Well-posedness and propagation of chaos for McKean-Vlasov equations with jumps and locally Lipschitz coefficients

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Well-posedness and propagation of chaos for McKean-Vlasov equations with jumps and locally

Lipschitz coefficients

Xavier Erny

To cite this version:

Xavier Erny. Well-posedness and propagation of chaos for McKean-Vlasov equations with jumps and

locally Lipschitz coefficients. 2021. �hal-03139922�

Well-posedness and propagation of chaos for McKean-Vlasov equations with jumps and locally

Lipschitz coecients

Xavier Erny

Université Paris-Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d'Evry, 91037, Evry, France

Abstract: We study McKean-Vlasov equations where the coecients are locally Lipschitz continuous. We prove the strong well-posedness and a propagation of chaos property in this framework. These questions can be treated with classical arguments under the assumptions that the coecients are globally Lipschitz continuous. In the locally Lipschitz case, we use truncation arguments and Osgood's lemma instead of Grönwall's lemma. This approach entails technical diculties in the proofs, in particular for the existence of solution of the McKean- Vlasov equations that are considered. This proof relies on a Picard iteration scheme that is not guaranteed to converge in an L

sense because the coecients are not Lipschitz continuous.

However, we still manage to prove its convergence in distribution, and the (strong) well- posedness of the equation using a generalization of Yamada and Watanabe results.

MSC2020 subject classications: 60J60, 60K35.

Keywords and phrases: McKean-Vlasov equations, Mean eld interaction, Interacting par- ticle systems, Propagation of chaos.

1. Introduction

The aim of this paper is to prove the strong well-posedness and a propagation of chaos property for Mckean-Vlasov equations. These are SDEs where the coecients depend on the solution of the equation and on the law of this solution. This type of equation arises naturally in the framework of N particle systems where the particles interact in a mean eld way: this phenomenon can be seen as a law of large numbers. Indeed, in examples where the dynamic of the N particle system is directed by an SDE, the mean eld interactions can be expressed as a dependency of the coecients on the empirical measure of the system. And as N goes to innity, this empirical measure converges to the law of any particle of the limit system. This entails natural dependencies of the coecients on the law of the solution of the limit SDE. For instance, see De Masi et al. (2015) and Fournier and Löcherbach (2016) for examples in neural network modeling, Fischer and Livieri (2016) for an example in portfolio modeling, and Carmona, Delarue and Lacker (2016) for an application in mean eld games.

The weak well-posedness and the propagation of chaos are classical for the McKean-Vlasov equations without jump term, even without assuming that the coecients are Lipschitz continuous.

Gärtner (1988) treats both questions in this frame. We can also mention more recent work on the well-posedness of McKean-Vlasov equations without jump term as Mishura and Veretennikov (2020) and Chaudru de Raynal (2020), with dierent assumptions on the smoothness of the coecients, and Lacker (2018) that investigates the well-posedness and the propagation of chaos.

In this paper, we consider McKean-Vlasov equations with jumps. The questions about the strong well-posedness and the propagation of chaos have also been studied in this framework under globally

1

Lipschitz assumptions on the coecients: see Graham (1992) for the well-posedness and Andreis, Dai Pra and Fischer (2018) for the propagation of chaos. Note that in Section 4 of Andreis, Dai Pra and Fischer (2018), these questions are treated in a multi-dimensional case, where the drift coef- cient is of the form ∇b 1 p x q b 2 p x, m q , where b 1 is C ¹ and convex, and b 2 , as well as the jump coecient and the volatility coecient, are globally Lipschitz.

The novelty of our results is to work on McKean-Vlasov equations with jumps and with generic locally Lipschitz assumptions (see Assumption 1 for a precise and complete statement of the hy- pothesis).

The rst main result is the strong well-posedness, under this locally Lipschitz assumption, of the following McKean-Vlasov equation

dX _t b p X _t , µ _t q dt σ p X _t , µ _t q dW _t

»

R

E

Φ p X _t , µ _t , u q1 _t z ¤ f p X

,µ

qu dπ p t, z, u q , where µ _t is the distribution of X _t , W a Brownian motion, π a Poisson measure and E some measurable space (see the beginning of Section 2 for details on the notation). To prove the well- posedness when the coecients are locally Lipschitz continuous, we adapt the computations of the proofs in the globally Lipschitz continuous case. We use a truncation argument to handle the dependency of the local Lipschitz constant w.r.t. to the variables. On the contrary of the globally Lipschitz case, Grönwall's lemma does not allow to conclude immediately. In the locally Lipschitz case, we have to use a generalization of this lemma: Osgood's lemma (see Lemma 4.1).

The uniqueness of solution of the McKean-Vlasov equation follows rather quickly from Osgood's lemma and the truncation argument, but other diculties emerge in the proof of the existence of solution. We construct a weak solution of the equation using a Picard iteration scheme, but the fact that the coecients are only locally Lipschitz continuous does not allow to prove that this scheme converges in an L ¹ sense. Instead, we prove that a subsequence converges in distribution to some limit that is shown to be a solution of the equation. Some technical diculties emerge in this part of the proof for two reasons. The rst one is that the Picard scheme is not shown to converge but only to have a converging subsequence. This implies that we need to control the variation between two consecutive steps of the scheme. The second one is that we only prove a convergence in distribution, thus it is not straightforward that the limit of the Picard scheme is solution to the equation. It is shown studying its semimartingale characteristics.

The second main result is a propagation of chaos property of McKean-Vlasov particle systems under the same locally Lipschitz assumptions. More precisely, it is the convergence of the following N particle system

dX _t ^N,i b p X _t ^N,i , µ ^N _t q dt σ p X _t ^N,i , µ ^N _t q dW _t ⁱ

»

R

F

Ψ p X _t ^N,i , µ ^N _t , v ⁱ q1 t ^{z ¤ f p X}

,µ

q u dπ ⁱ p t, z, v q 1

N

¸ N j 1

»

R

F

Θ p X _t ^N,j , X _t ^N,i , µ ^N _t , v ^j , v ⁱ q1 t ^{z ¤ f p X}

,µ

q u dπ ^j p t, z, v q , where µ ^N _t : N ¹ ° N

j 1 δ _X

t

, to the innite system d X ¯ _t ⁱ b p X ¯ _t ⁱ , µ ¯ t q dt σ p X ¯ _t ⁱ , µ ¯ t q dW _t ⁱ

»

R

F

Ψ p X ¯ _t ⁱ , µ ¯ t , v ⁱ q1 t ^{z ¤ f p X ¯}

,¯ µ

q u dπ ⁱ p t, z, v q

»

R

»

F

Θ p x, X ¯ _t ⁱ , µ ¯ t , v ¹ , v ² q f p x, µ ¯ t q dν p v q d¯ µ t p x q ,

where µ ¯ t : L p X ¯ t q, as N goes to innity. The W ⁱ ( i ¥ 1 ) are independent standard Brownian motions, the π ⁱ ( i ¥ 1 ) are independent Poisson measures, F is some measurable space and N denotes the set of the positive integers (see Section 3 for details on the notation). The proof of this second main result relies on a similar reasoning as the one used to prove the uniqueness of the McKean-Vlasov equation: a truncation argument and Osgood's lemma.

Let us note that this propagation of chaos property has already been proven under dierent hypothesis. Indeed, the N particle system and the limit system above are the same as in Andreis, Dai Pra and Fischer (2018). Note also that, in this model, for each N P N , the particles X _t ^N,i ( 1 ¤ i ¤ N ) do not only interact through the empirical measure µ ^N _t , but also through the simultaneous jumps term.

Let us mention the other conditions in Assumption 1. We assume that some boundedness condi- tions on the coecients hold true. To the best of our knowledge, in order to prove a priori estimates for solutions of Mckean-Vlasov equations, one needs to assume that the coecients are bounded w.r.t. the measure variable. In this paper, we need the coecients to be bounded w.r.t. to both variables, because for the truncation arguments mentioned above, we need a priori estimates on the exponential moments of the solutions of the McKean-Vlasov equation. This is also the reason why we need conditions on exponential moments.

Let us nally mention that, in this paper, we chose to work in dimension one to simplify the notation, but the results still hold in nite dimensions.

Organization. In Section 2, we state and prove our rst main result: the well-posedness of the McKean-Vlasov equation (1) with locally Lipschitz coecients. Section 3 is devoted to our second main result, the propagation of chaos in the same framework.

1.1. Notation

Let us introduce some notation we use throughout the paper:

• If X is random variable, we note L p X q its distribution.

• If X and X _n ( n P N ) are random variables, we note X _{n n} ÝÑ

Ñ 8 X for "p X _n q n converges in distribution to X ".

• P 1 pRq is the space of probability measures on R with nite rst moment. This space will always be endowed with the rst-order Wassertein metric W 1 dened by: for m 1 , m 2 P P 1 pRq ,

W ₁ p m ₁ , m ₂ q inf

X

m

,X

m

E r d p X ₁ , X ₂ qs sup

f P Lip

»

R

f p x q dm ₁ p x q

»

R

f p x q dm ₂ p x q , with Lip ₁ the space of Lipschitz continuous functions w.r.t. the metric d with Lipschitz con- stant non-greater than one. Let us note that characterizations of this convergence are given in Theorem 6.9 and Denition 6.8 of Villani (2008).

• For T ¡ 0 and p G, d q a Polish space, D pr 0, T s , G q (resp. D pR , G q) denotes the space of càdlàg G valued functions dened on r 0, T s (resp. R ) endowed with Skorohod topology, whence this space is Polish. Let us recall that the convergence of a sequence p x n q n of D pr 0, T s , G q to some x in Skorohod topology is equivalent to the existence of continuous increasing functions λ n satisfying λ n p 0 q 0, λ n p T q T and both

sup

0 ¤ t ¤ T | λ n p t q t | and sup

0 ¤ t ¤ T

d p x p λ n p t qq , x n p t qq

vanish as n goes to innity. In the following, we call such a sequence p λ n q n a sequence of time-changes.

• L denotes the Lipschitz constant of the coecients (see Assumption 1).

• C denotes any arbitrary positive constant, whose value can change from line to line in an equation. If the constant depends on some parameter θ, we write C θ instead.

2. Well-posedness of McKean-Vlasov equations

This section is dedicated to prove the well-posedness of the following McKean-Vlasov equation.

dX _t b p X _t , µ _t q dt σ p X _t , µ _t q dW _t

»

R

E

Φ p X _t , µ _t , u q1 _t z ¤ f p X

,µ

qu dπ p t, z, u q , (1) with µ t L p X t q , W a standard one-dimensional Brownian motion, π a Poisson measure on R R E having intensity dt dz dρ p u q , where p E, E, ρ q is a σ nite measure space. The assumptions on the coecients are specied in Assumption 1 below. Let us note here that f is assumed to be non-negative.

Assumption 1.

1. Locally Lipschitz conditions: there exists a ¡ 0, such that for all x 1 , x 2 P R , m 1 , m 2 P P 1 pRq ,

| b p x ₁ , m ₁ q b p x ₂ , m ₂ q|

»

E

»

R

| Φ p x ₁ , m ₁ , u q1 _t z ¤ f p x

,m

qu Φ p x ₂ , m ₂ , u q1 _t z ¤ f p x

,m

qu | dzdρ p u q

¤ L

1 | x ₁ | | x ₂ |

»

R

e ^{a | x |} dm ₁ p x q

»

R

e ^{a | x |} dm ₂ p x q

p| x ₁ x ₂ | W ₁ p m ₁ , m ₂ qq . 2. Globally Lipschitz condition for σ : for all x 1 , x 2 P R , m 1 , m 2 P P 1 pRq ,

| σ p x 1 , m 1 q σ p x 2 , m 2 q| ¤ L p| x 1 x 2 | W 1 p m 1 , m 2 qq .

3. Boundedness conditions: the functions b, σ and f are bounded (uniformly w.r.t. all the vari- ables), and for the same constant a ¡ 0 as in Item 1.,

sup

x PR ,m P

pRq

»

E

e ^{a | Φ p x,m,u q|} dρ p u q   8 . 4. Initial condition: for the same a ¡ 0 as in Items 1. and 3.,

E

e ^{a | X}

^|   8 .

Remark 2.1. If we consider equation (1) without the jump term (that is Φ 0 ), then, we can adapt the proof of Theorem 2.3 to the case where σ is also locally Lipschitz continuous. More precisely, we can replace the two rst Items of Assumption 1 by: for all x 1 , x 2 P R , m 1 , m 2 P P 1 pRq ,

| b p x 1 , m 1 q b p x 2 , m 2 q| | σ p x 1 , m 1 q σ p x 2 , m 2 q|

¤ L

1 a

| x 1 | a

| x 2 | d»

R

e ^{a | x |} dm 1 p x q d»

R

e ^{a | x |} dm 2 p x q

p| x 1 x 2 | W 1 p m 1 , m 2 qq .

See Remark 2.6 for more details on the adaptation of the proof.

Note that Item 3 of Assumption 1 implies that, for all n P N , sup

x PR ,m P

pRq

»

E

| Φ p x, m, u q| ⁿ dν p u q   8 .

Remark 2.2. Under the boundedness conditions of Assumption 1, a sucient condition to obtain the locally Lipschitz condition of the jump term in Assumption 1 is given by the following direct conditions on the functions f and Φ : for all x 1 , x 2 P R , m 1 , m 2 P P 1 pRq ,

| f p x 1 , m 1 q f p x 2 , m 2 q|

»

E

| Φ p x 1 , m 1 , u q Φ p x 2 , m 2 , u q| dρ p u q

¤ L

1 | x 1 | | x 2 |

»

R

e ^{a | x |} dm 1 p x q

»

R

e ^{a | x |} dm 2 p x q

p| x 1 x 2 | W 1 p m 1 , m 2 qq . Example 1. A natural form of the coecient of a McKean-Vlasov equation is given by the so-called

"true McKean-Vlasov" case. For simplicity, we only give the form for b , but a similar form can be considered for the other coecient.

b p x, m q

»

R

˜ b p x, y q dm p y q , with ˜ b : R ² Ñ R .

For b to satisfy the conditions of Assumption 1 in this example, it is sucient to assume that ˜ b is bounded and that: for all x, x ¹ , y, y ¹ P R ,

| ˜ b p x, y q ˜ b p x ¹ , y ¹ q| ¤ C p 1 | x | | x ¹ |qp| x x ¹ | | y y ¹ |q . Indeed, for any x, x ¹ P R , m, m ¹ P P ₁ pRq ,

| b p x, m q b p x ¹ , m ¹ q| ¤| b p x, m q b p x ¹ , m q| | b p x ¹ , m q b p x ¹ , m ¹ q|

¤

»

R

| ˜ b p x, y q ˜ b p x ¹ , y q| dm p y q »

R

˜ b p x ¹ , y q dm p y q

»

R

˜ b p x ¹ , y q dm ¹ p y q

¤ C p 1 | x | | x ¹ |q| x x ¹ | C p 1 2 | x ¹ |q W 1 p m, m ¹ q ,

where the second quantity of the last line has been obtained using Kantorovich-Rubinstein duality (see Remark 6.5 of Villani (2008)) and the fact that, for a xed x ¹ , the function y ÞÑ ˜ b p x ¹ , y q is Lipschitz continuous with Lipschitz constant C p 1 2 | x ¹ |q .

Theorem 2.3. Under Assumption 1, there exists a unique strong solution of (1).

The rest of this section is dedicated to prove Theorem 2.3.

2.1. A priori estimates for equation (1)

In this section, we prove the following a priori estimates for the solutions of the SDE (1).

Lemma 2.4. Grant the boundedness conditions and the initial condition of Assumption 1. Any

solution p X t q t ¥ 0 of (1) satises for all t ¡ 0,

(i) sup

0 ¤ s ¤ t E

e ^{a | X}

^|

  8 , with a ¡ 0 the same as constant as in Assumption 1, (ii) E

sup

0 ¤ s ¤ t | X s |

  8 .

Proof. Let us prove p i q. It is sucient to prove sup

0 ¤ s ¤ t E e ^aX

  8 , (2)

and

sup

0 ¤ s ¤ t E

e ^aX

  8 . (3)

To prove (2), let us apply Ito's formula e ^aX

e ^aX

a

» t 0

e ^aX

b p X s , µ s q ds a

» t 0

e ^aX

σ p X s , µ s q dW s

a ² 2

» t 0

e ^aX

σ p X s , µ s q ² ds

»

r 0,t sR E

e ^{a p X}

^{Φ p X}

^,µ

^{,u qq} e ^aX

1 t z ¤ f p X

,µ

qu dπ p s, z, u q . Introducing, for M ¡ 0 , the stopping time τ M : inf t t ¡ 0 : | X t | ¡ M u, we have

E

e ^aX

¤E e ^aX

a || b || 8

» t 0

E

e ^aX

ds 1

2 a ² || σ || ² ₈

» t 0

E

e ^aX

ds

» t 0

»

E

e ^aX

e ^{Φ p X}

^,µ

^{,u q} 1

f p X s ^ τ

, µ s ^ τ

q dρ p u q ds.

Then, introducing u ^M _t : E

e ^aX

, and using the boundedness condition of Φ from Assump- tion 1, we obtain, for all t ¡ 0,

u ^M _t ¤ E e ^aX

K

» t 0

u ^M _s ds, with

K : a || b || ₈ 1

2 a ² || σ || ² ₈ || f || ₈ sup

x PR ,m P

pRq

»

E

e ^{a | Φ p x,m,u q|} dρ p u q   8 . Consequently, Grönwall's lemma implies

sup

0 ¤ s ¤ t

u ^M _s ¤ E e ^aX

e ^Kt .

As the bound above does not depend on M , it implies that τ M goes to innity almost surely as M goes to innity. Fatou's lemma then implies

sup

0 ¤ s ¤ t E e ^aX

¤ E e ^aX

e ^Kt . This proves (2), and with the same reasoning we can prove

sup

0 ¤ s ¤ t E

e ^aX

¤ E

e ^aX

e ^Kt .

which proves (3), whence the point p i q of the lemma.

To prove the point p ii q, let us use the following bound, which is a direct consequence of the form of the SDE (1),

E

sup

0 ¤ s ¤ t | X s |

¤ E r| X 0 |s || b || 8 t E

sup

0 ¤ s ¤ t

» s 0

σ p X t , µ r q dW r

|| f || 8

» t 0

E

»

E

| Φ p X _s , µ _s , u q| dρ p u q

ds.

Then the result follows from Burkholder-Davis-Gundy's inequality and the boundedness condi- tions of σ and Φ from Assumption 1.

2.2. Pathwise uniqueness for equation (1)

Proposition 2.5. Grant Assumption 1. The pathwise uniqueness property holds true for (1).

Proof. Let p X ˆ t q t ¥ 0 and p X ˇ t q t ¥ 0 be two solutions of (1) dened w.r.t. the same initial condition X 0 , the same Brownian motion W and the same Poisson measure π . The proof consists in showing that the function

u p t q : E

sup

0 ¤ s ¤ t | X ˆ s X ˇ s |

is zero. This choice of function u is inspired of the proof of Theorem 2.1 of Graham (1992). It allows to treat equations with both a jump term and a Brownian term.

We know that, for all t ¥ 0, u t   8 by Lemma 2.4 . p ii q . Writing µ ˆ t : L p X ˆ t q and µ ˇ t : L p X ˇ t q , we have

X ˆ _s X ˇ _s ¤

» s 0

| b p X ˆ _r , µ ˆ _r q b p X ˇ _r , µ ˇ _r q| dr » s

0

p σ p X ˆ _r , µ ˆ _r q σ p X ˇ _r , µ ˇ _r qq dW _r

»

r 0,s sR R

Φ p X ˆ r , µ ˆ r , u q1 t ^{z ¤ f p X ˆ}

,ˆ µ

q u Φ p X ˇ r , µ ˇ r , u q1 t ^{z ¤ f p X ˇ}

,ˇ µ

q u dπ p r, z, u q . This implies that

u p t q ¤ L E

» t

0

| X ˆ s X ˇ s | W 1 p µ ˆ s , µ ˇ s q ² ds ¹ { 2

2L

» t 0

E

1 | X ˆ _s | | X ˇ _s |

»

R

e ^{a | x |} dˆ µ _s p x q

»

R

e ^{a | x |} dˇ µ _s p x q | X ˆ _s X ˇ _s | W ₁ p µ ˆ _s , µ ˇ _s q ds (4) where we have used Burkholder-Davis-Gundy's inequality to deal with the Brownian term that corresponds to the term at the rst line above. The term at the rst line corresponds to the controls of the drift term and the jump term.

By Lemma 2.4 . p i q, we have for all t ¥ 0, sup

0 ¤ s ¤ t

»

R

e ^{a | x |} dˆ µ _s p x q sup

0 ¤ s ¤ t

»

R

e ^{a | x |} dˇ µ _s p x q ¤ C _t   8 .

And, from the denition of W 1 , we have the following bound W ₁ p µ ˆ _s , µ ˇ _s q ¤ E X ˆ _s X ˇ _s

¤ u p s q . Then, (4) and Lemma 2.4 imply that, for all 0 ¤ t ¤ T,

u p t q ¤

» t 0

E

1 | X ˆ s | | X ˇ s | C T

| X ˆ s X ˇ s | u p s q ds

L E

» t

0

| X ˆ s X ˇ s | u p s q ² ds 1 { 2

¤ C T

» t 0

E

1 | X ˆ s | | X ˇ s |

| X ˆ s X ˇ s | u p s q ds 2L ? tu p t q

¤ C _T

» t 0

E

1 | X ˆ _s | | X ˇ _s |

| X ˆ _s X ˇ _s | ds C _T

» t 0

u p s q ds 2L ? tu p t q ,

where we have bounded the second integral of the RHS of the rst inequality above by t times the supremum of the integrand. Note that the value of C _T changes from line to line.

Now, to end the proof, we have to control a term of the type p 1 | x | | y |q| x y | . To do so, we use a truncation argument based on the following inequality: for all x, y P R , R ¡ 0,

p 1 | x | | y |q| x y | ¤ p 1 2R q| x y | p 1 | x | | y |q| x y | 1 t| x |¡ R u 1 t| y |¡ R u . Let R : s ÞÑ R _s ¡ 0 be the truncation function whose values will be chosen later. By Lemma 2.4, for any 0 ¤ s ¤ T,

E

1 | X ˆ s | | X ˇ s | X ˆ s X ˇ s

¤ p 1 2R s q u p s q C T

c P

| X ˆ s | ¡ R s C T

b

P | X ˇ s | ¡ R s

.

The exponential moments proven in Lemma 2.4 . p i q are used to control the two last term above.

Indeed, by Markov's inequality P

| X ˆ s | ¡ R s P | X ˇ s | ¡ R s

¤ C T e ^aR

. Consequently, dening r s : aR s { 2 , for any 0 ¤ t ¤ T,

u p t q ¤ C T

» t 0

p 1 r s q u p s q e ^r

ds 2L ? tu p t q . Now, let T 1 {p 16L ² q such that 2L ?

T ¤ 1 { 2. Then, we can rewrite the above inequality as, for all t P r 0, T s ,

u p t q ¤ C _T

» t 0

p 1 r _s q u p s q e ^r

ds.

Let us prove by contradiction that, for all t ¤ T, u p t q 0. To do so, let t ₀ : inf t t ¡ 0 : u p t q ¡ 0 u and assume that t ₀   T. Notice that, as u is non-decreasing, this implies that, for all t P r 0, t ₀ r , u p t q 0. In particular, for all t P r t ₀ , T s ,

u p t q ¤ C _T

» t t

p 1 r _s q u p s q e ^r

ds.

Besides u p t q is nite and bounded (see Lemma 2.4 . p ii q) on r 0, T s , say by a constant D ¡ 1. Let v p t q : u p t q{p De ² q   e ² . Obviously v satises the same inequality as u above. Now we dene r s : ln v p s q , so that, for all t 0   t ¤ T,

v p t q ¤ C _T

» t t

p 2 ln v p s qq v p s q ds ¤ 2C _T

» t t

v p s q ln v p s q ds, where we have used that 2 ln v p s q ¤ 2 ln v p s q since ln v p s q ¥ 2.

In particular, for any c Ps 0, e ² r , for all t 0 ¤ t ¤ T, v p t q ¤ c 2C _T

» t t

v p s q ln v p s q ds.

Then, introducing M : x Ps 0, e ² rÞÑ ³ e

x

sln 1 s ds, we may apply Osgood's lemma (see Lemma 4.1) with γ 2C T and µ p v q p ln v q v to obtain that

M p v p T qq M p c q ¤

» T t

2C T ds 2C T p T t 0 q or equivalently,

M p c q ¤ M p v p T qq 2C T T.

Recalling that we assumed v p T q ¡ 0 such that the right hand side of the above equality is nite, if we let c tend to 0, we obtain

M p 0 q

» e

0

1

s ln s ds ¤

» e

v p T q

1

s ln s ds 2C T T   8 , which is absurd since M p 0 q 8 .

A consequence of the above considerations is that for all t P r 0, T s, u p t q 0. Recalling the denition of u , we have proven that the processes X ˆ and X ˇ are equal on r 0, T s.

We can repeat this argument on the interval r T, 2T s and iterate up to any nite time interval r 0, T 0 s since T 1 {p 16L ² q does only depend on the coecients of the system but not on the initial condition. This proves the pathwise-uniqueness property for the McKean-Vlasov equation (1).

Let us complete our previous Remark 2.1.

Remark 2.6. The adaptation suggested in Remark 2.1 is the following: in the proof of Proposi- tion 2.5 above, one has to replace the distance E

sup _{s ¤ t} | X ˆ s X ˇ s | by E

p X ˆ t X ˇ t q ²

, and one has to do similar changes in the proof of Proposition 2.11 below.

2.3. Existence of a weak solution of equation (1)

Before proving the existence of solution of (1), let us state some elementary lemmas about series and Skorohod topology, whose proofs are postponed to the Appendix.

Lemma 2.7. Let p u n q n ¥ 0 be a sequence of non-negative real numbers, and S n ° n

k 0 u k ( n P N).

If there exists 0   ε   1, such that for all n P N , S n ¤ Cn ^{1 ε} , then there exists a subsequence of

p u n q n ¥ 0 that converges to 0.

Lemma 2.8. Let T ¡ 0, and p x n q n be a sequence of càdlàg functions converging to some càdlàg function x in D pr 0, T s , Rq. In addition, if p y n q n is a sequence of càdlàg functions that satises

sup

0 ¤ t ¤ T

| x n p t q y n p t q| ÝÑ

n Ñ8 0, then, the sequence p x n , y n q n converges to p x, x q in D pr 0, T s , R ² q .

Lemma 2.9. Let T ¡ 0, and x and x n p n P Nq be càdlàg functions. Let λ n ( n P N) be continuous, increasing functions satisfying λ n p 0 q 0, λ n p T q T , and that both

sup

0 ¤ t ¤ T | x _n p t q x p λ _n p t qq| and sup

0 ¤ t ¤ T | t λ _n p t q|

vanish as n goes to innity. Then, sup

0 ¤ t ¤ T

» t 0

x n p s q ds

» λ

p t q 0

x p s q ds

ÝÑ

n Ñ8 0.

Remark 2.10. An interesting consequence of the previous lemma is that, if p x _n , y _n q n converges to p x, y q in D pr 0, T s , R ² q , then p x n , ³

0 y n p s q ds q n converges to p x, ³

0 y p s q ds q in D pr 0, T s , R ² q . Note that it is important to have convergence in D pr 0, T s , R ² q instead that in D pr 0, T s , Rq ² . The dierence between these two topologies is that, the convergence in D pr 0, T s , R ² q means that the two coordinates have to share the same sequence of time-changes, whereas for the convergence in D pr 0, T s , Rq ² each of the coordinates has its own sequence.

The aim of this section is to construct a weak solution of the McKean-Vlasov equation (1), using a Picard iteration. The idea of the proof is to show that this scheme converges to a solution of (1).

However, because of our locally Lipschitz conditions, we cannot prove it directly. Instead, we prove that a subsequence converges in distribution by tightness. That is why, in a rst time, we only construct a weak solution.

Proposition 2.11. Grant Assumption 1. There exists a weak solution of (1) on r 0, T s , with T 1 {p 16L ² q .

Proof. As in the proof of the pathwise uniqueness of Section 2.2, we work on a time interval r 0, T s where T ¡ 0 is a number whose value can be xed at 1 {p 16L ² q.

Step 1. In this rst step, we introduce the iteration scheme, and state its basic properties at (5).

Let X _t ^{r 0 s} : X 0 , and dene the process X ^{r n 1 s} from X ^{r n s} and µ ^r _t ^{n s} : L p X _t ^{r n s} q by X _t ^{r n 1 s} : X ₀

» t 0

b p X _s ^{r n s} , µ ^r _s ^{n s} q ds

» t 0

σ p X _s ^{r n s} , µ ^r _s ^{n s} q dW _s

»

r 0,t sR R

Φ p X _s ^{r n s} , µ ^r _s ^{n s} , u q1 ^! _{z ¤ f p X}

s

,µ

q ) dπ p s, z, u q

Note that, thanks to the boundedness conditions of Assumption 1, using the same computations as in the proof of Lemma 2.4, we can prove that, for all t ¥ 0,

sup

n PN sup

0 ¤ s ¤ t E

e ^{a | X}

^|

  8 and sup

n PN E

sup

0 ¤ s ¤ t

| X _s ^{r n s} |

  8 . (5)

Step 2. Now let us show that p X ^{r n s} , X ^{r n 1 s} q n has a converging subsequence in distribution

in D pr 0, T s , R ² q , by showing that it satises Aldous' tightness criterion:

p a q for all ε ¡ 0 , lim δ Ó 0 lim sup _{N Ñ8} sup _{p S,S}

qP A

P p| X _S ^{r n}

^s X _S ^{r n s} | | X _S ^{r n}

^{1 s} X _S ^{r n s} | ¡ ε q 0 , where A _δ,T is the set of all pairs of stopping times p S, S ¹ q such that 0 ¤ S ¤ S ¹ ¤ S δ ¤ T a.s.,

p b q lim _{K Ò8} sup _n Pp sup _{t Pr 0,T s} | X _t ^{r n s} | | X _t ^{r n 1 s} | ¥ K q 0 .

Assertion p b q is a straightforward consequence of (5) and Markov's inequality. To check asser- tion p a q, notice that, for any p S, S ¹ q P A _δ,T , by BDG inequality,

E X _S ^{r n}

^{1 s} X _S ^{r n 1 s}

¤ || b || 8 δ || σ || 8

? δ δ || f || 8 sup

0 ¤ s ¤ T E

»

E

| Φ p X _s ^{r n s} , µ ^r _s ^{n s} , u q| dρ p u q

. (6) Then, by tightness, there exists a subsequence of p X ^{r n s} , X ^{r n 1 s} q n that converges in distribution to some p X, Y q in D pr 0, T s , R ² q . In the rest of the proof, we work on this subsequence without writing it explicitly for the sake of notation.

Step 3. In this step, we show that X Y almost surely. Note that, since we work on a subse- quence, this is not obvious. It is for this part of the proof that we need to restrict our processes to a time interval of the form r 0, T s . By Lemma 2.8, it is sucient to prove that, for a subsequence,

E

sup

0 ¤ s ¤ T

X _s ^{r n 1 s} X _s ^{r n s}

(7) vanishes as n goes to innity. Indeed, (7) implies that, for another subsequence, sup _{s ¤ T} | X s ^{r n 1 s} X s ^{r n s} | converges to zero almost surely. Then, we can apply Skorohod representation theorem (see Theorem 6.7 of Billingsley (1999)) to the following sequence

X ^{r n s} , X ^{r n 1 s}

n

that converges in distribution in D pr 0, T s , R ² q to p X, Y q . Thus we can consider, for n P N , random variables p X ˜ ^{r n s} , X ˜ ^{r n 1 s} q (resp. p X, ˜ Y ˜ q) having the same distribution as p X ^{r n s} , X ^{r n 1 s} q (resp. p X, Y q) for which the previous convergence is almost sure. In particular, we also know that sup _{s ¤ T} | X ˜ s ^{r n 1 s} X ˜ s ^{r n s} | vanishes almost surely. Hence, by Lemma 2.8, the representing r.v. p X ˜ ^{r n s} , X ˜ ^{r n 1 s} q converges a.s. to the representing r.v. p X, ˜ X ˜ q in D pr 0, T s , R ² q . As a consequence X ˜ Y ˜ almost surely, and so X Y almost surely.

Now let us prove (7). Let

u ^{r n s} p t q : E

sup

0 ¤ s ¤ t

X _s ^{r n 1 s} X _s ^{r n s}

. By (5),

sup

n PN u ^{r n s} p t q   8 .

Let us x some n P N and consider a truncation function r ^r _t ^{n s} ¡ 0 whose values will be xed later.

The same truncation argument used in Section 2.2 allows to prove that, for all 0 ¤ k ¤ n 1, t ¤ T, u ^{r k 1 s} p t q ¤ C T

» t 0

p 1 r ^r _s ^{n s} q u ^{r k s} p s q e ^r

ds 2L ?

T u ^{r k s} p t q

¤ C _T

» t 0

p 1 r ^r _s ^{n s} q u ^{r k s} p s q e ^r

ds 1 2 u ^{r k s} p t q .

where C _T ¡ 0 does not depend on n thanks to (5). The second inequality above comes from the fact that we x the value of T ¡ 0 such that L ?

T   1 { 4.

Now, introducing S n p t q : ° n

k 0 u ^r _t ^{k s} and summing the above inequality from k 0 to k n 1, we have, for all t ¤ T,

S n p t q ¤ C T C T

» t 0

p 1 r ^r _s ^{n s} q S n p s q ds ne ^r

s ds 1 2 S n p t q , where we have used that u ^r _t ^{0 s} ¤ C _T and S _{n 1} p t q ¤ S _n p t q . This implies

S _n p t q ¤ C _T C _T

» t 0

p 1 r ^r _s ^{n s} q S _n p s q ds ne ^r

ds.

Let D T : max p sup

k ¥ 0

sup

s ¤ T | u ^r s ^{k s} | , C T , 1 q   8 , and introduce R _n p t q : S _n p t q

p n 1 q D T e ² ¤ e ² . Consequently, for all t ¤ T,

R n p t q ¤ 1

n 1 C T

» t 0

p 1 r ^r _s ^{n s} q R n p s q e ^r

ds.

Finally we choose r _t ^{r n s} : ln R n p t q ¥ 2 and obtain for all t ¤ T, R n p t q ¤ 1

n 1 C T

» t 0

p 2 ln R n p s qq R n p s q ds ¤ 1

n 1 C T

» t 0

R n p s q ln R n p s q ds.

As before we apply Osgood's lemma. Let M p x q : ³ e

x

s 1 ln s ds ln p ln x q ln 2. Then M p R n p T qq M p 1 {p n 1 qq ¤ C T T

or equivalently

R n p T q ¤ p n 1 q ^e

such that S n p T q ¤ C T n ^{1 e}

.

Lemma 2.7 above then implies that there exists a subsequence of p u ^r _T ^{n s} q n that converges to 0 as n goes to innity. This proves (7).

Step 4. Let us prove that a subsequence of p µ ^{r n s} q n converges to some limit µ : t ÞÑ µ _t in the following sense

sup

0 ¤ t ¤ T

W ₁ p µ ^r _t ^{n s} , µ _t q ÝÑ

n Ñ8 0, where µ t : L p X t q for a.e. t ¤ T.

We prove this point by proving that the sequence of functions µ ^{r n s} : t ÞÑ µ ^r _t ^{n s} L p X _t ^{r n s} q P P 1 pRq

is relatively compact, using Arzelà-Ascoli's theorem.

To begin with, the denition of W 1 and the same computation as the one used to obtain (6) allows to prove that, for all s, t ¤ T, for all n P N ,

W 1 p µ ^r _t ^{n s} , µ ^r _s ^{n s} q ¤ E X _t ^{r n s} X _s ^{r n s}

¤ C

| t s | a

| t s | , (8)

for a constant C ¡ 0 independent of n .

This implies that the sequence µ ^{r n s} : t ÞÑ µ ^r _t ^{n s} is equicontinuous. In addition, by (5) we know that, for every t ¤ T, the set p µ ^r _t ^{n s} q n is tight, and whence relatively compact (in the topology of the weak convergence, but not in P 1 pRq a priori). Indeed, for any ε ¡ 0, considering M _ε : sup _n E

| X _t ^{r n s} | { ε, we have, for all n,

µ ^r _t ^{n s} pRzr M ε , M ε sq P

| X _t ^{r n s} | ¡ M ε ¤ 1 M _ε E

| X _t ^{r n s} |

¤ ε.

In particular, for every t ¤ T, we can consider a subsequence of p µ ^r _t ^{n s} q n that converges weakly.

To prove that this convergence holds for the metric W 1 , we rely on the characterization p iii q of W 1

given in Denition 6.8, and Theorem 6.9 of Villani (2008). According to this result, the convergence of the same subsequence of p µ ^r _t ^{n s} q n for W ₁ follows from (5), Markov's inequality, Cauchy-Schwarz's inequality and the fact that,

E

| X _t ^{r n s} |1 ^! _{| X}

|¡ R | )

¤ 1 R sup

k PN E

| X _t ^{r k s} | E

p X _t ^{r k s} q ² 1 { 2

R ÝÑ Ñ8 0.

We can then conclude that, for all t ¤ T, the sequence p µ ^r _t ^{n s} q n is also relatively compact on P 1 pRq . Then, thanks to (8), Arzelà-Ascoli's theorem implies that the sequence p µ ^{r n s} q n is relatively com- pact. As a consequence, there exists a subsequence of p µ ^{r n s} q n (as previously, we do not write this subsequence explicitly in the notation) that converges to some µ : t ÞÑ µ t P P 1 pRq in the following sense

sup

0 ¤ t ¤ T

W ₁ p µ ^r _t ^{n s} , µ _t q ÝÑ

n Ñ8 0.

The last thing to show in this step is that µ _t L p X _t q for a.e. t ¤ T. By construction, µ _t is the limit of µ ^r _t ^{n s} : L p X _t ^{r n s} q . Recalling that X ^{r n s} converges to X in distribution in Skorohod topology, we know that for all continuity point t of s ÞÑ L p X _s q, µ _t L p X _t q .

Step 5. Recall that, for a subsequence, p X ^{r n s} , X ^{r n 1 s} q n converges to p X, X q in distribution in D pr 0, T s , R ² q , and p µ ^{r n s} q n (which is a sequence of deterministic and continuous functions from R to P 1 pRq) converges uniformly to µ on r 0, T s. The aim of this step is to prove that p X ^{r n s} , X ^{r n 1 s} , µ ^{r n s} q n

converges to p X, X, µ q in distribution in D pr 0, T s , R ² P 1 pRqq . We consider µ ^{r n s} in the previous distribution even though it is deterministic, because the important point in the convergence we want to prove is that µ ^{r n s} must converge w.r.t. the same sequence of time-changes as the one of p X ^{r n s} , X ^{r n 1 s} q to be able to apply Lemma 2.9 almost surely. In particular, it is important to have convergence in the topology of D pr 0, T s , R ² P 1 pRqq rather than in the weaker topology D pr 0, T s , R ² q P 1 pRq .

As µ is continuous, we have that, for any sequence of time-changes p λ n q n the convergence sup

0 ¤ t ¤ T

W ₁ p µ ^r _t ^{n s} , µ _λ

_{p t q} q ÝÑ _n

Ñ8 0. (9)

By Skorohod's representation theorem, we can assume that some representative r.v. p X ˜ ^{r n s} , X ˜ ^{r n 1 s} q of p X ^{r n s} , X ^{r n 1 s} q converges a.s. to representative r.v. p X, ˜ X ˜ q of p X, X q in D pr 0, T s , R ² q . This implies that, almost surely, there exists a sequence of time-changes p λ n q n such that

sup

0 ¤ t ¤ T

X ˜ _t ^{r n s} X ˜ _λ

_{p t q} and sup

0 ¤ t ¤ T

X ˜ _t ^{r n 1 s} X ˜ _λ

_{p t q}

vanish as n goes to innity. So, by (9), almost surely, there exists a sequence of time-changes p λ n q n

such that

sup

0 ¤ t ¤ T

d

X ˜ _t ^{r n s} , X ˜ _t ^{r n 1 s} , µ ^{r n s} s t ,

X ˜ _λ

_{p t q} , X ˜ λ

p t q , µ _λ

_{p t q} ÝÑ

n Ñ8 0,

with d rp x, y, m q , p x ¹ , y ¹ , m ¹ qs | x x ¹ | | y y ¹ | W 1 p m, m ¹ q . In particular, we know that the sequence p X ˜ ^{r n s} , X ˜ ^{r n 1 s} , µ ^{r n s} q n converges to p X, ˜ X, µ ˜ q almost surely in D pr 0, T s , R ² P 1 pRqq . This implies that p X ^{r n s} , X ^{r n 1 s} , µ ^{r n s} q n converges to p X, X, µ q in distribution in D pr 0, T s , R ² P 1 pRqq .

Step 6. This step concludes the proof, showing that X is solution to (1). In order to prove that X is solution to (1), we use the fact that, using the notation of Denitions II.2.6 and II.2.16 of Jacod and Shiryaev (2003), X ^{r n 1 s} is a semimartingale with characteristics p B ^{r n 1 s} , C ^{r n 1 s} , ν ^{r n 1 s} q given by

B _t ^{r n 1 s}

» t 0

b p X _s ^{r n s} , µ ^r _s ^{n s} q ds, C _t ^{r n 1 s}

» t 0

σ p X _s ^{r n s} , µ ^r _s ^{n s} q ² ds, ν ^{r n 1 s} p dt, dx q f p X _t ^{r n s} , µ ^r _t ^{n s} q dt

»

E

δ _Φ

p X

,µ

,u q p dx q dρ p u q .

Let us note that, above, we have chosen as truncation function h 0, hence the modied second characteristics C ˜ ^{r n 1 s} is the same as C ^{r n 1 s} .

Recall that, in Step 5, we have shown that, for a subsequence, p X ^{r n s} , X ^{r n 1 s} , µ ^{r n s} q n converges in distribution in D pr 0, T s , R ² P 1 pRqq to p X, X, µ q . Using once again Skorohod's representation theorem, we can consider representative r.v. for which the previous convergence is almost sure.

Whence, by Lemma 2.9, for all g P C _b pRq , the following convergences hold almost surely for the representative r.v. and hence in distribution:

X ^{r n 1 s} , B ^{r n 1 s} , C ^{r n 1 s} ÝÑ

n Ñ 8

X,

»

0

b p X _s , µ _s q ds,

»

0

σ p X _s , µ _s q ² ds

,

X ^{r n 1 s} ,

»

r 0, sR

g p x q ν ^{r n 1 s} p ds, dx q

ÝÑ

n Ñ 8

X,

»

0

»

E

g p Φ p X s , µ s , u qq f p X s , µ s q dρ p u q ds

, where the convergences hold respectively in the spaces D pR , R ³ q and D pR , R ² q .

Then, Theorem IX.2.4 of Jacod and Shiryaev (2003) implies that X is a semimartingale with characteristics p B, C, ν q given by

B t

» t 0

b p X s , µ s q ds, C t

» t 0

σ p X s , µ s q ² ds,

ν p dt, dx q f p X t , µ t q dt

»

E

δ Φ p X

,µ

,u q p dx q dρ p u q .

Then, we can use the canonical representation of X (see Theorem II.2.34 of Jacod and Shiryaev (2003)): X X 0 B M ^c Id µ ^X , where M ^c is a continuous locale martingale, µ ^X

°

s 1 t ∆Y

0 u δ _{p s,X}

_q is the jump measure of X (let us recall that we chose the truncation func- tion h 0 ) and p Id µ ^X q t : ³ t

0

³

R

xdµ ^X p s, x q. By denition of the characteristics, x M ^c y t C _t . Whence, by Theorem II.7.1 of Ikeda and Watanabe (1989), there exists a Brownian motion W such that

M _t ^c

» t 0

σ p X s , µ s q dW s . (10)

In addition, we know that ν is the compensator of µ ^X . We rely on Theorem II.7.4 of Ikeda and Watanabe (1989). Using the notation therein, we introduce Z R E, m p dz, du q dzρ p du q and

θ p t, z, u q : Φ p X _t , µ _t , u q1 _t z ¤ f p X

,µ

qu .

According to Theorem II.7.4 of Ikeda and Watanabe (1989), there exists a Poisson measure π on R R E having intensity dt dz dρ p u q such that, for all A P B pRq ,

µ ^X pr 0, t s A q

» t 0

» ₈

0

»

E

1 _t θ p s,z,u qP A u dπ p s, z, u q . This implies that

p Id µ ^X q t

»

r 0,t sR E

Φ p X _s , µ _s , u q1 _t z ¤ f p X

,µ

qu dπ p s, z, u q . (11) Finally, recalling that X X 0 B M ^c Id µ ^X , (10) and (11), we have just shown that X is a weak solution to (1) on r 0, T s.

2.4. Proof of Theorem 2.3

In Section 2.2, we have proven the (global) pathwise uniqueness of solutions of (1), and, in Sec- tion 2.3, the existence of a weak solution of (1) on r 0, T s, with T 1 {p 16L ² q.

Then, generalizations of Yamada-Watanabe results allows to construct a strong solution on r 0, T s:

it is a consequence of Theorem 1.5 and Lemma 2.10 of Kurtz (2014) (see the discussion before Lemma 2.10 or Example 2.14 for more details).

More precisely, given a Brownian motion W, a Poisson random measure π and an initial condi- tion X 0 , there exists a strong solution p X t q 0 ¤ t ¤ T dened w.r.t. these W, π, X 0 . Then, one can con- struct a strong solution p X t q T ¤ t ¤ 2T on r T, 2T s dened w.r.t. the Brownian motion p W T t W T q t ¥ 0 , the Poisson measure π T dened by

π T p A B q π pt T x : x P A u B q ,

and the initial condition X T . Iterating this reasoning, we can construct a strong solution of (1) on r 0, kT s for any k P N , with T 1 {p 16L ² q ¡ 0. Hence, there exists a (global) strong solution of (1).

This proves Theorem 2.3.

3. Propagation of chaos

In this section, we prove a propagation of chaos for McKean-Vlasov systems: Theorem 3.3. This property in the globally Lipschitz case has been proven in Proposition 3.1 of Andreis, Dai Pra and Fischer (2018). Let us introduce the N particle system p X ^N,i q 1 ¤ i ¤ N

dX _t ^N,i b p X _t ^N,i , µ ^N _t q dt σ p X _t ^N,i , µ ^N _t q dW _t ⁱ

»

R

F

Ψ p X _t ^N,i , µ ^N _t , v ⁱ q1 t ^{z ¤ f p X}

,µ

q u dπ ⁱ p t, z, v q 1

N

¸ N j 1

»

R

F

Θ p X _t ^N,j , X _t ^N,i , µ ^N _t , v ^j , v ⁱ q1 t ^{z ¤ f p X}

,µ

q u dπ ^j p t, z, v q , (12) with µ ^N : N ¹ ° N

j 1 δ _X

, W ⁱ ( i ¥ 1 ) independent standard one-dimensional Brownian motions, and π ⁱ ( i ¥ 1 ) independent Poisson measures on R ² F

with intensity dt dz dν p v q , where F is a measurable space, and ν is a σ nite symmetric measure on F

(i.e. ν is invariant under nite permutations).

In the following, we assume that b, σ and f satisfy the same conditions as in Assumption 1, and that Ψ satises the same as Φ for some constant a ¡ 0 , with E F

and ρ ν. We also assume that Θ satises similar conditions: for all x 1 , x ¹ ₁ , x 2 , x ¹ ₂ P R , m 1 , m 2 P P 1 pRq ,

»

F

»

R

| Θ p x 1 , x ¹ ₁ , m 1 , v ¹ , v ² q1 _t z ¤ f p x

,m

qu Θ p x 2 , x ¹ ₂ , m 2 , v ¹ , v ² q1 _t z ¤ f p x

,m

qu | dzdν p v q

¤ L

1 | x ₁ | | x ¹ ₁ | | x ₂ | | x ¹ ₂ |

»

R

e ^{a | x |} dm ₁ p x q

»

R

e ^{a | x |} dm ₂ p x q

| x 1 x 2 | | x ¹ ₁ x ¹ ₂ | W 1 p m 1 , m 2 q , and

sup

x,x

PR ,m P

pRq

»

F

e ^{a | Θ p x,x}

^,m,v

^,v

^q| dν p v q   8 .

In addition, we assume that each X ₀ ^N,i ( i ¥ 1, N ¥ 1 ) satises the initial condition of Assump- tion 1, and that, for every N P N , the system p X ₀ ^N,i q 1 ¤ i ¤ N is i.i.d.

We prove that these N particles systems converge as N goes to innity to the following limit system.

d X ¯ _t ⁱ b p X ¯ _t ⁱ , µ ¯ t q dt σ p X ¯ _t ⁱ , µ ¯ t q dW _t ⁱ

»

R

F

Ψ p X ¯ _t ⁱ , µ ¯ t , v ⁱ q1 t ^{z ¤ f p X ¯}

,¯ µ

q u dπ ⁱ p t, z, v q

»

R

»

F

Θ p x, X ¯ _t ⁱ , µ ¯ _t , v ¹ , v ² q f p x, µ ¯ _t q dν p v q d¯ µ _t p x q , (13) where µ ¯ t L p X ¯ t q . We assume that the variables X ¯ ₀ ⁱ ( i ¥ 1 ) are i.i.d. and satisfy the initial condition of Assumption 1.

Let us remark that the (strong) well-posedness of equation (13) is a consequence of Theorem 2.3 for the same σ, Ψ Φ and for the drift

b p x, m q

»

R

»

F

Θ p y, x, m, v ¹ , v ² q f p y, m q dν p v q dm p y q .

One can also prove the (strong) well-posedness of equation (12) using a similar reasoning as the one used in the proof of Theorem 2.3. The only dierence is for the Step 4 of the proof of Proposition 2.11, since, for (12) the measure µ ^N is not deterministic. Instead of proving that the sequence of measures p µ ^{r n s} q n constructed in the Picard scheme is relatively compact by Arzelà- Ascoli's theorem, we rely exclusively on the following lemma whose proof is postponed to Appendix.

Lemma 3.1. Let N P N , T ¡ 0, and p x ^k q 1 ¤ k ¤ N and p x ^k _n q 1 ¤ k ¤ N p n P Nq be càdlàg functions.

Dene

µ n p t q : N ¹

¸ N k 1

δ _x

n

p t q and µ p t q :

¸ N k 1

δ _x

p t q .

Let λ n ( n P N) be continuous, increasing functions satisfying λ n p 0 q 0, λ n p T q T , and that, for any 1 ¤ k ¤ N ,

sup

0 ¤ t ¤ T

x ^k _n p t q x ^k p λ n p t qq and sup

0 ¤ t ¤ T

| t λ n p t q|

vanish as n goes to innity. Then, sup

0 ¤ t ¤ T

W 1 p µ n p t q , µ p λ n p t qqq ÝÑ

n Ñ8 0.

Remark 3.2. This lemma allows to prove that, if p x n , y n q n converges to p x, y q in D pr 0, T s , pR ^N q ² q , then, the sequence p x n , y n , µ n q n converges to p x, y, µ q in D pr 0, T s , pR ^N q ² P 1 pRqq .

In the following, we assume that p X ^N,i q 1 ¤ i ¤ N and p X ¯ ⁱ q i ¥ 1 are strong solutions of respec- tively (12) and (13) dened w.r.t. the same Brownian motions W ⁱ ( i ¥ 1 ) and the same Poisson measures π ⁱ ( i ¥ 1 ) such that all the systems are dened on the same space. In addition, we assume that the following condition holds true

ε ^N ₀ : E X ₀ ^N,1 X ¯ ₀ ¹

N ÝÑ Ñ8 0. (14)

Now let us state the main result of this section: the propagation of chaos of the N particle systems, that is, the convergence of the systems p X ^N,i q 1 ¤ i ¤ N to the i.i.d. system p X ¯ ⁱ q i ¥ 1 as N goes to innity. We comment the convergence speed in Remark 3.5.

Theorem 3.3. We have, for all T 0 ¡ 0, E

sup

0 ¤ t ¤ T

X _t ^N,1 X ¯ _t ¹

N ÝÑ Ñ8 0.

Consequently, for all k ¥ 1, the following weak convergence holds true:

L p X ^N,1 , X ^N,2 , ..., X ^N,k q ÝÑ

N Ñ8 L p X ¯ ¹ q b L p X ¯ ² q b ... b L p X ¯ ^k q ,

in the product topology of the topology of the uniform convergence on every compact set.

Remark 3.4. We just state the result of Theorem 3.3 for the rst coordinate because both systems

p X ^N,i q 1 ¤ i ¤ N and p X ¯ ⁱ q i ¥ 1 are exchangeable.

Remark 3.5. In the proof of Theorem 3.3, we obtain a convergence speed for E

sup

0 ¤ t ¤ T

X _t ^N,1 X ¯ _t ¹

N ÝÑ Ñ8 0

that depends on T ₀ . Indeed, if T : 1 {p 16L ² q , the formula (16) below gives a convergence speed for

E

sup

0 ¤ t ¤ T

X _t ^N,1 X ¯ _t ¹

of the form

S ₀ ^N : C ¹

ε ^N ₀ N ^{1 { 2}

C

,

for some positive constants C ¹ , C ² , where ε ^N ₀ is given at (14). And, for all k P N , the convergence speed S ^N _k of

E

sup

kT ¤ t ¤p k 1 q T

X _t ^N,1 X ¯ _t ¹

can be obtained inductively by

S ^N _k C ¹

S _k ^N ₁ N ^{1 { 2}

C

for the same constants C ¹ , C ² for all k.

Before proving Theorem 3.3, let us state a lemma about some a priori estimates of the pro- cess p X _t ^N,1 q t ¥ 0 .

Lemma 3.6. For every N P N , let p X ^N,i q 1 ¤ i ¤ N be the solution of (12). For any t ¥ 0, (i) sup

N PN

sup

0 ¤ s ¤ t E

e ^{a | X}

^|

  8 and sup

0 ¤ s ¤ t E

e ^{a | X ¯}

^|   8 ,

(ii) sup

N PN

E

sup

0 ¤ s ¤ t | X _s ^N,1 |

  8 and E

sup

0 ¤ s ¤ t | X ¯ _s ¹ |

  8 .

Sketch of proof of Lemma 3.6. We only give the main steps of the proof of the rst part of the point p i q since the main part of the proof relies on the same computations as in the proof of Lemma 2.4. The only new property of the lemma is the fact that the bounds has to be uniform in N . This property is easy to prove for the point p ii q of the lemma in this model because of the normalization in N ¹ of the jump term Θ of (12). For the point p i q , as in the proof of Lemma 2.4, we apply Ito's formula to both functions x ÞÑ e ^ax and x ÞÑ e ^ax , and then, one has to note that, by exchangeability,

E

e ^aX

¤E

e ^X

a || b || 8

» t 0

E

e ^aX

ds 1

2 a ² || σ || ² ₈

» t 0

E

e ^aX

ds

|| f || ₈

» t 0

E

e ^aX

»

F

e ^{aΨ p X}

^,µ

^,v

^q 1 dν p v q

ds p N 1 q|| f || 8

» t 0

E

e ^aX

»

F

e ^aN

^{Θ p X}

^,X

^,µ

^,v

^,v

^q 1 dν p v q

ds.