^{p}(\Omega;\mathcal{H}^{\otimes q}),\quad
LF=\lim_{n\rightarrow\infty}LF_{n}\quad in\quad L^{p}(\Omega).$ (36)

We still associate the same norms introduced above for
$F\in\mathcal{D}_{\infty}$.

######  Lemma 3.1.

The triplet $(\mathcal{D}_{\infty},D,L)$ is an IbP framework.

###### Proof.

Here we just show that $D$ verifies (29): For $F\in\mathcal{D}_{\infty}$ and
$h\in\mathcal{H}$, we have $\langle
DF,h\rangle_{\mathcal{H}}\in\mathcal{D}_{\infty}$.

In fact, for any $k\geq 1,p\geq 2$, any $F\in\mathcal{D}_{k+1,p}$, there is a
sequence $F_{n}\in\mathcal{S}$ such that $\|F_{n}-F\|_{k+1,p}\rightarrow 0$.
Then for any $u_{1},\cdots,u_{k}\in
L^{p}(\Omega;\mathcal{H}),h\in\mathcal{H}$, any $n,m\in\mathbb{N},$

$\displaystyle\mathbb{E}\langle D^{k}(\langle
DF_{m},h\rangle_{\mathcal{H}}-\langle
DF_{n},h\rangle_{\mathcal{H}}),u_{1}\otimes\cdots\otimes
u_{k}\rangle_{\mathcal{H}^{\otimes
k}}^{\frac{p}{2}}=\mathbb{E}|D_{u_{k}}D_{u_{k-1}}\cdots D_{u_{1}}\langle
D(F_{m}-F_{n}),h\rangle_{\mathcal{H}}|^{\frac{p}{2}}$
$\displaystyle=\mathbb{E}|D_{u_{k}}D_{u_{k-1}}\cdots
D_{u_{1}}D_{h}(F_{m}-F_{n})|^{\frac{p}{2}}=\mathbb{E}|\langle
D^{k+1}(F_{m}-F_{n}),h\otimes u_{1}\otimes\cdots\otimes
u_{k}\rangle_{\mathcal{H}^{\otimes(k+1)}}|^{\frac{p}{2}}$
$\displaystyle\leq\mathbb{E}|D^{k+1}(F_{m}-F_{n})|_{\mathcal{H}^{\otimes(k+1)}}^{\frac{p}{2}}|h\otimes
u_{1}\otimes\cdots\otimes
u_{k}|_{\mathcal{H}^{\otimes(k+1)}}^{\frac{p}{2}}\rightarrow 0,$

which yields that $\mathbb{E}|D^{k}(\langle
DF_{m},h\rangle_{\mathcal{H}}-\langle
DF_{n},h\rangle_{\mathcal{H}})|_{\mathcal{H}^{\otimes k}}^{p}\rightarrow 0.$
Therefore, $\langle DF,h\rangle_{\mathcal{H}}\in\mathcal{D}_{k,p}$ and (29) is
verified. ∎

The following lemma is useful in order to control the Sobolev norms
$\left\|F\right\|_{L,l,q}$.

######  Lemma 3.2.

We fix $p\geq 2,l\geq 2.$ Let $F\in L^{1}(\Omega)$ and let
$F_{n}\in\mathcal{S},n\in\mathbb{N}$ such that

$\displaystyle i)\quad\mathbb{E}\left|F_{n}-F\right|$
$\displaystyle\rightarrow$ $\displaystyle 0,$ $\displaystyle
ii)\quad\sup_{n}\left\|F_{n}\right\|_{L,l,p}$ $\displaystyle\leq$
$\displaystyle K_{l,p}<\infty.$

Then for every $1\leq\bar{p}<p,$ we have $F\in\mathcal{D}_{l,\bar{p}}$ and
$\left\|F\right\|_{L,l,\bar{p}}\leq K_{l,\bar{p}}$ .

###### Proof.

The Hilbert space $\mathcal{H}_{l}=\mathcal{D}_{l,2}$ equipped with the scalar
product

$\displaystyle\left\langle U,V\right\rangle_{L,l,2}$ $\displaystyle:=$
$\displaystyle\sum_{q=1}^{l}\mathbb{E}\langle
D^{q}U,D^{q}V\rangle_{\mathcal{H}^{\otimes q}}+\mathbb{E}|UV|$
$\displaystyle+$ $\displaystyle\sum_{q=1}^{l-2}\mathbb{E}\langle
D^{q}LU,D^{q}LV\rangle_{\mathcal{H}^{\otimes q}}+\mathbb{E}|LU\times LV|$

is the space of the functionals which are $l-$times differentiable in $L^{2}$
sense. By $ii)$, for $p\geq 2$,
$\left\|F_{n}\right\|_{L,l,2}\leq\left\|F_{n}\right\|_{L,l,p}\leq K_{l,p}$.
Then, applying Banach Alaoglu’s theorem, there exists a functional
$G\in\mathcal{H}_{l}$ and a subsequence (we still denote it by $n$), such that
$F_{n}\rightarrow G$ weakly in the Hilbert space $\mathcal{H}_{l}$. This means
that for every $Q\in\mathcal{H}_{l}$, $\langle
F_{n},Q\rangle_{L,l,2}\rightarrow\langle G,Q\rangle_{L,l,2}$. Therefore, by
Mazur’s theorem, we can construct some convex combination

$G_{n}=\sum_{i=n}^{m_{n}}\lambda_{i}^{n}\times F_{i}\in\mathcal{S}$

with $\lambda_{i}^{n}\geq 0,i=n,....,m_{n}$ and
$\sum\limits_{i=n}^{m_{n}}\lambda_{i}^{n}=1$, such that

$\left\|G_{n}-G\right\|_{L,l,2}\rightarrow 0.$

In particular we have

$\mathbb{E}\left|G_{n}-G\right|\leq\left\|G_{n}-G\right\|_{L,l,2}\rightarrow
0.$

Also, we notice that by i),

$\mathbb{E}\left|G_{n}-F\right|\leq\sum_{i=n}^{m_{n}}\lambda_{i}^{n}\times\mathbb{E}\left|F_{i}-F\right|\rightarrow
0.$

So we conclude that $F=G\in\mathcal{H}_{l}.$ Thus, we have

$\mathbb{E}(\left|G_{n}-F\right|_{l}^{2})+\mathbb{E}(\left|LG_{n}-LF\right|_{l-2}^{2})\leq\left\|G_{n}-F\right\|_{L,l,2}^{2}\rightarrow
0.$

By passing to a subsequence, we have
$\left|G_{n}-F\right|_{l}+\left|LG_{n}-LF\right|_{l-2}\rightarrow 0$ almost
surely. Now, for every $\bar{p}\in[1,p)$, we denote
$Y_{n}:=\left|G_{n}\right|_{l}^{\bar{p}}+\left|LG_{n}\right|_{l-2}^{\bar{p}}$
and $Y:=\left|F\right|_{l}^{\bar{p}}+\left|LF\right|_{l-2}^{\bar{p}}$. Then,
$Y_{n}\rightarrow Y$ almost surely, and for any $\tilde{q}\in[\bar{p},p]$,

$\displaystyle\mathbb{E}|G_{n}|_{l}^{\tilde{q}}+\mathbb{E}|LG_{n}|_{l-2}^{\tilde{q}}$
$\displaystyle\leq$
$\displaystyle\left\|G_{n}\right\|_{L,l,\tilde{q}}^{\tilde{q}}=\left\|\sum_{i=n}^{m_{n}}\lambda_{i}^{n}\times
F_{i}\right\|_{L,l,\tilde{q}}^{\tilde{q}}\leq(\sum_{i=n}^{m_{n}}\lambda_{i}^{n}\times\left\|F_{i}\right\|_{L,l,\tilde{q}})^{\tilde{q}}$
$\displaystyle\leq$
$\displaystyle(\sup_{i}\left\|F_{i}\right\|_{L,l,\tilde{q}}\times\sum_{i=n}^{m_{n}}\lambda_{i}^{n})^{\tilde{q}}=\sup_{i}\left\|F_{i}\right\|_{L,l,\tilde{q}}^{\tilde{q}}\leq
K_{l,\tilde{q}}^{\tilde{q}}.$

So $(Y_{n})_{n\in\mathbb{N}}$ is uniformly integrable, and we have

$\left\|F\right\|_{L,l,\bar{p}}^{\bar{p}}=\mathbb{E}(\left|F\right|_{l}^{\bar{p}})+\mathbb{E}(\left|LF\right|_{l-2}^{\bar{p}})=\mathbb{E}(Y)=\lim_{n\rightarrow\infty}\mathbb{E}(Y_{n})\leq
K_{l,\bar{p}}^{\bar{p}},$

∎

###  3.1 Main consequences: Convergence in total variation distance

We will use the abstract framework in [6] for the IbP framework
$(\mathcal{D}_{\infty},D,L)$, with $D$ and $L$ defined in (36). Using
Malliavin type arguments, [6] proves the following results. The first result,
concerning the density, is classical:

######  Lemma 3.3.

Let $F\in\mathcal{D}_{\infty}$. If $\Sigma_{6p+4}(F)<\infty$, then the law of
random variable $F$ has a density $p_{F}(x)$ which is $p-$times
differentiable.

In the following, we define the distances between random variables
$F,G:\Omega\rightarrow\mathbb{R}$:

$d_{k}(F,G)=\sup\\{\left|\mathbb{E}(f(F))-\mathbb{E}(f(G))\right|:\sum_{0\leq
i\leq k}\left\|f^{(i)}\right\|_{\infty}\leq 1\\}$

For $k=1$, this is the Fortet Mourier distance (which is a variant of the
Wasserstein distance), while for $k=0$, this is the total variation distance
and we denote it by $d_{TV}$. Now we present the second result concerning the
total variation distance:

######  Lemma 3.4.

We fix some index $l$, some $r\in\mathbb{N}$ and some $\delta>0.$ We define
$p_{1}=2(r(\frac{1}{\delta}-1)+2)$,
$p_{2}=\max\\{6l+4,2(\frac{r+l}{\delta}-r+2)\\}$, $q_{1}\geq
r(\frac{1}{\delta}-1)+4$, $q_{2}\geq\frac{r+l}{\delta}-r+4$. Let
$F,G\in\mathcal{D}_{\infty}.$ Then one may find $C\in\mathbb{R}_{+}$ ,
$p\in\mathbb{N}$ (depending on $r$ and $\delta)$ such that

$i)\quad d_{TV}(F,G)\leq
C(1+\Sigma_{p_{1}}(F)+\Sigma_{p_{1}}(G)+\left\|F\right\|_{L,q_{1},p}+\left\|G\right\|_{L,q_{1},p})\times
d_{r}(F,G)^{1-\delta},$ (37)

and

$ii)\quad\|p_{F}-p_{G}\|_{l,\infty}\leq
C(1+\Sigma_{p_{2}}(F)+\Sigma_{p_{2}}(G)+\left\|F\right\|_{L,q_{2},p}+\left\|G\right\|_{L,q_{2},p})\times
d_{r}(F,G)^{1-\delta},$ (38)

where $p_{F}(x)$ and $p_{G}(x)$ denote the density functions of $F$ and $G$
respectively.

Comment The significance of this lemma is the following. Suppose that one has
already obtained an estimate of a "smooth" distance $d_{r}$ between two random
vectors $F$ and $G$ (in our case $r=3$ in (17)). But we would like to control
the total variation distance between them. In order to do this, one employs
some integration by parts techniques which are developed in [6] and conclude
the following. We need to assume that both $F$ and $G$ are "smooth" in the
sense that $\left\|F\right\|_{L,q,p}+\left\|G\right\|_{L,q,p}<\infty$ for
sufficiently large $q,p.$ Moreover, we need some non degeneracy condition:
both $F$ and $G$ are non-degenerated, that is
$\Sigma_{p}(F)+\Sigma_{p}(G)<\infty$, with $p$ large enough. Then (37) asserts
that one may control $d_{TV}$ by $d_{r},$ and the control is quasi optimal: we
loose just a power $\delta>0$ which we may take as small as we want. And (38)
says that we may also control the distance between the derivatives of density
functions by $d_{r}$.

Then we can get the following corollary.

######  Corollary 3.4.1.

We fix some index $l$, some $r\in\mathbb{N}$ and some $\delta>0.$ We define
$p_{1},p_{2},q_{1},q_{2}$ as in Lemma 3.4. Let
$F_{M}\in\mathcal{D}_{\infty},M\in\mathbb{N}$ such that for every $p\geq 1$,

$\sup_{M}(\left\|F_{M}\right\|_{L,q_{1},p}+\Sigma_{p_{1}}(F_{M}))\leq
Q_{q_{1},p,p_{1}}<\infty,$

with $Q_{q_{1},p,p_{1}}$ a constant not dependent on $M$. Consider moreover
some random variable $F$ such that $d_{r}(F,F_{M})\rightarrow 0.$ Then there
exists a constant $C>0$ such that

$i)\quad d_{TV}(F,F_{M})\leq Cd_{r}(F,F_{M})^{1-\delta}.$

Moreover, if
$\sup\limits_{M}(\left\|F_{M}\right\|_{L,q_{2},p}+\Sigma_{p_{2}}(F_{M}))\leq
Q_{q_{2},p,p_{2}}<\infty$, the law of $F$ is absolutely continuous with smooth
density $p_{F}$ and one has

$ii)\quad\left\|p_{F}-p_{F_{M}}\right\|_{l,\infty}\leq
Cd_{r}(F,F_{M})^{1-\delta}.$

###### Proof.

We take $C$ to be a constant depending on $p,p_{1},q_{1},r$ and $\delta$ which
can change from one line to another. By Lemma 3.4, for every $M<M^{\prime}$,
one has

$\displaystyle d_{TV}(F_{M},F_{M^{\prime}})\leq
Cd_{r}(F_{M},F_{M^{\prime}})^{1-\delta}\leq
C[d_{r}(F_{M},F)^{1-\delta}+d_{r}(F,F_{M^{\prime}})^{1-\delta}].$ (39)

So $(F_{M})_{M\in\mathbb{N}}$ is a Cauchy sequence in $d_{TV}$. It follows
that it has a limit $G$. But since $d_{r}(F_{M},F)\rightarrow 0$, it follows
that $F=G$. Passing to the limit $M^{\prime}\rightarrow\infty$ in (39), we get

$d_{TV}(F_{M},F)\leq Cd_{r}(F_{M},F)^{1-\delta}.$

The proof of $ii)$ is analogous. ∎

##  4 Malliavin calculus and stochastic differential equations with jumps

In this section we present the integration by parts framework that will be
used in the following. To begin we give a quick informal presentation of our
strategy. We will work with the solution of the equation (10), but, for
technical reasons, we make the change of variable $z\mapsto\frac{1}{z}$ so the
equation of interest is now the equation (49). We use the notation from that
section. The intensity measure for our random measure is
$\mathbbm{1}_{[1,M)}(z)\nu(dz)ds$ and this is a finite measure. Then the
corresponding Poisson Point measure $N_{\nu}$ may be represented by means of a
compound Poisson process. For some technical reasons, we produce the
representation on each set $\\{z\in I_{k}=[k,k+1)\\},k\in\mathbb{N}$, so the
equation (49) reads

$\displaystyle\widehat{X}_{t}^{M}$ $\displaystyle=$ $\displaystyle
x+\int_{0}^{t}\sum_{k=1}^{M-1}\int_{\\{z\in
I_{k}\\}}\widetilde{c}(s,z,\widehat{X}_{s-}^{M})N_{\nu}(ds,dz)$
$\displaystyle+\int_{0}^{t}b_{M}(s,\widehat{X}_{s}^{M})ds+\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(s,z,\widehat{X}_{s}^{M})W_{\nu}(ds,dz)$ $\displaystyle=$
$\displaystyle
x+\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\widetilde{c}(T_{i}^{k},Z_{i}^{k},\widehat{X}_{T_{i}^{k}-}^{M})$
$\displaystyle+\int_{0}^{t}b_{M}(s,\widehat{X}_{s}^{M})ds+\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(s,z,\widehat{X}_{s}^{M})W_{\nu}(ds,dz).$

Here $T_{i}^{k},k,i\in\mathbb{N}$ are the jump times of the Poisson process
$(J_{t}^{k})_{t\in[0,T]}$ of parameter $\nu(I_{k})$, and
$Z_{i}^{k},k,i\in\mathbb{N}$ are independent random variables of law
$\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{\nu(I_{k})}$, which are independent of
$J^{k}$ as well. We will work conditionally to $T_{i}^{k},k,i\in\mathbb{N}$,
so the randomness in the system comes from $W_{\nu}$ on one hand and from
$Z_{i}^{k},k,i\in\mathbb{N}$ on the other hand. Concerning $W_{\nu}$ we will
use the standard Malliavin calculus (which fits in the framework presented in
Section 3). But we will also use this integration by parts calculus with
respect to the amplitude of the jumps given by $Z_{i}^{k},k,i\in\mathbb{N}.$
We present this kind of calculus now.

Suppose for a moment (just for simplicity) the law of $Z_{i}^{k}$ is
absolutely continuous with respect to the Lebesgue measure and has a smooth
density $h_{k}(z)$ which has compact support. We also assume that the
logarithm of the density $\ln h_{k}$ is smooth. Then we look to
$\widehat{X}_{t}^{M}$ as to a functional
$F(Z_{1}^{1},...,Z^{M-1}_{J_{t}^{k}})$ and we define the derivative operators

$D_{k,i}^{Z}F=\frac{\partial}{\partial
z_{i}^{k}}F(Z_{1}^{1},...,Z^{M-1}_{J_{t}^{k}}),$

and

$L^{Z}F=-\sum_{k,i}D_{k,i}^{Z}D_{k,i}^{Z}F+D_{k,i}^{Z}F\times\partial_{z}\ln
h_{k}(Z^{k}_{i}).$

And we check that these operators verify the conditions in Section 3. Since we
want to use integration by parts with respect to both $W_{\nu}$ and
$Z_{i}^{k},k,i\in\mathbb{N}$, we will consider the derivative operator
$D=(D^{W},D^{Z})$ and the operator $L=(L^{W},L^{Z})$ where $D^{W}$ and $L^{W}$
are the derivative and Ornstein Uhlenbeck operators from the standard
Malliavin calculus for Gaussian random variables. With these operators at hand
we check the hypotheses of Lemma 3.4 and of Corollary 3.4.1, and these are the
results which allow as to prove our Theorem 2.2.

Roughly speaking this is our strategy. But there is one more point: the
hypotheses we raise for the law of $Z^{k}_{i}$ that it has a smooth density
with compact support and has a smooth logarithm density, is rather strong and
we want to weaken it. This is the aim of the "splitting method". This amounts
to produce three independent random variables $V_{i}^{k},U_{i}^{k}$ and
$\xi_{i}^{k}$ such that $Z^{k}_{i}$ has the same law as
$\xi_{i}^{k}V_{i}^{k}+(1-\xi_{i}^{k})U_{i}^{k}$ with $\xi_{i}^{k}$ a Bernoulli
random variable and $V_{i}^{k}$ a random variable with good properties. So we
split $Z^{k}_{i}$ in two parts, $V_{i}^{k}$ and $U_{i}^{k}.$ We may do it in
such a way that $V_{i}^{k}$ has the law $\psi_{k}(v)dv$ with $\psi_{k}\in
C_{c}^{\infty}(\mathbb{R})$ (see Section 4.1 for the precise procedure). And
we perform the Malliavin calculus with respect to $V_{i}^{k}$ instead of
$Z^{k}_{i}$ (we work conditionally to $\xi_{i}^{k}$ and $U_{i}^{k}$ which
appear as constants).

###  4.1 The splitting method

We consider a Poisson point measure $N_{\nu}(ds,dz)$ with compensator
$\widehat{N}_{\nu}(ds,dz)=\nu(dz)ds$ on the state space $[1,\infty)$. We will
make use of the noise $z\in[1,\infty)$ in order to apply the results from the
previous section. We recall that $I_{k}=[k,k+1)$ and $m_{k}=\nu(I_{k})$, and
we suppose that for every $k$, there exists $\varepsilon_{k}>0$, such that

$\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}\geq\mathbbm{1}_{I_{k}}(z)\varepsilon_{k}\times
dz.$ (40)

######  Remark.

Under Hypothesis 2.4 $(a)$, the splitting condition (40) is satisfied with
$\varepsilon_{k}=\frac{\varepsilon_{\ast}}{(k+1)^{1-{\alpha}}}$, for any
${\alpha}\leq\alpha_{1}$. If instead we assume Hypothesis 2.4 $(b)$, (40) is
also satisfied, with $\varepsilon_{k}=\frac{\varepsilon_{\ast}}{k+1}$.

When (40) is satisfied, we are able to use the "splitting method" as follows.
To begin we define the functions

$\displaystyle a(y)$ $\displaystyle=$ $\displaystyle
1-\frac{1}{1-(4y-1)^{2}}\quad for\quad y\in[\tfrac{1}{4},\tfrac{1}{2})$ (41)
$\displaystyle\psi(y)$ $\displaystyle=$
$\displaystyle\mathbbm{1}_{\\{\left|y\right|\leq\frac{1}{4}\\}}+\mathbbm{1}_{\\{\frac{1}{4}<\left|y\right|\leq\frac{1}{2}\\}}e^{a(\left|y\right|)}.$
(42)

We notice that $\psi\in C_{c}^{\infty}(\mathbb{R})$ and that its support is
included in $[-\frac{1}{2},\frac{1}{2}]$. We also notice that for every
$q,p\in\mathbb{N}$ the function $y\mapsto|a^{(q)}(y)|^{p}\psi(y)$ is
continuous and has support included in $[-\frac{1}{2},\frac{1}{2}],$ so it is
bounded: one may find $C_{q,p}$ such that

$\big{|}a^{(q)}(y)\big{|}^{p}\psi(y)\leq C_{q,p}\quad\forall y\in\mathbb{R}.$
(43)

We denote

$\psi_{k}(y)=\psi(y-(k+\tfrac{1}{2})),\quad\theta_{k}(y):=\partial_{y}\ln\psi_{k}(y).$
(44)

By (43) (which is uniform with respect to $y$), we have

$\sup_{k}\big{|}(\ln\psi_{k})^{(q)}(y)\big{|}^{p}\psi_{k}(y)\leq
C_{q,p}\quad\forall y\in\mathbb{R}.$ (45)

We denote

$m(\psi)=\int_{-1/2}^{1/2}\psi(y)dy.$ (46)

We consider a sequence of independent random variables $Z^{k}$ such that

$Z^{k}\sim\mathbbm{1}_{I_{k}}(z)\frac{1}{m_{k}}\nu(dz).$

This is the sequence of random variables which are involved in the
representation of the measure $N_{\nu}(ds,dz)$ as long as $z\in[1,\infty)$ is
concerned. We notice that, according to our hypothesis (40),

$\mathbb{P}(Z^{k}\in
dz)=\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}\geq\mathbbm{1}_{I_{k}}(z)\varepsilon_{k}\times
dz.$

Then we construct some independent random variables $V^{k},U^{k},\xi^{k}$ with
laws

$\displaystyle\mathbb{P}(V^{k}\in dz)$ $\displaystyle=$
$\displaystyle\frac{1}{m(\psi)}\psi({z-(k+\frac{1}{2})})dz$
$\displaystyle\mathbb{P}(U^{k}\in dz)$ $\displaystyle=$
$\displaystyle\frac{1}{1-\varepsilon_{k}m(\psi)}(\mathbb{P}(Z^{k}\in
dz)-\varepsilon_{k}\psi({z-(k+\frac{1}{2})})dz)$ (47)
$\displaystyle\mathbb{P}(\xi^{k}=1)$ $\displaystyle=$
$\displaystyle\varepsilon_{k}m(\psi),\quad\mathbb{P}(\xi^{k}=0)=1-\varepsilon_{k}m(\psi).$

We choose $\varepsilon_{k}<1/m(\psi)$ so that $1-\varepsilon_{k}m(\psi)>0.$
Using (40), one may check that $\mathbb{P}(U^{k}\in dz)$ is a positive measure
and has mass one. So it is a probability measure. And finally one can easily
check the identity of laws:

$Z^{k}\sim\xi^{k}V^{k}+(1-\xi^{k})U^{k}.$ (48)

In the following, we will work directly with
$Z^{k}=\xi^{k}V^{k}+(1-\xi^{k})U^{k}.$ This is possible because all the
results that we discuss here concern the law of the random variables, and the
law remains unchanged.

The Poisson point measure $N_{\nu}$ can be written as the following sum:

$N_{\nu}(ds,dz)=\sum_{k=1}^{\infty}\mathbbm{1}_{I_{k}}(z)N_{\nu}(ds,dz)=\sum_{k=1}^{\infty}N_{\nu_{k}}(ds,dz),$

where $\nu_{k}(dz)=\mathbbm{1}_{I_{k}}(z)\nu(dz)$ and $N_{\nu_{k}}$ is a
Poisson point measure with intensity $\nu_{k}(dz)ds$.

The Poisson point measure $N_{\nu_{k}}$ can be represented by means of
compound Poisson processes as follows. For each $k\in\mathbb{N}$, we denote by
$T_{i}^{k},i\in\mathbb{N}$ the jump times of a Poisson process
$(J_{t}^{k})_{t\in[0,T]}$ of parameter $m_{k}$, and we consider a sequence of
independent random variables
$Z_{i}^{k}\sim\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}},i\in\mathbb{N}$,
which are independent of $J^{k}$ as well. Then, for any $t>0$ and
$A\in\mathcal{B}([k,k+1))$, $N_{\nu_{k}}([0,t]\times
A)=\sum\limits_{i=1}^{J_{t}^{k}}\mathbbm{1}_{A}(Z_{i}^{k}).$ And for each
$k,i\in\mathbb{N}$, we will split $Z_{i}^{k}$ as
$Z_{i}^{k}=\xi_{i}^{k}V_{i}^{k}+(1-\xi_{i}^{k})U_{i}^{k}.$

######  Remark.

The law of $Z^{k}_{i}$ could be very irregular and it is not possible to make
integration by parts based on it. So we construct the $V^{k}_{i}$, which has
all the good regularity properties in order to make Malliavin calculus. This
is the idea of the splitting method. The splitting method presented here is
analogous to the one in [10]. Therein, Bichteler, Gravereau and Jacod deal
with 2 kinds of independent Poisson point measures. One is very regular, and
smooth enough to make Malliavin calculus on it (in our paper, $V^{k}_{i}$ play
the same role). The other one can be arbitrary, and it may be very irregular
(in our paper, it corresponds to $U^{k}_{i}$). But the difference is that
instead of splitting the Poisson point measure, we split the random variables,
and so this method can also be applied in a large class of different problems.
For example, Bally, Caramellino and Poly use the splitting method to show the
convergence in total variation distance in the central limit theorem in [5].
Other possible approaches to the Malliavin calculus for jump processes are
given in the papers
$\cite[cite]{[\@@bibref{}{ref33}{}{}]},\cite[cite]{[\@@bibref{}{ref34}{}{}]},\cite[cite]{[\@@bibref{}{ref36}{}{}]},\cite[cite]{[\@@bibref{}{ref35}{}{}]}$
and the book [19] for example.

###  4.2 Malliavin calculus for Poisson point measures and space-time Brownian
motions

In this section we present the IbP framework on a space where we have the
Poisson point measure $N_{\nu}$ presented in the previous section and moreover
we have a space-time Brownian motion $W_{\nu}(ds,dz)$ with covariance measure
$\nu(dz)ds$, which is independent of $N_{\nu}$. We recall that in Section 2.2
we have introduced the random variables $W_{\nu}(\varphi),N_{\nu}(\varphi)$
and the filtrations $(\mathcal{F}_{t}^{W})_{t\in[0,T]}$,
$(\mathcal{F}_{t}^{N})_{t\in[0,T]}$, and we denote
$\mathcal{F}_{t}=\mathcal{F}_{t}^{W}\bigvee\mathcal{F}_{t}^{N}$. We present
now the Malliavin calculus. We recall the random variables $T^{k}_{i}$, and
$Z_{i}^{k}=\xi_{i}^{k}V_{i}^{k}+(1-\xi_{i}^{k})U_{i}^{k}$ introduced in the
previous section and we take
$\mathcal{G}=\sigma(U^{k}_{i},\xi^{k}_{i},T_{i}^{k}:k,i\in\mathbb{N})$ to be
the $\sigma-$algebra associated to the noise from
$U^{k}_{i},\xi^{k}_{i},T_{i}^{k},\ k,i\in\mathbb{N}.$ These are the noises
which will not be involved in the Malliavin calculus. We denote by
$C_{\mathcal{G},p}$ the space of the functions
$f:\Omega\times\mathbb{R}^{m^{\prime}\times
m}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$ such that $f$ is
$\mathcal{F}_{T}-$measurable, and for each $\omega,$ the function
$(v_{1}^{1},...,v^{m}_{m^{\prime}},w_{1},\cdots,w_{n})\mapsto
f(\omega,v_{1}^{1},...,v^{m}_{m^{\prime}},w_{1},\cdots,w_{n})$ belongs to
$C_{p}^{\infty}(\mathbb{R}^{m^{\prime}\times m}\times\mathbb{R}^{n})$, and for
each $(v_{1}^{1},...,v^{m}_{m^{\prime}},w_{1},\cdots,w_{n})$, the function
$\omega\mapsto f(\omega,v_{1}^{1},...,v^{m}_{m^{\prime}},w_{1},\cdots,w_{n})$
is $\mathcal{G}$-measurable. Then we define the space of simple functionals

$\mathcal{S}=\\{F=f(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq i\leq
m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},(W_{\nu}(\varphi_{j}))_{j=1}^{n}):f\in
C_{\mathcal{G},p},\varphi_{1},\cdots,\varphi_{n}\in
L^{2}([1,\infty)\times[0,T],\nu\times Leb),m^{\prime},m,n\in\mathbb{N}\\}.$

On the space $\mathcal{S}$ we define the derivative operators

$\displaystyle D^{Z}_{(k_{0},i_{0})}F=\mathbbm{1}_{\\{k_{0}\leq
m\\}}\mathbbm{1}_{\\{i_{0}\leq m^{\prime}\\}}\xi_{i_{0}}^{k_{0}}\frac{\partial
f}{\partial v^{k_{0}}_{i_{0}}}(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq
i\leq m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},(W_{\nu}(\varphi_{j}))_{j=1}^{n}),\quad
k_{0},i_{0}\in\mathbb{N}$ $\displaystyle
D^{W}_{(s,z)}F=\sum_{r=1}^{n}\frac{\partial f}{\partial
w_{r}}(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq i\leq m^{\prime}\\\ 1\leq
k\leq
m\end{subarray}},(W_{\nu}(\varphi_{j}))_{j=1}^{n})\varphi_{r}(s,z),\quad(s,z)\in[0,T]\times[1,\infty).$

We regard $D^{Z}F$ as an element of the Hilbert space $l_{2}$ (the space of
the sequences $h=(h^{k}_{i})_{k,i\in\mathbb{N}}$ with
$\left|h\right|_{l_{2}}^{2}=\sum_{k=1}^{\infty}\sum_{i=1}^{\infty}|h^{k}_{i}|^{2}<\infty$)
and $D^{W}F$ as an element of the Hilbert space
$L^{2}([1,\infty)\times[0,T],\nu\times Leb).$ Then

$DF:=(D^{Z}F,D^{W}F)\in l_{2}\otimes L^{2}([1,\infty)\times[0,T],\nu\times
Leb).$

We also denote $D^{Z,W}F=DF$ and $\mathcal{H}=l_{2}\otimes
L^{2}([1,\infty)\times[0,T],\nu\times Leb)$. And we have

$\left\langle
DF,DG\right\rangle_{\mathcal{H}}=\sum_{k=1}^{\infty}\sum_{i=1}^{\infty}D^{Z}_{(k,i)}F\times
D^{Z}_{(k,i)}G+\int_{[0,T]\times[1,\infty)}D^{W}_{(s,z)}F\times
D^{W}_{(s,z)}G\ \nu(dz)ds.$

Moreover, we define the derivatives of order $q\in\mathbb{N}$ recursively:

$D^{Z,W,q}_{(k_{1},i_{1})\cdots(k_{q},i_{q}),(s_{1},z_{1})\cdots(s_{q},z_{q})}F:=D^{Z,W}_{(k_{q},i_{q}),(s_{q},z_{q})}D^{Z,W}_{(k_{q-1},i_{q-1}),(s_{q-1},z_{q-1})}\cdots
D^{Z,W}_{(k_{1},i_{1}),(s_{1},z_{1})}F,$

and we denote $D^{q}F=D^{Z,W,q}F$. We also denote $D^{Z,q}$ (respectively
$D^{W,q}$) as the derivative $D^{Z}$ (respectively $D^{W}$) of order $q$.

We recall the function $\theta_{k}$ defined in (44) and we define the
Ornstein-Uhlenbeck operators $L^{Z}$, $L^{W}$ and $L=L^{Z}+L^{W}$ (which
verify the duality relation), with

$\displaystyle L^{Z}F$ $\displaystyle=$
$\displaystyle-\sum_{k=1}^{m}\sum_{i=1}^{m^{\prime}}(D^{Z}_{(k,i)}D^{Z}_{(k,i)}F+\xi^{k}_{i}D^{Z}_{(k,i)}F\times\theta_{k}(V^{k}_{i})),$
$\displaystyle L^{W}F$ $\displaystyle=$
$\displaystyle\sum_{r=1}^{n}\frac{\partial f}{\partial
w_{r}}(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq i\leq m^{\prime}\\\ 1\leq
k\leq m\end{subarray}},(W_{\nu}(\varphi_{j}))_{j=1}^{n})W_{\nu}(\varphi_{r})$
$\displaystyle-$ $\displaystyle\sum_{l,r=1}^{n}\frac{\partial^{2}f}{\partial
w_{l}\partial w_{r}}(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq i\leq
m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},(W_{\nu}(\varphi_{j}))_{j=1}^{n})\langle\varphi_{l},\varphi_{r}\rangle_{L^{2}([1,\infty)\times[0,T],\nu\times
Leb)}.$

One can check that the triplet $(\mathcal{S},D,L)$ is consistent with the IbP
framework given in Section 3. The proof is left to Appendix 5.3.

In the following, we will close the operator $D^{q}$ and $L$, so we will use
the IbP framework $(\mathcal{D}_{\infty},D,L)$ associated to
$(\mathcal{S},D,L)$ in Lemma 3.1.

###  4.3 Malliavin calculus applied to stochastic differential equations with
jumps

Now we will use the IbP framework presented in Section 4.2 for the equation
(10).

Let $\theta:(0,1]\rightarrow[1,\infty)$ be a function such that
$\theta(z)=\frac{1}{z}$. By a change of variables, instead of dealing with
equation (10), it is equivalent to consider the following equation.

$\displaystyle\widehat{X}_{t}^{M}$ $\displaystyle=$ $\displaystyle
x+\int_{0}^{t}\int_{[1,M)}\widetilde{c}(s,z,\widehat{X}_{s-}^{M})N_{\nu}(ds,dz)$
(49)
$\displaystyle+\int_{0}^{t}b_{M}(s,\widehat{X}_{s}^{M})ds+\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(s,z,\widehat{X}_{s}^{M})W_{\nu}(ds,dz),$

where $M=\frac{1}{\varepsilon}$, $\nu(dz)=\mu\circ\theta^{-1}(dz)$,
$\widetilde{c}(s,z,x)=c(s,\frac{1}{z},x)$,

$\displaystyle b_{M}(s,x)=\int_{\\{z\geq M\\}}\widetilde{c}(s,z,x)\nu(dz),$
(50)

and $W_{\nu}$ is the space-time Brownian motion with covariance measure
$\nu(dz)ds$. One can check that $\widehat{X}_{t}^{M}$ has the same law as
$X_{t}^{\varepsilon}$.

Here we give two lemmas, concerning the Malliavin-Sobolev norms and the
Malliavin covariance. We recall that $\varepsilon_{\ast}$ and $\alpha$ are
introduced in Hypothesis 2.4 (b), and $q^{\ast}$ is introduced in Hypothesis
2.1.

######  Lemma 4.1.

Assuming Hypothesis 2.1 with $q^{\ast}\geq 2$ and Hypothesis 2.4 (either
2.4(a) or 2.4(b)), we have $\widehat{X}_{t}^{M}\in\mathcal{D}_{\infty}$, and
for all $p\geq 1,2\leq l\leq q^{\ast}$, there exists a constant $C_{l,p}(T)$
depending on $l,p,x$ and $T$, such that
$\sup\limits_{M}\|\widehat{X}_{t}^{M}\|_{L,l,p}\leq C_{l,p}(T)$.

######  Lemma 4.2.

Assume that Hypothesis 2.1 with $q^{\ast}\geq 1$ and Hypothesis 2.2, 2.3 hold
true.

a) If we also assume Hypothesis 2.4 (a), then for every $p\geq 1$,
$t\in[0,T]$, we have

$\displaystyle\sup\limits_{M}\mathbb{E}(1/\sigma_{\widehat{X}_{t}^{M}})^{p}\leq
C_{p},$ (51)

with $C_{p}$ a constant only depending on $p$ and $T$.

b) If we assume Hypothesis 2.4 (b), then for every $p\geq 1$, $t\in[0,T]$ such
that $t>\frac{4p\alpha}{\varepsilon_{\ast}}$, we have
$\sup\limits_{M}\mathbb{E}(1/\sigma_{\widehat{X}_{t}^{M}})^{p}\leq C_{p}$.

The proofs of these lemmas are rather technical and are postponed for the
Appendix (Section 5.1 and 5.2).

###  4.4 Proof of the main result (Theorem 2.2)

###### Proof.

$(a)$ By Lemma 4.1 and Lemma 4.2 $a)$, we know that for any $\delta>0$, for
any $p,p_{1}\geq 1,2\leq q\leq q^{\ast}$, with
$q^{\ast}\geq\frac{3}{\delta}+1$, there exists a constant $C_{q,p,p_{1}}(T)$
such that for any $M\geq 1$, $t\in[0,T]$, we have

$\Sigma_{p_{1}}(\widehat{X}_{t}^{M})+\|\widehat{X}_{t}^{M}\|_{L,q,p}\leq
C_{q,p,p_{1}}(T).$

By Lemma 2.1, we know that

$d_{3}(\widehat{X}_{t},\widehat{X}_{t}^{M})=d_{3}(X_{t},X_{t}^{\varepsilon})\leq
C\eta_{3}(\varepsilon).$

Then applying Corollary 3.4.1 $i)$ for $r=3$, we have

$d_{TV}(X_{t},X_{t}^{\varepsilon})=d_{TV}(\widehat{X}_{t},\widehat{X}_{t}^{M})\leq
Cd_{3}(\widehat{X}_{t},\widehat{X}_{t}^{M})^{1-\delta}\leq
C_{\delta}{\eta_{3}}(\varepsilon)^{1-\delta}.$

So we obtain (23). The proof of (24) is obtained by Corollary 3.4.1 $ii)$,
since $q^{\ast}\geq\frac{3+l}{\delta}+1$.
$(b)$ The proof is almost the same. If
$t>\frac{8\alpha(\frac{3}{\delta}-1)}{\varepsilon_{\ast}},$ then by Lemma 4.2
$b)$, $\Sigma_{p_{1}}(\widehat{X}_{t}^{M})<\infty$ for
${p_{1}}=2(\frac{3}{\delta}-1)$. So Corollary 3.4.1 $i)$ still holds, and we
can obtain (25). For

$t>\max\\{\frac{8\alpha}{\varepsilon_{\ast}}(\frac{3+l}{\delta}-1),\frac{8\alpha(3l+2)}{\varepsilon_{\ast}}\\},$

by Lemma 4.2 $b)$, $\Sigma_{p_{2}}(\widehat{X}_{t}^{M})<\infty$ for
${p_{2}}=\max\\{2(\frac{3+l}{\delta}-1),6l+4\\}$. So Corollary 3.4.1 $ii)$
still holds, and we obtain (26). ∎

##  5 Appendix

###  5.1 Proof of Lemma 4.1

In the following, we will only work with the measure $\nu$ supported on
$[1,\infty)$ and with the processes $(\widehat{X}_{t})_{t\in[0,T]}$ and
$(\widehat{X}_{t}^{M})_{t\in[0,T]}$. So in order to simplify the notation,
from now on we denote $\widehat{X}_{t}=X_{t}$ and
$\widehat{X}_{t}^{M}=X_{t}^{M}$. We remark that $M=\frac{1}{\varepsilon}$ is
generally not an integer, but for simplicity, we assume in the following that
$M$ is an integer.

Here is the idea of the proof. Since $X_{t}^{M}$ is not a simple functional,
we construct first the Euler scheme $(X_{t}^{n,M})_{t\in[0,T]}$ in subsection
5.1.1 and check that $X_{t}^{n,M}\rightarrow X_{t}^{M}$ in $L^{1}$ when
$n\rightarrow\infty$. We will prove that $\mathbb{E}|X_{t}^{n,M}|_{l}^{p}$ and
$\mathbb{E}|LX_{t}^{n,M}|_{l}^{p}$ are bounded (uniformly in $n,M$) in
subsection 5.1.3. Then based on Lemma 3.2, we obtain that
$X^{M}_{t}\in\mathcal{D}_{\infty}$ and the norms $\|X_{t}^{M}\|_{L,l,p}$ are
bounded (uniformly in $M$).

####  5.1.1 Construction of the Euler scheme

We take a time-partition
$\mathcal{P}_{t}^{n}=\\{r_{j}=\frac{jt}{n},j=0,\cdots,n\\}$ and a space-
partition
$\tilde{\mathcal{P}}_{M}^{n}=\\{z_{j}=M+\frac{j}{n},j=0,1,\cdots\\}$. We
denote $\tau_{n}(r)=r_{j}$ when $r\in[r_{j},r_{j+1})$, and denote
$\gamma_{n}(z)=z_{j}$ when $z\in[z_{j},z_{j+1})$. Let

$\displaystyle X_{t}^{n,M}$ $\displaystyle=$ $\displaystyle
x+\int_{0}^{t}\int_{[1,M)}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})N_{\nu}(dr,dz)$
(52)
$\displaystyle+\int_{0}^{t}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})dr+\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})W_{\nu}(dr,dz).$

Then we can obtain the following lemma.

######  Lemma 5.1.

Assume that the Hypothesis 2.1 holds true with $q^{\ast}\geq 1$. Then for any
$p\geq 1,M\geq 1$, we have $\mathbb{E}|X_{t}^{n,M}-X_{t}^{M}|^{p}\rightarrow
0$ as $n\rightarrow\infty$.

###### Proof.

We first notice that since $\bar{c}(z)$ (in Hypothesis 2.1) is decreasing,
$\sup\limits_{n\in\mathbb{N}}\bar{c}(\gamma_{n}(z))\leq\bar{c}(\gamma_{1}(z)).$
So

$\int_{1}^{\infty}\sup\limits_{n\in\mathbb{N}}|\bar{c}(\gamma_{n}(z))|^{2}\nu(dz)\leq\int_{1}^{\infty}|\bar{c}(\gamma_{1}(z))|^{2}\nu(dz)\leq|\bar{c}(1)|^{2}\nu[1,2]+\int_{1}^{\infty}|\bar{c}(z)|^{2}\nu(dz)\leq
C<\infty.$ (53)

Then by the Lebesgue dominated convergence theorem, (53) implies that

$\displaystyle\lim\limits_{n\rightarrow\infty}\sup\limits_{x\in\mathbb{R}}\int_{0}^{T}\int_{[1,\infty)}|\widetilde{c}(s,z,x)-\widetilde{c}(\tau_{n}(s),\gamma_{n}(z),x)|^{2}\nu(dz)ds=0,$
(54)

and

$\displaystyle\sup\limits_{n\in\mathbb{N}}\sup\limits_{x\in\mathbb{R}}\int_{0}^{T}\int_{[1,\infty)}|\widetilde{c}(\tau_{n}(s),\gamma_{n}(z),x)|^{2}\nu(dz)ds\leq
C.$ (55)

In the following proof, $C_{p}(T)$ will be a constant depending on $p$ and $T$
which may be changed from line to line. For $p\geq 2$, we write
$\mathbb{E}|X_{t}^{n,M}-X_{t}^{M}|^{p}\leq C_{p}(T)(E_{1}+E_{2}+E_{3})$, where

$\displaystyle
E_{1}=\mathbb{E}|\int_{0}^{t}\int_{[1,M)}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})-\widetilde{c}(r,z,X_{r-}^{M})N_{\nu}(dr,dz)|^{p},$
$\displaystyle
E_{2}=\mathbb{E}|\int_{0}^{t}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})-b_{M}(r,X_{r}^{M})dr|^{p},$
$\displaystyle E_{3}=\mathbb{E}|\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})-\widetilde{c}(r,z,X_{r}^{M})W_{\nu}(dr,dz)|^{p}.$

Then, compensating $N_{\nu}$ and using Burkholder’s inequality (see for
example the Theorem 2.11 in [26]),

$\displaystyle E_{1}$ $\displaystyle\leq$ $\displaystyle
C_{p}(T)[\mathbb{E}\int_{0}^{t}(\int_{[1,M)}|\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})-\widetilde{c}(r,z,X_{r-}^{M})|^{2}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{t}\int_{[1,M)}|\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})-\widetilde{c}(r,z,X_{r-}^{M})|^{p}\nu(dz)dr$
$\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{t}|\int_{[1,M)}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})-\widetilde{c}(r,z,X_{r-}^{M})\nu(dz)|^{p}dr]$
$\displaystyle\leq$ $\displaystyle
C_{p}(T)[R_{n}^{1}+((\bar{c}_{2})^{\frac{p}{2}}+\bar{c}_{p}+(\bar{c}_{1})^{p})\int_{0}^{t}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}-X_{r}^{M}|^{p}dr],$

with

$\displaystyle R_{n}^{1}$ $\displaystyle=$
$\displaystyle\mathbb{E}\int_{0}^{t}(\int_{[1,M)}|\widetilde{c}(\tau_{n}(r),z,X_{r-}^{M})-\widetilde{c}(r,z,X_{r-}^{M})|^{2}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{t}\int_{[1,M)}|\widetilde{c}(\tau_{n}(r),z,X_{r-}^{M})-\widetilde{c}(r,z,X_{r-}^{M})|^{p}\nu(dz)dr$
$\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{t}|\int_{[1,M)}\widetilde{c}(\tau_{n}(r),z,X_{r-}^{M})-\widetilde{c}(r,z,X_{r-}^{M})\nu(dz)|^{p}dr.$

Since
$|\widetilde{c}(\tau_{n}(r),z,X_{r-}^{M})-\widetilde{c}(r,z,X_{r-}^{M})|^{p}\leq|2\bar{c}(z)|^{p}\in
L^{1}(\Omega\times[1,\infty)\times[0,T],\mathbb{P}\times\nu\times Leb)$, we
apply the Lebesgue’s dominated convergence theorem and we obtain that
$R_{n}^{1}\rightarrow 0$. Next,

$\displaystyle E_{2}$ $\displaystyle\leq$
$\displaystyle{C}_{p}(T)\mathbb{E}\int_{0}^{t}|\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)}^{n,M})-\widetilde{c}(r,z,X_{r}^{M})\nu(dz)|^{p}dr$
$\displaystyle\leq$
$\displaystyle{C}_{p}(T)[R_{n}^{2}+(\bar{c}_{1})^{p}\int_{0}^{t}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}-X_{r}^{M}|^{p}dr],$

with

$R_{n}^{2}=\mathbb{E}\int_{0}^{t}|\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),z,X_{r}^{M})-\widetilde{c}(r,z,X_{r}^{M})\nu(dz)|^{p}dr\rightarrow
0.$

Finally, using Burkholder’s inequality,

$\displaystyle E_{3}$ $\displaystyle\leq$
$\displaystyle{C}_{p}(T)\mathbb{E}|\int_{0}^{t}\int_{\\{z\geq
M\\}}|\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})-\widetilde{c}(r,z,X_{r}^{M})|^{2}\nu(dz)|^{\frac{p}{2}}dr$
$\displaystyle\leq$
$\displaystyle{C}_{p}(T)[R_{n}^{3}+(\bar{c}_{2})^{\frac{p}{2}}\int_{0}^{t}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}-X_{r}^{M}|^{p}dr],$

where (by (54)),

$\displaystyle R_{n}^{3}=\mathbb{E}|\int_{0}^{t}\int_{\\{z\geq
M\\}}|\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{r}^{M})-\widetilde{c}(r,z,X_{r}^{M})|^{2}\nu(dz)|^{\frac{p}{2}}dr\rightarrow
0.$

Therefore,
$\mathbb{E}|X_{t}^{n,M}-X_{t}^{M}|^{p}\leq{C_{p}}(T)[R_{n}+\int_{0}^{t}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}-X_{r}^{M}|^{p}dr]$,
with $R_{n}=R_{n}^{1}+R_{n}^{2}+R_{n}^{3}\rightarrow 0$ as
$n\rightarrow\infty$. One can easily check that
$\mathbb{E}|X_{t}^{n,M}-X_{\tau_{n}(t)}^{n,M}|^{p}\rightarrow 0$. Also there
exists a constant $C_{p}(T)$ depending on $p$ and $T$ such that for any
$n,M\in\mathbb{N}$ and any $t\in[0,T]$, $\mathbb{E}|X_{t}^{n,M}|^{p}\leq
C_{p}(T)$ (see (71) for details). Then, by the dominated convergence theorem,
these yield
$\int_{0}^{t}\mathbb{E}|X_{r}^{n,M}-X_{\tau_{n}(r)}^{n,M}|^{p}dr\rightarrow
0$. So we have
$\mathbb{E}|X_{t}^{n,M}-X_{t}^{M}|^{p}\leq{C_{p}}(T)[\widetilde{R_{n}}+\int_{0}^{t}\mathbb{E}|X_{r}^{n,M}-X_{r}^{M}|^{p}dr]$,
with $\widetilde{R_{n}}\rightarrow 0$ as $n\rightarrow\infty$. We conclude by
using Gronwall’s lemma. ∎

######  Remark.

Some results on the convergence of the Euler scheme of a jump-diffusion can be
found for example in
$\cite[cite]{[\@@bibref{}{ref38}{}{}]},\cite[cite]{[\@@bibref{}{ref30}{}{}]}$.
The special thing in our paper is that we deal with the space-time Brownian
motion instead of the classical Brownian motion, and this is why we need to
assume (54).

Now we represent the jump’s part of $(X_{t}^{n,M})_{t\in[0,T]}$ by means of
compound Poisson processes. We recall that for each $k\in\mathbb{N}$, we
denote by $T_{i}^{k},i\in\mathbb{N}$ the jump times of a Poisson process
$(J_{t}^{k})_{t\in[0,T]}$ of parameter $m_{k}$, and we consider a sequence of
independent random variables
$Z_{i}^{k}\sim\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}},i\in\mathbb{N}$,
which are independent of $J^{k}$ as well. Then we write

$\displaystyle X_{t}^{n,M}$ $\displaystyle=$ $\displaystyle
x+\int_{0}^{t}\sum_{k=1}^{M-1}\int_{\\{z\in
I_{k}\\}}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n.M})N_{\nu}(dr,dz)$
(56)
$\displaystyle+\int_{0}^{t}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})dr+\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})W_{\nu}(dr,dz)$
$\displaystyle=$ $\displaystyle
x+\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X_{\tau_{n}(T_{i}^{k})-}^{n,M})$
$\displaystyle+\int_{0}^{t}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})dr+\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})W_{\nu}(dr,dz).$

So for every $t\in[0,T]$, $X_{t}^{n,M}$ is a simple functional.

####  5.1.2 Preliminary estimates

In order to estimate the Sobolev norms of the Euler scheme, we need the
following preliminary lemmas.

######  Lemma 5.2.

We fix $M\geq 1$. Let
$y:\Omega\times[0,T]\times[M,\infty)\rightarrow\mathbb{R}$ be a function which
is piecewise constant with respect to both $t$ and $z$. We assume that
$y_{t}(z)$ is progressively measurable with respect to $\mathcal{F}_{t}$
(defined in (11)), $y_{t}(z)\in\mathcal{S}$, and
$\mathbb{E}(\int_{0}^{t}\int_{\\{z\geq
M\\}}\left|y_{r}(z)\right|^{2}\nu(dz)dr)<\infty$. We denote
$I_{t}({y})=\int_{0}^{t}\int_{\\{z\geq M\\}}{y}_{r}(z)W_{\nu}(dr,dz)$. Then
for any $l\geq 1,p\geq 2$, there exists a constant $C_{l,p}(T)$ such that

$a)\quad\mathbb{E}|I_{t}(y)|_{l}^{p}\leq
C_{l,p}(T)\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|y_{r}(z)|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr,$ $\displaystyle
b)\quad\mathbb{E}|LI_{t}(y)|_{l}^{p}\leq
C_{l,p}(T)[\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|Ly_{r}(z)|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr+\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|y_{r}(z)|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr].$

###### Proof.

Proof of $a)$: Let $C_{l,p}(T)$ be a constant depending on $l,p$ and $T$ which
may change from one line to another. For any $l\geq 1$, we take $l_{W}\geq 0$
and $l_{Z}\geq 0$ such that $0<l_{W}+l_{Z}\leq l$.
It is easy to check that

$D^{Z,l_{Z}}_{(k_{1},i_{1})\cdots(k_{l_{Z}},i_{l_{Z}})}I_{t}(y)=\int_{0}^{t}\int_{\\{z\geq
M\\}}D^{Z,{l_{Z}}}_{(k_{1},i_{1})\cdots(k_{l_{Z}},i_{l_{Z}})}y_{r}(z)W_{\nu}(dr,dz).$

And by recurrence, one can show that

$D^{W,{l_{W}}}_{(s_{1},z_{1})\cdots(s_{l_{W}},z_{l_{W}})}I_{t}(y)=\int_{0}^{t}\int_{\\{z\geq
M\\}}D^{W,{l_{W}}}_{(s_{1},z_{1})\cdots(s_{l_{W}},z_{l_{W}})}y_{r}(z)W_{\nu}(dr,dz)+\sum_{j=1}^{l_{W}}D^{W,{{l_{W}}-1}}_{\widehat{(s_{j},z_{j})}^{{l_{W}}-1}}y_{s_{j}}(z_{j})\mathbbm{1}_{s_{j}\leq
t},$

with

$\widehat{(s_{j},z_{j})}^{{l_{W}}-1}:=(s_{1},z_{1})\cdots(s_{j-1},z_{j-1})(s_{j+1},z_{j+1})\cdots(s_{l_{W}},z_{l_{W}}).$

We denote

$\bar{y}_{r}(z)(k_{1},i_{1},\cdots,k_{l_{Z}},i_{l_{Z}}):=D^{Z,{l_{Z}}}_{(k_{1},i_{1})\cdots(k_{l_{Z}},i_{l_{Z}})}y_{r}(z),\quad\bar{y}_{r}^{l_{Z}}(z):=D^{Z,{l_{Z}}}y_{r}(z)\in
l_{2}^{\otimes l_{Z}}.$

Then $D^{Z,{l_{Z}}}I_{t}(y)=I_{t}(\bar{y}^{l_{Z}})$, and

$\displaystyle
D^{W,l_{W}}_{(s_{1},z_{1})\cdots(s_{l_{W}},z_{l_{W}})}D^{Z,l_{Z}}_{(k_{1},i_{1})\cdots(k_{l_{Z}},i_{l_{Z}})}I_{t}(y)$
$\displaystyle=$ $\displaystyle\int_{0}^{t}\int_{\\{z\geq
M\\}}D^{W,l_{W}}_{(s_{1},z_{1})\cdots(s_{l_{W}},z_{l_{W}})}\bar{y}_{r}(z)(k_{1},i_{1},\cdots,k_{l_{Z}},i_{l_{Z}})W_{\nu}(dr,dz)$
$\displaystyle+$
$\displaystyle\sum_{j=1}^{l_{W}}D^{W,{l_{W}-1}}_{\widehat{(s_{j},z_{j})}^{l_{W}-1}}\bar{y}_{s_{j}}(z_{j})(k_{1},i_{1},\cdots,k_{l_{Z}},i_{l_{Z}})\mathbbm{1}_{s_{j}\leq
t}.$

Let ${H}_{{l_{Z}},{l_{W}},T}=l_{2}^{\otimes{l_{Z}}}\otimes
L^{2}([0,T]\times[M,\infty),Leb\times\nu)^{\otimes{l_{W}}}.$ We have

$\displaystyle|D^{W,{l_{W}}}D^{Z,{l_{Z}}}I_{t}(y)|_{H_{l,\bar{l},T}}^{2}=\int_{[0,T]^{l_{W}}}\int_{[M,\infty)^{l_{W}}}|D^{W,{l_{W}}}_{(s_{1},z_{1})\cdots(s_{l_{W}},z_{l_{W}})}I_{t}(\bar{y}^{l_{Z}})|_{l_{2}^{\otimes{l_{Z}}}}^{2}\nu(dz_{1})ds_{1}\cdots\nu(dz_{l_{W}})ds_{l_{W}}$
$\displaystyle\leq 2|\int_{0}^{t}\int_{\\{z\geq
M\\}}D^{W,{l_{W}}}\bar{y}^{l_{Z}}_{r}(z)W_{\nu}(dr,dz)|_{H_{{l_{Z}},{l_{W}},T}}^{2}+{l_{W}}2^{l_{W}}\int_{0}^{t}\int_{\\{z\geq
M\\}}|D^{W,{{l_{W}}-1}}\bar{y}^{l_{Z}}_{r}(z)|_{H_{{l_{Z}},{l_{W}}-1,T}}^{2}\nu(dz)dr.$

Using Burkholder’s inequality for Hilbert-space-valued martingales (see [30]
for example), we have

$\displaystyle\mathbb{E}|D^{W,{l_{W}}}D^{Z,{l_{Z}}}I_{t}(y)|_{H_{{l_{Z}},{l_{W}},T}}^{p}$
$\displaystyle\leq$ $\displaystyle
C_{l,p}(T)[\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|D^{W,{l_{W}}}D^{Z,{l_{Z}}}y_{r}(z)|_{H_{{l_{Z}},{l_{W}},T}}^{2}\nu(dz))^{\frac{p}{2}}dr$
(57) $\displaystyle+$ $\displaystyle\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|D^{W,{{l_{W}}-1}}D^{Z,{l_{Z}}}y_{r}(z)|_{H_{{l_{Z}},{l_{W}}-1,T}}^{2}\nu(dz))^{\frac{p}{2}}dr].$

We recall that for $F\in\mathcal{D}_{\infty}$, we have
$|D^{W,{l_{W}}}D^{Z,{l_{Z}}}F|_{H_{{l_{Z}},{l_{W}},T}}\leq|F|_{{l_{Z}}+{l_{W}}}$
(see the definition in (33)). Then (57) gives

$\displaystyle\mathbb{E}|I_{t}(y)|_{1,l}^{p}\leq
C_{l,p}(T)\sum\limits_{{l_{Z}}+{l_{W}}\leq
l}\mathbb{E}|D^{W,{l_{W}}}D^{Z,{l_{Z}}}I_{t}(y)|_{H_{{l_{Z}},{l_{W}},T}}^{p}\leq{C}_{l,p}(T)\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|y_{r}(z)|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr.$ (58)

Finally, using Burkholder’s inequality, we have

$\displaystyle\mathbb{E}|I_{t}(y)|^{p}\leq{C}_{l,p}(T)\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|y_{r}(z)|^{2}\nu(dz))^{\frac{p}{2}}dr.$ (59)

So $a)$ is proved.

Proof of $b)$: We first show that

$\displaystyle LI_{t}(y)=I_{t}(Ly)+I_{t}(y).$ (60)

We denote

$I_{k}^{t}(f_{k})=k!\int_{0}^{t}\int_{0}^{s_{1}}\cdots\int_{0}^{s_{k-1}}\int_{[M,+\infty)^{k}}f_{k}(s_{1},\cdots,s_{k},z_{1},\cdots,z_{k})W_{\nu}(ds_{k},dz_{k})\cdots
W_{\nu}(ds_{1},dz_{1})$

the multiple stochastic integral for a deterministic function $f_{k}$, which
is square integrable with respect to $(\nu(dz)ds)^{\otimes k}$ and is
symmetric with respect to the time variation $(s_{1},\cdots,s_{k})$ for each
fixed $(z_{1},\cdots,z_{k})$. Notice that $L^{Z}I_{k}^{t}(f_{k})=0$ and
$L^{W}I_{k}^{t}(f_{k})=kI_{k}^{t}(f_{k})$. So,
$LI_{k}^{t}(f_{k})=kI_{k}^{t}(f_{k})$. Then by the duality relation (32),

$\displaystyle\mathbb{E}(I_{k}^{t}(f_{k})L(I_{t}(y)))=\mathbb{E}(I_{t}(y)\times
LI_{k}^{t}(f_{k}))=k\mathbb{E}(I_{t}(y)\times I_{k}^{t}(f_{k})).$ (61)

On the other hand, using the isometry property and the duality relation,

$\displaystyle\mathbb{E}(I_{k}^{t}(f_{k})\times
I_{t}(Ly))=k\mathbb{E}\int_{0}^{t}\int_{\\{z\geq
M\\}}I_{k-1}^{r}(f_{k}(r,z,\cdot))Ly_{r}(z)\nu(dz)dr$
$\displaystyle=k\int_{0}^{t}\int_{\\{z\geq M\\}}\mathbb{E}[y_{r}(z)\times
LI_{k-1}^{r}(f_{k}(r,z,\cdot))]\nu(dz)dr=k(k-1)\mathbb{E}\int_{0}^{t}\int_{\\{z\geq
M\\}}y_{r}(z)I_{k-1}^{r}(f_{k}(r,z,\cdot))\nu(dz)dr$
$\displaystyle=k(k-1)\mathbb{E}(I_{t}(y)\times\int_{0}^{t}\int_{\\{z\geq
M\\}}I_{k-1}^{r}(f_{k}(r,z,\cdot))W_{\nu}(dr,dz))$
$\displaystyle=(k-1)\mathbb{E}(I_{t}(y)\times I_{k}^{t}(f_{k})).$

Combining this with (61), we get

$\displaystyle\mathbb{E}(I_{k}^{t}(f_{k})(I_{t}(y)+I_{t}(Ly)))=k\mathbb{E}(I_{k}^{t}(f_{k})I_{t}(y))=\mathbb{E}[I_{k}^{t}(f_{k})\times
LI_{t}(y)].$ (62)

Since every element in $L^{2}(W)$ (defined by (12)) can be represented as the
direct sum of multiple stochastic integrals, we have for any $F\in L^{2}(W)$,

$\displaystyle\mathbb{E}[FLI_{t}(y)]=\mathbb{E}[F(I_{t}(Ly)+I_{t}(y))].$ (63)

For $G\in L^{2}(N)$, one has $L^{W}G=0$ and $L^{Z}G\in L^{2}(N)$. Then by
using duality and (13),

$\mathbb{E}[GLI_{t}(y)]=\mathbb{E}[I_{t}(y)LG]=\mathbb{E}[I_{t}(y)L^{Z}G]=0,$

and by (13),

$\mathbb{E}[G(I_{t}(Ly)+I_{t}(y))]=0.$

So,

$\displaystyle\mathbb{E}[GLI_{t}(y)]=\mathbb{E}[G(I_{t}(Ly)+I_{t}(y))].$ (64)

Combining (63) and (64), for any $\tilde{G}\in L^{2}(W)\otimes L^{2}(N)$, we
have
$\mathbb{E}[\tilde{G}LI_{t}(y)]=\mathbb{E}[\tilde{G}(I_{t}(Ly)+I_{t}(y))]$,
which proves (60).
Then, by Lemma 5.2 $a)$,

$\displaystyle\mathbb{E}|LI_{t}(y)|_{l}^{p}$ $\displaystyle\leq$
$\displaystyle 2^{p-1}(\mathbb{E}|\int_{0}^{t}\int_{\\{z\geq
M\\}}Ly_{r}(z)W_{\nu}(dr,dz)|_{l}^{p}+\mathbb{E}|\int_{0}^{t}\int_{\\{z\geq
M\\}}y_{r}(z)W_{\nu}(dr,dz)|_{l}^{p})$ $\displaystyle\leq$ $\displaystyle
C_{l,p}(T)[\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|Ly_{r}(z)|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr+\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|y_{r}(z)|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr].$

∎

We will also need the following lemma from [7] (Lemma 8 and Lemma 10), which
is a consequence of the chain rule for $D^{q}$ and $L$.

######  Lemma 5.3.

Let $F\in\mathcal{S}^{d}$. For every $l\in\mathbb{N},$ if
$\phi:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a $C^{l}(\mathbb{R}^{d})$
function ($l-$times differentiable function), then

$a)\quad|\phi(F)|_{1,l}\leq|\nabla\phi(F)||F|_{1,l}+C_{l}\sup_{2\leq|\beta|\leq
l}|\partial^{\beta}\phi(F)||F|_{1,l-1}^{l}.$

If $\phi\in C^{l+2}(\mathbb{R}^{d})$, then

$b)\quad|L\phi(F)|_{l}\leq|\nabla\phi(F)||LF|_{l}+C_{l}\sup_{2\leq|\beta|\leq
l+2}|\partial^{\beta}\phi(F)|(1+|F|_{l+1}^{l+2})(1+|LF|_{l-1}).$

For $l=0$, we have

$c)\quad|L\phi(F)|\leq|\nabla\phi(F)||LF|+\sup_{|\beta|=2}|\partial^{\beta}\phi(F)||F|_{1,1}^{2}.$

We finish this section with a first estimate concerning the operator $L$.

######  Lemma 5.4.

Under the Hypothesis 2.4 (either 2.4(a) or 2.4(b)), for every $p\geq
2,\tilde{p}\geq 1,l\geq 0$, there exists a constant $C_{l,p,\tilde{p}}(T)$
such that

$\displaystyle\sup_{M\in\mathbb{N}}\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\bar{c}(Z_{i}^{k})|LZ_{i}^{k}|_{l}^{\tilde{p}})^{p}\leq
C_{l,p,\tilde{p}}(T).$ (65)

###### Proof.

We notice that (with $\psi_{k}$ given in (44)),
$LZ_{i}^{k}=\xi_{i}^{k}(\ln{\psi_{k}})^{\prime}(V_{i}^{k})$ and
$D^{W,l}LZ^{k}_{i}=0$. Moreover,

$D^{Z,l}_{(r_{1},m_{1})\cdots(r_{l},m_{l})}LZ_{i}^{k}\\\
=\prod\limits_{j=1}^{l}(\delta_{r_{j}k}\delta_{m_{j}i})\xi_{i}^{k}(\ln{\psi_{k}})^{(l+1)}(V_{i}^{k}),$

with $\delta_{rk}$ the Kronecker delta, so that

$\displaystyle|LZ_{i}^{k}|_{l}=\xi_{i}^{k}\sum\limits_{0\leq\tilde{l}\leq
l}|(\ln{\psi_{k}})^{(\tilde{l}+1)}(V_{i}^{k})|.$ (66)

It follows that

$\displaystyle\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\bar{c}(Z_{i}^{k})|LZ_{i}^{k}|_{l}^{\tilde{p}})^{p}\leq
C_{l,p,\tilde{p}}\sum\limits_{0\leq\tilde{l}\leq
l}\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\bar{c}(Z_{i}^{k})\xi_{i}^{k}|(\ln{\psi_{k}})^{(\tilde{l}+1)}(V_{i}^{k})|^{\tilde{p}})^{p}.$

Since $\bar{c}(Z_{i}^{k})\xi_{i}^{k}=\bar{c}(V_{i}^{k})\xi_{i}^{k}$, we may
replace $Z_{i}^{k}$ by $V_{i}^{k}$ in the right hand side of the above
estimate. This gives

$C_{l,p,\tilde{p}}\sum\limits_{0\leq\tilde{l}\leq
l}\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\bar{c}(V_{i}^{k})\xi_{i}^{k}|(\ln{\psi_{k}})^{(\tilde{l}+1)}(V_{i}^{k})|^{\tilde{p}})^{p}=C_{l,p,\tilde{p}}\sum\limits_{0\leq\tilde{l}\leq
l}\mathbb{E}|\int_{0}^{t}\int_{[1,M)}\int_{\\{0,1\\}}\bar{c}(v)\xi|(\ln{\bar{\psi}})^{(\tilde{l}+1)}(v)|^{\tilde{p}}\Lambda(ds,d\xi,dv)|^{p},$

where
$\bar{\psi}(v):=\sum\limits_{k=1}^{\infty}\mathbbm{1}_{I_{k}}(v)\psi(v-(k+\frac{1}{2}))$
and $\Lambda$ is a Poisson point measure on $\\{0,1\\}\times[1,\infty)$ with
compensator

$\widehat{\Lambda}(ds,d\xi,dv)=\sum_{k=1}^{\infty}[\frac{\psi(v-(k+\frac{1}{2}))}{m(\psi)}\mathbbm{1}_{I_{k}}(v)dv\times
b(v,d\xi)]ds,$

with $b(v,d\xi)$ the Bernoulli probability measure on $\\{0,1\\}$ with
parameter $\varepsilon_{k}m(\psi)$, if $v\in I_{k}$. Then by compensating
$\Lambda$ and using Burkholder’s inequality (the same proof as for (5)),

$\displaystyle C_{l,p,\tilde{p}}\sum\limits_{0\leq\tilde{l}\leq
l}\mathbb{E}|\int_{0}^{t}\int_{[1,M)}\int_{\\{0,1\\}}\bar{c}(v)\xi|(\ln{\bar{\psi}})^{(\tilde{l}+1)}(v)|^{\tilde{p}}\Lambda(ds,d\xi,dv)|^{p}$
$\displaystyle\leq C_{l,p,\tilde{p}}(T)\sum\limits_{0\leq\tilde{l}\leq
l}[(\int_{0}^{t}\int_{[1,M)\times\\{0,1\\}}|\bar{c}(v)|^{2}\xi|(\ln{\bar{\psi}})^{(\tilde{l}+1)}(v)|^{2\tilde{p}}\widehat{\Lambda}(ds,d\xi,dv))^{\frac{p}{2}}$
$\displaystyle+\int_{0}^{t}\int_{[1,M)\times\\{0,1\\}}|\bar{c}(v)|^{p}\xi|(\ln{\bar{\psi}})^{(\tilde{l}+1)}(v)|^{p\tilde{p}}\widehat{\Lambda}(ds,d\xi,dv)+|\int_{0}^{t}\int_{[1,M)\times\\{0,1\\}}\bar{c}(v)\xi|(\ln{\bar{\psi}})^{(\tilde{l}+1)}(v)|^{\tilde{p}}\widehat{\Lambda}(ds,d\xi,dv)|^{p}].\quad\quad$
(67)

We notice that by (45),

$\displaystyle\int_{0}^{t}\int_{[1,M)\times\\{0,1\\}}|\bar{c}(v)|^{p}\xi|(\ln{\bar{\psi}})^{(\tilde{l}+1)}(v)|^{p\tilde{p}}\widehat{\Lambda}(ds,d\xi,dv)=t\sum_{k=1}^{M-1}\int_{I_{k}}\varepsilon_{k}m(\psi)|\bar{c}(v)|^{p}|(\ln{{\psi_{k}}})^{(\tilde{l}+1)}(v)|^{p\tilde{p}}\frac{\psi_{k}(v)}{m(\psi)}dv$
$\displaystyle\leq
C_{\tilde{l},p,\tilde{p}}(T)\sum_{k=1}^{M-1}\varepsilon_{k}\int_{I_{k}}|\bar{c}(v)|^{p}dv.$

Similar upper bounds hold for the two other terms in the right hanf side of
(67), so (67) is upper bounded by

$\displaystyle
C_{l,p,\tilde{p}}(T)[(\sum_{k=1}^{M-1}\varepsilon_{k}\int_{I_{k}}|\bar{c}(v)|^{2}dv)^{\frac{p}{2}}+\sum_{k=1}^{M-1}\varepsilon_{k}\int_{I_{k}}|\bar{c}(v)|^{p}dv+(\sum_{k=1}^{M-1}\varepsilon_{k}\int_{I_{k}}|\bar{c}(v)|dv)^{p}].$
(68)

If we assume the Hypothesis 2.4 (a), then we have
$\varepsilon_{k}=\varepsilon_{\ast}/(k+1)^{1-\alpha_{0}}$, with $\alpha_{0}$
given in (7). So the above term is less than

$C_{l,p,\tilde{p}}(T)[(\int_{1}^{\infty}\frac{|\bar{c}(v)|^{2}}{v^{1-\alpha_{0}}}dv)^{\frac{p}{2}}+\int_{1}^{\infty}\frac{|\bar{c}(v)|^{p}}{v^{1-\alpha_{0}}}dv+(\int_{1}^{\infty}\frac{|\bar{c}(v)|}{v^{1-\alpha_{0}}}dv)^{p}],$

which is upper bounded by a constant $C_{l,p,\tilde{p}}(T)$ thanks to (7). On
the other hand, if we assume the Hypothesis 2.4 (b), then
$\varepsilon_{k}=\varepsilon_{\ast}/(k+1)$. So (68) is upper bounded by a
constant $C_{l,p,\tilde{p}}(T)$ thanks to (9).

∎

####  5.1.3 Estimations of $\|X_{t}^{n,M}\|_{L,l,p}$

In this section, our aim is to prove the following lemma.

######  Lemma 5.5.

Under the Hypothesis 2.1 with $q^{\ast}\geq 2$ and Hypothesis 2.4 (either
2.4(a) or 2.4(b)), for all $p\geq 2,0\leq l\leq q^{\ast}$, there exists a
constant $C_{l,p}(T)$ depending on $l,p,x$ and $T$, such that

$\displaystyle
a)\quad\sup\limits_{n}\sup\limits_{M}\mathbb{E}|X_{t}^{n,M}|_{l}^{p}\leq
C_{l,p}(T),$ (69)

and for $0\leq l\leq q^{\ast}-2$,

$\displaystyle
b)\quad\sup\limits_{n}\sup\limits_{M}\mathbb{E}|LX_{t}^{n,M}|_{l}^{p}\leq
C_{l,p}(T).$ (70)

###### Proof.

In the following proof, $C_{l,p}(T)$ will be a constant which depends on
$l,p,x$ and $T$, and which may change from a line to another. $q^{\ast}\geq 2$
is fixed throughout the proof.

$a)$ We prove (69) for $0\leq l\leq q^{\ast}$ by recurrence on $l$.

Step 1 For $l=0$, using Burkholder’s inequality, Hypothesis 2.1 and (55),

$\displaystyle\mathbb{E}|X_{t}^{n,M}|^{p}$ $\displaystyle\leq$ $\displaystyle
C_{0,p}(T)[x^{p}+\mathbb{E}|\int_{0}^{t}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})dr|^{p}+\mathbb{E}|\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})W_{\nu}(dr,dz)|^{p}$
(71) $\displaystyle+$
$\displaystyle\mathbb{E}|\int_{0}^{t}\int_{[1,M)}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})N_{\nu}(dr,dz)|^{p}]$
$\displaystyle\leq$ $\displaystyle
C_{0,p}(T)[1+\mathbb{E}\int_{0}^{t}|\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)}^{n,M})\nu(dz)|^{p}dr$
$\displaystyle+$ $\displaystyle\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|^{2}\nu(dz))^{\frac{p}{2}}dr+\mathbb{E}\int_{0}^{t}(\int_{[1,M)}|\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})|^{2}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{t}\int_{[1,M)}|\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})|^{p}\nu(dz)dr+\mathbb{E}\int_{0}^{t}|\int_{[1,M)}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})\nu(dz)|^{p}dr]$
$\displaystyle\leq$ $\displaystyle C_{0,p}(T).$

Step 2 Now we assume that (69) holds for $l-1$, with $l\geq 1$ and for every
$p\geq 2$, and we prove that it holds for $l$ and for every $p\geq 2$. We
write $\mathbb{E}|X_{t}^{n,M}|_{l}^{p}\leq C_{l,p}(T)(A_{1}+A_{2}+A_{3})$,
with

$A_{1}=\mathbb{E}|\int_{0}^{t}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})dr|_{l}^{p},$
$A_{2}=\mathbb{E}|\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})W_{\nu}(dr,dz)|_{l}^{p},$
$A_{3}=\mathbb{E}|\int_{0}^{t}\int_{[1,M)}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})N_{\nu}(dr,dz)|_{l}^{p}.$

We notice that by Hypothesis 2.1, $\|b_{M}\|_{l,\infty}\leq\bar{c}_{1}.$ Then
using Lemma 5.3 $a)$ and the recurrence hypothesis, we get

$\displaystyle A_{1}$ $\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)\mathbb{E}\int_{0}^{t}|b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})|_{l}^{p}dr$
(72) $\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)[(\bar{c}_{1})^{p}+\mathbb{E}\int_{0}^{t}|\partial_{x}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})|^{p}|X_{\tau_{n}(r)}^{n,M}|_{1,l}^{p}dr$
$\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{t}\sup\limits_{2\leq|\beta|\leq
l}|\partial_{x}^{\beta}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})|^{p}|X_{\tau_{n}(r)}^{n,M}|_{1,l-1}^{lp}dr]$
$\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)[1+\int_{0}^{t}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr].$

Next, we estimate $A_{2}$. By Hypothesis 2.1, for every $n$,
$\|\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),\cdot)\|_{l,\infty}\leq|\bar{c}(\gamma_{n}(z))|$.
Then using Lemma 5.2 $a)$, Lemma 5.3 $a)$, (53) and the recurrence hypothesis,
we get

$\displaystyle A_{2}$ $\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)[\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr$
(73) $\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)[\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|\partial_{x}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|^{2}|X_{\tau_{n}(r)}^{n,M}|_{1,l}^{2}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle+$ $\displaystyle\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}\sup\limits_{2\leq|\beta|\leq
l}|\partial_{x}^{\beta}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|^{2}|X_{\tau_{n}(r)}^{n,M}|_{1,l-1}^{2l}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle+$ $\displaystyle\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|^{2}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)[1+\int_{0}^{t}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr].$

Finally we estimate $A_{3}$. We notice that
$D^{Z}_{(r,m)}Z_{i}^{k}=\xi_{i}^{k}\delta_{rk}\delta_{mi}$,
$D^{W}_{(s,z)}Z_{i}^{k}=0,$ and for $l\geq 2$,
$D^{Z,W,l}_{(r_{1},m_{1})\cdots(r_{l},m_{l}),(s_{1},z_{1})\cdots(s_{l},z_{l})}Z_{i}^{k}=0$.
So we have $|Z_{i}^{k}|_{1,l}^{p}=|\xi_{i}^{k}|^{p}\leq 1$. By Lemma 5.3 $a)$
for $d=2$, Hypothesis 2.1, for any $k,i\in\mathbb{N}$,

$\displaystyle|\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X^{n,M}_{\tau_{n}(T_{i}^{k})-})|_{l}\leq|\bar{c}(Z_{i}^{k})|$
$\displaystyle+(|\partial_{z}\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X_{\tau_{n}(T_{i}^{k})-}^{n,M})|+|\partial_{x}\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X_{\tau_{n}(T_{i}^{k})-}^{n,M})|)(|Z_{i}^{k}|_{1,l}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{1,l})$
$\displaystyle+{C}_{l,p}(T)\sup\limits_{2\leq|\beta_{1}+\beta_{2}|\leq
l}(|\partial_{z}^{\beta_{2}}\partial_{x}^{\beta_{1}}\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X_{\tau_{n}(T_{i}^{k})-}^{n,M})|)(|Z_{i}^{k}|_{1,l-1}^{l}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{1,l-1}^{l})$
$\displaystyle\leq{C}_{l,p}(T)\bar{c}(Z_{i}^{k})(1+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l-1}^{l}).$

It follows that

$\displaystyle A_{3}$ $\displaystyle\leq$
$\displaystyle\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J^{k}_{t}}|\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X^{n,M}_{\tau_{n}(T_{i}^{k})-})|_{l})^{p}\leq{C}_{l,p}(T)\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J^{k}_{t}}\bar{c}(Z_{i}^{k})(1+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l-1}^{l}))^{p}$
(74) $\displaystyle=$
$\displaystyle{C}_{l,p}(T)\mathbb{E}(\int_{0}^{t}\int_{[1,M)}{\bar{c}}(z)(1+|X_{\tau_{n}(r)-}^{n,M}|_{l}+|X_{\tau_{n}(r)-}^{n,M}|_{l-1}^{l})N_{\nu}(dr,dz))^{p}$
$\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)[1+\int_{0}^{t}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr],$

where the last inequality is obtained by using (5) and recurrence hypothesis.
Then combining (72),(73) and (74),

$\displaystyle\mathbb{E}|X_{t}^{n,M}|_{l}^{p}\leq{C}_{l,p}(T)[1+\int_{0}^{t}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr].$
(75)

So
$\mathbb{E}|X_{\tau_{n}(t)}^{n,M}|_{l}^{p}\leq{C}_{l,p}(T)[1+\int_{0}^{\tau_{n}(t)}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr]\leq{C}_{l,p}(T)[1+\int_{0}^{t}\mathbb{E}|X_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr]$.
We denote temporarily $g(t)=\mathbb{E}|X_{\tau_{n}(t)}^{n,M}|_{l}^{p}$, then
we have $g(t)\leq{C}_{l,p}(T)[1+\int_{0}^{t}g(r)dr]$. By Gronwall’s lemma,
$g(t)\leq{C}_{l,p}(T)e^{T{C}_{l,p}(T)}$, which means that

$\mathbb{E}|X_{\tau_{n}(t)}^{n,M}|_{l}^{p}\leq{C}_{l,p}(T)e^{T{C}_{l,p}(T)}.$

Substituting into (75), we conclude that

$\displaystyle\sup_{n,M}\mathbb{E}|X_{t}^{n,M}|_{l}^{p}\leq C_{l,p}(T).$ (76)

As a summary of the recurrence argument, we remark that the uniform bound in
$n,M$ of the operator $D$ for $l=0$ is due to the Hypothesis 2.1, and it
propagates to larger $l$ thanks to Lemma 5.3 $a)$.

$b)$ Now we prove (70) for $0\leq l\leq q^{\ast}-2$, by recurrence on $l$.

Step 1 One has to check that (70) holds for $l=0$. The proof is analogous to
that in the following Step 2, but simpler. It is done by using Lemma 5.3 $c)$,
(60), Burkholder’s inequality, Hypothesis 2.1,2.4, (53), (69) and Gronwall’s
lemma. So we skip it.

Step 2 Now we assume that (70) holds for $l-1$, with $l\geq 1$ and for any
$p\geq 2$ and we prove that it holds for $l$ and for any $p\geq 2$. We write
$\mathbb{E}|LX_{t}^{n,M}|_{l}^{p}\leq C_{l,p}(T)(B_{1}+B_{2}+B_{3})$, with

$B_{1}=\mathbb{E}|L\int_{0}^{t}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})dr|_{l}^{p},$
$B_{2}=\mathbb{E}|L\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})W_{\nu}(dr,dz)|_{l}^{p},$
$B_{3}=\mathbb{E}|L\int_{0}^{t}\int_{[1,M)}\widetilde{c}(\tau_{n}(r),z,X_{\tau_{n}(r)-}^{n,M})N_{\nu}(dr,dz)|_{l}^{p}.$

Using Lemma 5.3 $b)$, Hypothesis 2.1, the recurrence hypothesis and (69), we
get

$\displaystyle B_{1}$ $\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)\mathbb{E}\int_{0}^{t}|Lb_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})|_{l}^{p}dr$
(77) $\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)[\mathbb{E}\int_{0}^{t}|\partial_{x}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})|^{p}|LX_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr$
$\displaystyle+$
$\displaystyle\mathbb{E}\int_{0}^{t}\sup\limits_{2\leq|\beta|\leq
l+2}|\partial_{x}^{\beta}b_{M}(\tau_{n}(r),X_{\tau_{n}(r)}^{n,M})|^{p}(1+|X_{\tau_{n}(r)}^{n,M}|_{l+1}^{(l+2)p})(1+|LX_{\tau_{n}(r)}^{n,M}|_{l-1}^{p})dr]$
$\displaystyle\leq$
$\displaystyle{C}_{l,p}(T)[1+\int_{0}^{t}\mathbb{E}|LX_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr].$

Then by Lemma 5.2 $b)$, we get

$\displaystyle B_{2}$ $\displaystyle\leq$ $\displaystyle
C_{l,p}(T)[\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|L\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle+$ $\displaystyle\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr]$
$\displaystyle:=$ $\displaystyle C_{l,p}(T)[B_{2,1}+B_{2,2}].$

As a consequence of Lemma 5.3 $b)$, we have

$\displaystyle B_{2,1}$ $\displaystyle\leq$ $\displaystyle
C_{l,p}(T)[\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|\partial_{x}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|^{2}|LX_{\tau_{n}(r)}^{n,M}|_{l}^{2}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle+$ $\displaystyle\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}\sup\limits_{2\leq|\beta|\leq
l+2}|\partial_{x}^{\beta}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|^{2}(1+|X_{\tau_{n}(r)}^{n,M}|_{l+1}^{2(l+2)})(1+|LX_{\tau_{n}(r)}^{n,M}|_{l-1}^{2})\nu(dz))^{\frac{p}{2}}dr].$

And using Lemma 5.3 $a)$,

$\displaystyle B_{2,2}$ $\displaystyle\leq$ $\displaystyle
C_{l,p}(T)[\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|^{2}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle+$ $\displaystyle\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}|\partial_{x}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|^{2}|X_{\tau_{n}(r)}^{n,M}|_{1,l}^{2}\nu(dz))^{\frac{p}{2}}dr$
$\displaystyle+$ $\displaystyle\mathbb{E}\int_{0}^{t}(\int_{\\{z\geq
M\\}}\sup\limits_{2\leq|\beta|\leq
l}|\partial_{x}^{\beta}\widetilde{c}(\tau_{n}(r),\gamma_{n}(z),X_{\tau_{n}(r)}^{n,M})|^{2}|X_{\tau_{n}(r)}^{n,M}|_{1,l-1}^{2l}\nu(dz))^{\frac{p}{2}}dr].$

Then by Hypothesis 2.1, (53), (69) and the recurrence hypothesis,

$\displaystyle B_{2}\leq
C_{l,p}(T)[1+\int_{0}^{t}\mathbb{E}|LX_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr].$ (78)

Now we estimate $B_{3}$. By Lemma 5.3 $b)$ for $d=2$, Hypothesis 2.1, for any
$k,i\in\mathbb{N}$,

$\displaystyle|L\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X^{n,M}_{\tau_{n}(T_{i}^{k})-})|_{l}\leq(|\partial_{z}\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X_{\tau_{n}(T_{i}^{k})-}^{n,M})|+|\partial_{x}\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X_{\tau_{n}(T_{i}^{k})-}^{n,M})|)(|LZ_{i}^{k}|_{l}+|LX_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l})$
$\displaystyle+C_{l,p}(T)\sup\limits_{2\leq|\beta_{1}+\beta_{2}|\leq
l+2}(|\partial_{z}^{\beta_{2}}\partial_{x}^{\beta_{1}}\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X_{\tau_{n}(T_{i}^{k})-}^{n,M})|)$
$\displaystyle\times(1+|Z_{i}^{k}|_{l+1}^{l+2}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{l+2})(1+|LZ_{i}^{k}|_{l-1}+|LX_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l-1})$
$\displaystyle\leq
C_{l,p}(T){\bar{c}}(Z_{i}^{k})(1+|LZ_{i}^{k}|_{l}+|LX_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{l+2}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{l+2}\times(|LZ_{i}^{k}|_{l-1}+|LX_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l-1})).$

Then

$\displaystyle
B_{3}\leq\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}|L\widetilde{c}(\tau_{n}(T_{i}^{k}),Z_{i}^{k},X^{n,M}_{\tau_{n}(T_{i}^{k})-})|_{l})^{p}$
$\displaystyle\leq
C_{l,p}(T)\mathbb{E}|\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}{\bar{c}}(Z_{i}^{k})(1+|LZ_{i}^{k}|_{l}+|LX_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{l+2}$
$\displaystyle+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{l+2}\times(|LZ_{i}^{k}|_{l-1}+|LX_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l-1}))|^{p}$
$\displaystyle\leq C_{l,p}(T)(B_{3,1}+B_{3,2}+B_{3,3}),$

where

$B_{3,1}=\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}{\bar{c}}(Z_{i}^{k})|LX_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l})^{p},$
$B_{3,2}=\mathbb{E}|\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}{\bar{c}}(Z_{i}^{k})(|LZ_{i}^{k}|_{l}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{l+2}\times|LZ_{i}^{k}|_{l-1})|^{p},$
$B_{3,3}=\mathbb{E}|\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}{\bar{c}}(Z_{i}^{k})(1+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{l+2}+|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{l+2}\times|LX_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l-1})|^{p}.$

By (5),

$\displaystyle B_{3,1}$ $\displaystyle=$
$\displaystyle\mathbb{E}|\int_{0}^{t}\int_{[1,M)}{\bar{c}}(z)|LX_{\tau_{n}(r)-}^{n,M}|_{l}N_{\nu}(dr,dz)|^{p}$
(79) $\displaystyle\leq$ $\displaystyle
C_{l,p}(T)\int_{0}^{t}\mathbb{E}|LX_{\tau_{n}(r)-}^{n,M}|_{l}^{p}dr.$

Using Schwartz’s inequality, (5) and (69), we have

$\displaystyle\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}{\bar{c}}(Z_{i}^{k})|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{l+2}\times|LZ_{i}^{k}|_{l-1})^{p}$
$\displaystyle\leq[\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}{\bar{c}}(Z_{i}^{k})|X_{\tau_{n}(T_{i}^{k})-}^{n,M}|_{l+1}^{2(l+2)})^{p}]^{\frac{1}{2}}\times[\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}{\bar{c}}(Z_{i}^{k})|LZ_{i}^{k}|_{l-1}^{2})^{p}]^{\frac{1}{2}}$
$\displaystyle=[\mathbb{E}|\int_{0}^{t}\int_{[1,M)}{\bar{c}}(z)|X_{\tau_{n}(r)-}^{n,M}|_{l+1}^{2(l+2)}N_{\nu}(dr,dz)|^{p}]^{\frac{1}{2}}\times[\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}{\bar{c}}(Z_{i}^{k})|LZ_{i}^{k}|_{l-1}^{2})^{p}]^{\frac{1}{2}}$
$\displaystyle\leq
C_{l,p}(T)[\mathbb{E}(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}{\bar{c}}(Z_{i}^{k})|LZ_{i}^{k}|_{l-1}^{2})^{p}]^{\frac{1}{2}}.$

Then applying Lemma 5.4, we get

$B_{3,2}\leq{C}_{l,p}(T).$ (80)

By (5), (69) and recurrence hypothesis, we have

$\displaystyle B_{3,3}$ $\displaystyle=$
$\displaystyle\mathbb{E}|\int_{0}^{t}\int_{[1,M)}{\bar{c}}(z)(1+|X_{\tau_{n}(r)-}^{n,M}|_{l+1}^{l+2}+|X_{\tau_{n}(r)-}^{n,M}|_{l+1}^{l+2}\times|LX_{\tau_{n}(r)-}^{n,M}|_{l-1})N_{\nu}(dr,dz)|^{p}$
(81) $\displaystyle\leq$ $\displaystyle{C}_{l,p}(T).$

So by (79),(80) and (81),

$B_{3}\leq{C}_{l,p}(T)[1+\int_{0}^{t}\mathbb{E}|LX_{\tau_{n}(r)-}^{n,M}|_{l}^{p}dr].$
(82)

Then combining (77),(78) and (82),

$\displaystyle\mathbb{E}|LX_{t}^{n,M}|_{l}^{p}\leq{C}_{l,p}(T)[1+\int_{0}^{t}\mathbb{E}|LX_{\tau_{n}(r)}^{n,M}|_{l}^{p}dr],$
(83)

Using Gronwall’s lemma for (83) as for (75), we conclude that

$\displaystyle\sup_{n,M}\mathbb{E}|LX_{t}^{n,M}|_{l}^{p}\leq C_{l,p}(T).$ (84)

As a summary of the recurrence argument, we remark that the uniform bound in
$n,M$ of the operator $L$ for $l=0$ is due to the Hypothesis 2.1,2.4 and Lemma
5.3 $c)$, and it propagates to larger $l$ thanks to Lemma 5.3 $b)$. ∎

Proof of Lemma 4.1.

By Lemma 5.1 and Lemma 5.5, as a consequence of Lemma 3.2, we have
$X_{t}^{M}\in\mathcal{D}_{l,p}$ and $\sup\limits_{M}\|X_{t}^{M}\|_{L,l,p}\leq
C_{l,p}(T)$.∎

###  5.2 Proof of Lemma 4.2

In the following, we turn to the non-degeneracy of $X_{t}^{M}$. We consider
the approximate equation (49)

$X_{t}^{M}=x+\int_{0}^{t}\int_{[1,M)}\widetilde{c}(r,z,X_{r-}^{M})N_{\nu}(dr,dz)+\int_{0}^{t}b_{M}(r,X_{r}^{M})dr+\int_{0}^{t}\int_{\\{z\geq
M\\}}\widetilde{c}(r,z,X_{r}^{M})W_{\nu}(dr,dz).$

We can calculate the Malliavin derivatives of the Euler scheme and then by
passing to the limit, we have

$\displaystyle D^{Z}_{(k,i)}X_{t}^{M}=\mathbbm{1}_{\\{k\leq
M-1\\}}\mathbbm{1}_{\\{i\leq
J_{t}^{k}\\}}\xi^{k}_{i}\partial_{z}\widetilde{c}(T_{i}^{k},Z_{i}^{k},X_{T_{i}^{k}-}^{M})+\int_{T_{i}^{k}}^{t}\int_{[1,M)}\partial_{x}\widetilde{c}(r,z,X^{M}_{r-})D^{Z}_{(k,i)}X_{r-}^{M}N_{\nu}(dr,dz)$
$\displaystyle+\int_{T_{i}^{k}}^{t}\partial_{x}b_{M}(r,X_{r}^{M})D^{Z}_{(k,i)}X_{s}^{M}dr+\int_{T_{i}^{k}}^{t}\int_{\\{z\geq
M\\}}\partial_{x}\widetilde{c}(r,z,X_{r}^{M})D^{Z}_{(k,i)}X_{s}^{M}W_{\nu}(dr,dz).$
(85) $\displaystyle
D^{W}_{(s,z_{0})}X_{t}^{M}=\int_{s}^{t}\int_{[1,M)}\partial_{x}\widetilde{c}(r,z,X^{M}_{r-})D^{W}_{(s,z_{0})}X_{r-}^{M}N_{\nu}(dr,dz)+\int_{s}^{t}\partial_{x}b_{M}(r,X_{r}^{M})D^{W}_{(s,z_{0})}X_{r}^{M}dr$
$\displaystyle+\mathbbm{1}_{\\{s\leq t\\}}\mathbbm{1}_{\\{z_{0}\geq
M\\}}\widetilde{c}(s,z_{0},X_{s}^{M})+\int_{s}^{t}\int_{\\{z\geq
M\\}}\partial_{x}\widetilde{c}(r,z,X_{r}^{M})D^{W}_{(s,z_{0})}X_{r}^{M}W_{\nu}(dr,dz).$
(86)

We obtain now some explicit expressions for the Malliavin derivatives. We
consider the tangent flow $(Y^{M}_{t})_{t\in[0,T]}$ which is the solution of
the linear equation

$Y_{t}^{M}=1+\int_{0}^{t}\int_{[1,M)}\partial_{x}\widetilde{c}(r,z,X^{M}_{r-})Y_{r-}^{M}N_{\nu}(dr,dz)+\int_{0}^{t}\partial_{x}b_{M}(r,X_{r}^{M})Y_{r}^{M}dr+\int_{0}^{t}\int_{\\{z\geq
M\\}}\partial_{x}\widetilde{c}(r,z,X_{r}^{M})Y_{r}^{M}W_{\nu}(dr,dz).$

And using It$\hat{o}$’s formula, $\overline{Y}_{t}^{M}=1/Y^{M}_{t}$ verifies
the equation

$\displaystyle\overline{Y}_{t}^{M}=1-\int_{0}^{t}\int_{[1,M)}\partial_{x}\widetilde{c}(r,z,X^{M}_{r-})(1+\partial_{x}\widetilde{c}(r,z,X^{M}_{r-}))^{-1}\overline{Y}_{r-}^{M}N_{\nu}(dr,dz)-\int_{0}^{t}\partial_{x}b_{M}(r,X_{r}^{M})\overline{Y}_{r}^{M}dr$
$\displaystyle-\int_{0}^{t}\int_{\\{z\geq
M\\}}\partial_{x}\widetilde{c}(r,z,X_{r}^{M})\overline{Y}_{r}^{M}W_{\nu}(dr,dz)+\frac{1}{2}\int_{0}^{t}\int_{\\{z\geq
M\\}}|\partial_{x}\widetilde{c}(r,z,X_{r}^{M})|^{2}\overline{Y}_{r}^{M}\nu(dz)dr.$

Applying Hypothesis 2.1 with $q^{\ast}\geq 1$ and Hypothesis 2.2, with $K_{p}$
a constant only depending on $p$, one also has (the proof is standard)

$\mathbb{E}(\sup_{s\leq
t}(\left|Y_{s}^{M}\right|^{p}+\left|\overline{Y}_{s}^{M}\right|^{p}))\leq
K_{p}<\infty.$ (87)

######  Remark.

Due to (4), we have

$\max\Big{\\{}\int_{[1,M)}|\bar{c}(z)|^{p}\nu(dz),\int_{[M,\infty)}|\bar{c}(z)|^{p}\nu(dz)\Big{\\}}\leq\int_{[1,\infty)}|\bar{c}(z)|^{p}\nu(dz)=\bar{c}_{p},$

so the constant in (87) is uniform with respect to $M$.

Then using the uniqueness of solution to the equation (85) and (86), one
obtains

$\displaystyle D^{Z}_{(k,i)}X_{t}^{M}$ $\displaystyle=$
$\displaystyle\mathbbm{1}_{\\{k\leq M-1\\}}\mathbbm{1}_{\\{i\leq
J_{t}^{k}\\}}\xi^{k}_{i}Y_{t}^{M}\overline{Y}^{M}_{T^{k}_{i}-}\partial_{z}\widetilde{c}(T^{k}_{i},Z^{k}_{i},X^{M}_{T^{k}_{i}-}),$
$\displaystyle D^{W}_{(s,z_{0})}X_{t}^{M}$ $\displaystyle=$
$\displaystyle\mathbbm{1}_{\\{s\leq t\\}}\mathbbm{1}_{\\{z_{0}\geq
M\\}}Y_{t}^{M}\overline{Y}^{M}_{s}\widetilde{c}(s,z_{0},X_{s}^{M}).$ (88)

And the Malliavin covariance of $X_{t}^{M}$ is

$\displaystyle\sigma_{X_{t}^{M}}=\left\langle
DX_{t}^{M},DX_{t}^{M}\right\rangle_{\mathcal{H}}=\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}|D^{Z}_{(k,i)}X_{t}^{M}|^{2}+\int_{0}^{T}\int_{\\{z\geq
M\\}}|D^{W}_{(s,z)}X_{t}^{M}|^{2}\nu(dz)ds.$ (89)

In the following, we denote $\lambda_{t}^{M}=\sigma_{X_{t}^{M}}$. So the aim
is to prove that for every $p\geq 1$,

$\mathbb{E}(|\lambda_{t}^{M}|^{-p})\leq C_{p}.$ (90)

We proceed in 5 steps.

Step 1 We notice that by (88) and (89)

$\displaystyle\lambda_{t}^{M}=\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\xi_{i}^{k}|Y_{t}^{M}|^{2}|\overline{Y}_{T_{i}^{k}-}^{M}|^{2}|\partial_{z}\widetilde{c}(T_{i}^{k},Z_{i}^{k},X_{T_{i}^{k}-}^{M})|^{2}+|Y_{t}^{M}|^{2}\int_{0}^{t}|\overline{Y}_{s}^{M}|^{2}\int_{\\{z\geq
M\\}}|\widetilde{c}(s,z,X_{s}^{M})|^{2}\nu(dz)ds.$

We recall the ellipticity hypothesis (Hypothesis 2.3): There exists a function
$\underline{c}(z)$ such that

$\left|\partial_{z}\widetilde{c}(s,z,x)\right|^{2}\geq\underline{c}(z)\quad
and\quad\left|\widetilde{c}(s,z,x)\right|^{2}\geq\underline{c}(z).$

In particular

$\int_{\\{z\geq M\\}}|\widetilde{c}(s,z,x)|^{2}\nu(dz)\geq\int_{\\{z\geq
M\\}}\underline{c}(z)\nu(dz),$

so that

$\lambda_{t}^{M}\geq
Q_{t}^{-2}\times(\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\xi_{i}^{k}\underline{c}(Z_{i}^{k})+t\int_{\\{z\geq
M\\}}\underline{c}(z)\nu(dz))\quad with\quad Q_{t}=\inf_{s\leq
t}|Y_{s}^{M}\overline{Y}_{t}^{M}|.$

We denote

$\rho_{t}^{M}=\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\xi_{i}^{k}\underline{c}(Z_{i}^{k}),\quad\bar{\rho}_{t}^{M}=\sum_{k=M}^{\infty}\sum_{i=1}^{J_{t}^{k}}\xi_{i}^{k}\underline{c}(Z_{i}^{k}),\quad\alpha^{M}=\int_{\\{z\geq
M\\}}\underline{c}(z)\nu(dz).$

By (87), $(\mathbb{E}\sup\limits_{s\leq
t}\left|Y_{s}^{M}\overline{Y}_{t}^{M}\right|^{4p})^{1/2}\leq C<\infty,$ so
that

$\displaystyle\mathbb{E}(|\lambda_{t}^{M}|^{-p})\leq
C(\mathbb{E}(|\rho_{t}^{M}+t\alpha^{M}|^{-2p}))^{\frac{1}{2}}.$ (91)

Step 2 Let $\Gamma(p)=\int_{0}^{\infty}s^{p-1}e^{-s}ds$. By a change of
variables, we have the numerical equality

$\frac{1}{(\rho_{t}^{M}+t\alpha^{M})^{p}}=\frac{1}{\Gamma(p)}\int_{0}^{\infty}s^{p-1}e^{-s(\rho_{t}^{M}+t\alpha^{M})}ds$

which, by taking expectation, gives

$\displaystyle\mathbb{E}(\frac{1}{(\rho_{t}^{M}+t\alpha^{M})^{p}})=\frac{1}{\Gamma(p)}\int_{0}^{\infty}s^{p-1}\mathbb{E}(e^{-s(\rho_{t}^{M}+t\alpha^{M})})ds.$
(92)

Step 3 (splitting). In order to compute
$\mathbb{E}(e^{-s(\rho_{t}^{M}+t\alpha^{M})})$ we have to interpret
$\rho_{t}^{M}$ in terms of Poisson measures. We recall that we suppose the
"splitting hypothesis" (40):

$\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}\geq\mathbbm{1}_{I_{k}}(z)\varepsilon_{k}dz,$

with $I_{k}=[k,k+1),\ m_{k}=\nu(I_{k})$. We also have the function $\psi$ and
$m(\psi)=\int_{\mathbb{R}}\psi(t)dt.$ And we use the basic decomposition

$Z_{i}^{k}=\xi_{i}^{k}V_{i}^{k}+(1-\xi_{i}^{k})U_{i}^{k}$

where $V_{i}^{k},U_{i}^{k},\xi_{i}^{k},k,i\in\mathbb{N}$ are some independent
random variables with laws given in (47).

For every $k$ we consider a Poisson point measure $N_{k}(ds,d\xi,dv,du)$ with
$\xi\in\\{0,1\\},v,u\in[1,\infty),s\in[0,T]$ with compensator

$\displaystyle\widehat{N_{k}}(ds,d\xi,dv,du)$ $\displaystyle=$
$\displaystyle\widehat{M}_{k}(d\xi,dv,du)\times ds$ $\displaystyle
with\quad\widehat{M}_{k}(d\xi,dv,du)$ $\displaystyle=$ $\displaystyle
b_{k}(d\xi)\times\mathbbm{1}_{I_{k}}(v)\frac{1}{m(\psi)}\psi(v-(k+\frac{1}{2}))dv$
$\displaystyle\times$
$\displaystyle\frac{1}{1-\varepsilon_{k}m(\psi)}\mathbbm{1}_{I_{k}}(u)(\mathbb{P}(Z_{1}^{k}\in
du)-\varepsilon_{k}\psi(u-(k+\frac{1}{2}))du).$

Here $b_{k}(d\xi)$ is the Bernoulli law of parameter $\varepsilon_{k}m(\psi)$.
The intervals $I_{k},k\in\mathbb{N}$ are disjoint so the Poisson point
measures $N_{k},k=1,\cdots,M-1$ are independent. Then

$\sum_{i=1}^{J_{t}^{k}}\xi_{i}^{k}{\underline{c}}(Z_{i}^{k})=\sum_{i=1}^{J_{t}^{k}}\xi_{i}^{k}{\underline{c}}(\xi_{i}^{k}V_{i}^{k}+(1-\xi_{i}^{k})U_{i}^{k})=\int_{0}^{t}\int_{\\{0,1\\}}\int_{[1,\infty)^{2}}\xi{\underline{c}}(\xi
v+(1-\xi)u)N_{k}(ds,d\xi,dv,du).$

In order to get compact notation, we put together all the measures
$N_{k},k\leq M-1.$ Since they are independent we get a new Poisson point
measure that we denote by $\Theta.$ And we have

$\displaystyle\rho_{t}^{M}$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{M-1}\sum_{i=1}^{J_{t}^{k}}\xi_{i}^{k}{\underline{c}}(Z_{i}^{k})=\int_{0}^{t}\int_{\\{0,1\\}}\int_{[1,\infty)^{2}}\xi{\underline{c}}(\xi
v+(1-\xi)v)\Theta(ds,d\xi,dv,du).$

Step 4 Using Itô’s formula,

$\displaystyle\mathbb{E}(e^{-s\rho_{t}^{M}})$ $\displaystyle=$ $\displaystyle
1+\mathbb{E}\int_{0}^{t}\int_{\\{0,1\\}}\int_{[1,\infty)^{2}}(e^{-s(\rho_{r-}^{M}+\xi{\underline{c}}(\xi
v+(1-\xi)v))}-e^{-s\rho_{r-}^{M}})\widehat{\Theta}(dr,d\xi,dv,du)$
$\displaystyle=$ $\displaystyle
1-\int_{0}^{t}\mathbb{E}(e^{-s\rho_{r-}^{M}})dr\int_{\\{0,1\\}}\int_{[1,\infty)^{2}}(1-e^{-s\xi{\underline{c}}(\xi
v+(1-\xi)v)})\sum_{k=1}^{M-1}\widehat{M}_{k}(d\xi,dv,du).$

Solving the above equation we obtain

$\displaystyle\mathbb{E}(e^{-s\rho_{t}^{M}})$ $\displaystyle=$
$\displaystyle\exp(-t\sum_{k=1}^{M-1}\int_{\\{0,1\\}}\int_{[1,\infty)^{2}}(1-e^{-s\xi{\underline{c}}(\xi
v+(1-\xi)u)})\widehat{M}_{k}(d\xi,dv,du)).$

We compute

$\displaystyle\int_{\\{0,1\\}\times[1,\infty)^{2}}(1-e^{-s\xi{\underline{c}}(\xi
v+(1-\xi)u)})\widehat{M}_{k}(d\xi,dv,du)=\varepsilon_{k}m(\psi)\int_{k}^{k+1}(1-e^{-s{\underline{c}}(v)})\frac{1}{m(\psi)}\psi(v-(k+\frac{1}{2}))dv.$

Since $\psi\geq 0$ and $\psi(z)=1$ if $\left|z\right|\leq\frac{1}{4}$ it
follows that the above term is larger than

$\varepsilon_{k}\int_{k+\frac{1}{4}}^{k+\frac{3}{4}}(1-e^{-s{\underline{c}}(v)})dv.$

Finally this gives

$\displaystyle\mathbb{E}(e^{-s\rho_{t}^{M}})$ $\displaystyle\leq$
$\displaystyle\exp(-t\sum_{k=1}^{M-1}\varepsilon_{k}\int_{k+\frac{1}{4}}^{k+\frac{3}{4}}(1-e^{-s{\underline{c}}(v)})dv)$
$\displaystyle=$
$\displaystyle\exp(-t\int_{1}^{M}(1-e^{-s{\underline{c}}(v)})m(dv)),$

with

$\displaystyle
m(dv)=\sum_{k=1}^{\infty}\varepsilon_{k}1_{(k+\frac{1}{4},k+\frac{3}{4})}(v)dv.$
(93)

In the same way, we get

$\mathbb{E}(e^{-s\bar{\rho}_{t}^{M}})\leq\exp(-t\int_{M}^{\infty}(1-e^{-s\underline{{c}}(v)})m(dv)).$

Notice that $t\alpha^{M}\geq\mathbb{E}(\bar{\rho}_{t}^{M})$. Then using
Jensen’s inequality for the convex function $f(x)=e^{-sx},s,x>0$, we have

$\displaystyle e^{-st\alpha^{M}}\leq
e^{-s\mathbb{E}\bar{\rho}_{t}^{M}}\leq\mathbb{E}(e^{-s\bar{\rho}_{t}^{M}})\leq\exp(-t\int_{M}^{\infty}(1-e^{-s\underline{{c}}(v)})m(dv)).$

So for every $M\in\mathbb{N}$, we get

$\displaystyle\mathbb{E}(e^{-s(\rho_{t}^{M}+t\alpha^{M})})$ $\displaystyle=$
$\displaystyle e^{-st\alpha^{M}}\times\mathbb{E}(e^{-s\rho_{t}^{M}})$ (94)
$\displaystyle\leq$
$\displaystyle\exp(-t\int_{M}^{\infty}(1-e^{-s\underline{{c}}(v)})m(dv))\times\exp(-t\int_{1}^{M}(1-e^{-s\underline{{c}}(v)})m(dv))$
$\displaystyle=$
$\displaystyle\exp(-t\int_{1}^{\infty}(1-e^{-s\underline{{c}}(v)})m(dv)),$

and the last term does not depend on $M$.

Now we will use the Lemma 14 from [7], which states the following.

######  Lemma 5.6.

We consider an abstract measurable space $E$, a $\sigma$-finite measure $\eta$
on this space and a non-negative measurable function
$f:E\rightarrow\mathbb{R}_{+}$ such that $\int_{E}fd\eta<\infty.$ For $t>0$
and $p\geq 1$, we note

$\alpha_{f}(t)=\int_{E}(1-e^{-tf(a)})\eta(da)\quad and\quad
I_{t}^{p}(f)=\int_{0}^{\infty}s^{p-1}e^{-t\alpha_{f}(s)}ds.$

We suppose that for some $t>0$ and $p\geq 1$,

$\underline{\lim}_{u\rightarrow\infty}\frac{1}{\ln
u}\eta(f\geq\frac{1}{u})>p/t,$ (95)

then $I_{t}^{p}(f)<\infty.$

We will use the above lemma for $\eta=m$ and $f=\underline{c}$. So if we have

$\underline{\lim}_{u\rightarrow\infty}\frac{1}{\ln
u}m({\underline{c}}\geq\frac{1}{u})=\infty,$ (96)

then for every $p\geq 1,t>0,M\geq 1$, (92),(94) and Lemma 5.6 give

$\displaystyle\mathbb{E}(\frac{1}{\rho_{t}^{M}+t\alpha^{M}})^{2p}$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(2p)}\int_{0}^{\infty}s^{2p-1}\mathbb{E}(e^{-s(\rho_{t}^{M}+t\alpha^{M})})ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{\Gamma(2p)}\int_{0}^{\infty}s^{2p-1}\exp(-t\int_{1}^{\infty}(1-e^{-s{\underline{c}}(v)})m(dv))ds<\infty.$

Finally using (91), we conclude that if (96) holds, then

$\displaystyle\sup_{M}\mathbb{E}(\lambda_{t}^{M})^{-p}<\infty.$ (98)

Step 5 Now the only problem left is to compute
$m({\underline{c}}\geq\frac{1}{u}).$ It seems difficult to discuss this in a
completely abstract framework. So we suppose Hypothesis 2.4 (a): There exists
a constant $\varepsilon_{\ast}>0$ and there are some $\alpha_{1}>\alpha_{2}>0$
such that for every $k\in\mathbb{N}$,

$\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}\geq\mathbbm{1}_{I_{k}}(z)\varepsilon_{k}dz\quad
with\quad\varepsilon_{k}=\frac{\varepsilon_{\ast}}{{(k+1)}^{1-{{\alpha}}}},\
for\ any\ {\alpha}\in(\alpha_{2},\alpha_{1}],\quad
and\quad{\underline{c}}(z)\geq e^{-z^{\alpha_{2}}},$

Then $\\{z:{\underline{c}}(z)\geq\frac{1}{u}\\}\supseteq\\{z:(\ln
u)^{1/{\alpha_{2}}}\geq z\\}.$ In particular, for $k\leq\lfloor(\ln
u)^{1/\alpha_{2}}\rfloor-1:=k(u)$, one has
$I_{k}\subseteq\\{z:\underline{c}(z)\geq\frac{1}{u}\\}$. Then for $u$ large
enough, we compute

$\displaystyle m({\underline{c}}\geq\frac{1}{u})$ $\displaystyle\geq$
$\displaystyle\sum_{k=1}^{k(u)}m(I_{k})\geq\frac{1}{2}\sum_{k=1}^{k(u)}\varepsilon_{k}\geq\frac{1}{2}\varepsilon_{\ast}\sum_{k=1}^{k(u)}\frac{1}{(k+1)^{1-{{\alpha}}}}\geq\frac{1}{2}\varepsilon_{\ast}\int_{2}^{(\ln
u)^{1/{\alpha_{2}}}}\frac{1}{z^{1-{{\alpha}}}}dz$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{\ast}}{{2{\alpha}}}((\ln
u)^{{\alpha}/{\alpha_{2}}}-2^{{{\alpha}}}).$

Since ${\alpha}>\alpha_{2}$, (96) is verified and we obtain (98).

Now we consider Hypothesis 2.4 (b): We suppose that there exists a constant
$\varepsilon_{\ast}>0$ and there are some $\alpha>0$ such that for every
$k\in\mathbb{N}$,

$\mathbbm{1}_{I_{k}}(z)\frac{\nu(dz)}{m_{k}}\geq\mathbbm{1}_{I_{k}}(z)\varepsilon_{k}dz\quad
with\quad\varepsilon_{k}=\frac{\varepsilon_{\ast}}{k+1},\quad
and\quad{\underline{c}}(z)\geq\frac{1}{z^{\alpha}}.$

Now $\\{z:{\underline{c}}(z)\geq\frac{1}{u}\\}\supseteq\\{z:z\leq
u^{1/\alpha}\\}$. Then for $u$ large enough,

$m({\underline{c}}\geq\frac{1}{u})\geq\frac{1}{2}\varepsilon_{\ast}\sum_{k=1}^{\lfloor
u^{1/\alpha}\rfloor-1}\frac{1}{k+1}\geq\frac{1}{2}\varepsilon_{\ast}\int_{2}^{u^{1/\alpha}}\frac{dz}{z}=\frac{1}{2}\varepsilon_{\ast}(\frac{1}{\alpha}\ln
u-\ln{2}).$

And consequently

$\underline{\lim}_{u\rightarrow\infty}\frac{1}{\ln
u}m({\underline{c}}\geq\frac{1}{u})\geq\frac{\varepsilon_{\ast}}{2\alpha}.$

Using Lemma 5.6, this gives: if

$\frac{2p}{t}<\frac{\varepsilon_{\ast}}{2\alpha}\quad\Leftrightarrow\quad
t>\frac{4p\alpha}{\varepsilon_{\ast}}$

then

$\sup_{M}\mathbb{E}(\frac{1}{\rho_{t}^{M}+t\alpha^{M}})^{2p}<\infty,$

and we have $\sup\limits_{M}\mathbb{E}(\lambda_{t}^{M})^{-p}<\infty$.∎

###  5.3 Some proofs concerning Section 4.2

We will prove that the triplet $(\mathcal{S},D,L)$ defined in Section 4.2 is
an IbP framework. Here, we only show that $D^{q}$ is closable and $L$ verifies
the duality formula (32). To do so, we introduce the divergence operator
$\delta$. We denote the space of simple processes by

$\displaystyle\mathcal{P}=\\{u=((\bar{u}^{k}_{i})_{\begin{subarray}{c}1\leq
i\leq m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},\sum_{r=1}^{n}u_{r}\varphi_{r}):\bar{u}^{k}_{i},u_{r}\in\mathcal{S},\varphi_{r}\in
L^{2}(\mathbb{R}_{+}\times\mathbb{R}_{+},\nu\times
Leb),m^{\prime},m,n\in\mathbb{N}\\}.$

For $u=((\bar{u}^{k}_{i})_{\begin{subarray}{c}1\leq i\leq m^{\prime}\\\ 1\leq
k\leq m\end{subarray}},\sum_{r=1}^{n}u_{r}\varphi_{r})\in\mathcal{P}$, we
denote $u^{Z}=(\bar{u}^{k}_{i})_{\begin{subarray}{c}1\leq i\leq m^{\prime}\\\
1\leq k\leq m\end{subarray}}$ and $u^{W}=\sum_{r=1}^{n}u_{r}\varphi_{r}$, so
that $u=(u^{Z},u^{W})$.
We notice that $\mathcal{P}$ is dense in $L^{2}(\Omega;\mathcal{H})$, with
$\mathcal{H}=l_{2}\otimes L^{2}(\mathbb{R}_{+}\times\mathbb{R}_{+},\nu\times
Leb)$.

Then we define the divergence operator
$\delta:\mathcal{P}\rightarrow\mathcal{S}$ by

$\displaystyle\delta(u)=\delta^{Z}(u^{Z})+\delta^{W}(u^{W})$ $\displaystyle
with\ $
$\displaystyle\delta^{Z}(u^{Z})=-\sum_{k=1}^{m}\sum_{i=1}^{m^{\prime}}(D^{Z}_{(k,i)}\bar{u}^{k}_{i}+\xi^{k}_{i}\bar{u}^{k}_{i}\times\theta_{k}(V^{k}_{i}))$
$\displaystyle\delta^{W}(u^{W})=\sum_{r=1}^{n}u_{r}W_{\nu}(\varphi_{r})-\sum_{r=1}^{n}\langle
D^{W}u_{r},\varphi_{r}\rangle_{L^{2}(\mathbb{R}_{+}\times\mathbb{R}_{+},\nu\times
Leb)}.$

We will show that $\delta$ satisfies the following duality formula: For every
$F\in\mathcal{S},u\in\mathcal{P}$,

$\mathbb{E}\langle DF,u\rangle_{\mathcal{H}}=\mathbb{E}F\delta(u).$ (99)

In fact, if we denote $\hat{V}_{i}^{k}(x)$ the sequence
$(V_{i_{0}}^{k_{0}})_{\begin{subarray}{c}1\leq i_{0}\leq m^{\prime}\\\ 1\leq
k_{0}\leq m\end{subarray}}$ after replacing $V_{i}^{k}$ by $x$, then for any
$m^{\prime},m\in\mathbb{N}$,

$\displaystyle\mathbb{E}\langle
D^{Z}F,u^{Z}\rangle_{l_{2}}=\mathbb{E}\sum_{k=1}^{m}\sum_{i=1}^{m^{\prime}}D_{(k,i)}^{Z}F\times\bar{u}^{k}_{i}$
$\displaystyle=\sum_{k=1}^{m}\sum_{i=1}^{m^{\prime}}\mathbb{E}\xi^{k}_{i}\partial_{v^{k}_{i}}f(\omega,(V_{i_{0}}^{k_{0}})_{\begin{subarray}{c}1\leq
i_{0}\leq m^{\prime}\\\ 1\leq k_{0}\leq
m\end{subarray}},(W_{\nu}(\varphi_{j}))_{j=1}^{n})\bar{u}^{k}_{i}(\omega,(V_{i_{0}}^{k_{0}})_{\begin{subarray}{c}1\leq
i_{0}\leq m^{\prime}\\\ 1\leq k_{0}\leq
m\end{subarray}},(W_{\nu}(\varphi_{j}))_{j=1}^{n})$
$\displaystyle=\sum_{k=1}^{m}\sum_{i=1}^{m^{\prime}}\mathbb{E}\int_{\mathbb{R}}\xi^{k}_{i}\partial_{v^{k}_{i}}f(\omega,\hat{V}_{i}^{k}(x),(W_{\nu}(\varphi_{j}))_{j=1}^{n})\times\bar{u}_{k}(\omega,\hat{V}_{i}^{k}(x),(W_{\nu}(\varphi_{j}))_{j=1}^{n})\frac{\psi_{k}(x)}{m(\psi)}dx$
$\displaystyle=-\sum_{k=1}^{m}\sum_{i=1}^{m^{\prime}}\mathbb{E}\int_{\mathbb{R}}\xi^{k}_{i}f(\omega,\hat{V}_{i}^{k}(x),(W_{\nu}(\varphi_{j}))_{j=1}^{n})\times[\partial_{v^{k}_{i}}\bar{u}^{k}_{i}(\omega,\hat{V}_{i}^{k}(x),(W_{\nu}(\varphi_{j}))_{j=1}^{n})$
$\displaystyle+\bar{u}^{k}_{i}(\omega,\hat{V}_{i}^{k}(x),(W_{\nu}(\varphi_{j}))_{j=1}^{n})\frac{\partial_{x}\psi_{k}(x)}{\psi_{k}(x)}]\frac{\psi_{k}(x)}{m(\psi)}dx$
$\displaystyle=-\sum_{k=1}^{m}\sum_{i=1}^{m^{\prime}}\mathbb{E}F[D^{Z}_{(k,i)}\bar{u}^{k}_{i}+\xi^{k}_{i}\bar{u}^{k}_{i}\partial_{x}(\ln\psi_{k}(V^{k}_{i}))]=\mathbb{E}(F\delta^{Z}(u^{Z})).$

On the other hand, since $L^{2}(\mathbb{R}_{+}\times\mathbb{R}_{+},\nu\times
Leb)$ is a separable Hilbert space, we can assume without loss of generality
that, in the definition of simple functionals,
$(\varphi_{1},\cdots,\varphi_{m},\cdots)$ is the orthogonal basis of the space
$L^{2}(\mathbb{R}_{+}\times\mathbb{R}_{+},\nu\times Leb)$.
Then with
$p_{r}=\int_{\mathbb{R}_{+}\times\mathbb{R}_{+}}\varphi_{r}^{2}(s,z)\nu(dz)ds$,
for any $n\in\mathbb{N}$,

$\displaystyle\mathbb{E}\langle
D^{W}F,u^{W}\rangle_{L^{2}(\mathbb{R}_{+}\times\mathbb{R}_{+},\nu\times
Leb)}=\mathbb{E}\int_{\mathbb{R}_{+}\times\mathbb{R}_{+}}D^{W}_{(s,z)}F\times\sum_{r=1}^{n}u_{r}\varphi_{r}(s,z)\
\nu(dz)ds$
$\displaystyle=\mathbb{E}\sum_{r=1}^{n}\partial_{w_{r}}f(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq
i\leq m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},(W_{\nu}(\varphi_{j}))_{j=1}^{n})u_{r}(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq
i\leq m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},(W_{\nu}(\varphi_{j}))_{j=1}^{n})p_{r}$
$\displaystyle=\sum_{r=1}^{n}\mathbb{E}\int_{\mathbb{R}}\partial_{w_{r}}f(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq
i\leq m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},W_{\nu}(\varphi_{1}),\cdots,W_{\nu}(\varphi_{r-1}),y,W_{\nu}(\varphi_{r+1}),\cdots,W_{\nu}(\varphi_{n}))$
$\displaystyle\times u_{r}(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq i\leq
m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},W_{\nu}(\varphi_{1}),\cdots,W_{\nu}(\varphi_{r-1}),y,W_{\nu}(\varphi_{r+1}),\cdots,W_{\nu}(\varphi_{n}))\frac{1}{\sqrt{2\pi
p_{r}}}e^{-\frac{y^{2}}{2p_{r}}}dy\times p_{r}$
$\displaystyle=-\sum_{r=1}^{n}\mathbb{E}\int_{\mathbb{R}}f(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq
i\leq m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},W_{\nu}(\varphi_{1}),\cdots,W_{\nu}(\varphi_{r-1}),y,W_{\nu}(\varphi_{r+1}),\cdots,W_{\nu}(\varphi_{n}))$
$\displaystyle\times[\partial_{w_{r}}u_{r}(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq
i\leq m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},W_{\nu}(\varphi_{1}),\cdots,W_{\nu}(\varphi_{r-1}),y,W_{\nu}(\varphi_{r+1}),\cdots,W_{\nu}(\varphi_{n}))$
$\displaystyle-\frac{y}{p_{r}}u_{r}(\omega,(V_{i}^{k})_{\begin{subarray}{c}1\leq
i\leq m^{\prime}\\\ 1\leq k\leq
m\end{subarray}},W_{\nu}(\varphi_{1}),\cdots,W_{\nu}(\varphi_{r-1}),y,W_{\nu}(\varphi_{r+1}),\cdots,W_{\nu}(\varphi_{n}))]\frac{1}{\sqrt{2\pi
p_{r}}}e^{-\frac{y^{2}}{2p_{r}}}dy\times p_{r}$
$\displaystyle=\mathbb{E}F(\sum_{r=1}^{n}u_{r}W_{\nu}(\varphi_{r})-\sum_{r=1}^{n}\langle
D^{W}u_{r},\varphi_{r}\rangle_{L^{2}(\mathbb{R}_{+}\times\mathbb{R}_{+},\nu\times
Leb)})=\mathbb{E}(F\delta^{W}(u^{W})).$

Then (99) is proved. Using this duality formula recursively, we can show the
closability of $D^{q}$. If there exists $u\in
L^{2}(\Omega;\mathcal{H}^{\otimes q})$ such that $F_{n}\rightarrow 0$ in
$L^{2}(\Omega)$ and $D^{q}F_{n}\rightarrow u$ in
$L^{2}(\Omega;\mathcal{H}^{\otimes q})$, then for any
$h_{1},\cdots,h_{q}\in\mathcal{P}$, $\mathbb{E}\langle
u,h_{1}\otimes\cdots\otimes h_{q}\rangle_{\mathcal{H}^{\otimes
q}}=\lim\limits_{n\rightarrow\infty}\mathbb{E}\langle
D^{q}F_{n},h_{1}\otimes\cdots\otimes h_{q}\rangle_{\mathcal{H}^{\otimes
q}}=\lim\limits_{n\rightarrow\infty}\mathbb{E}F_{n}\delta(h_{1}\delta(h_{2}(\cdots\delta(h_{q}))))=0$.
Since $\mathcal{P}^{\otimes q}$ is dense in $L^{2}(\Omega;\mathcal{H}^{\otimes
q})$, we conclude that $u=0$. This implies that $D^{q}$ is closable.

We notice that from the definition of $\delta$ and $L$, we get immediately
that $LF=\delta(DF),\ \forall F\in\mathcal{S}$. And if we replace $u$ by $DG$
in (99) for $G\in\mathcal{S}$, we get the duality formula of $L$ (32).

Data avaibility statement. Data sharing is not applicable to this article as
no datasets were generated or analyzed during the current study.

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