WELL-POSEDNESS AND APPROXIMATION OF SOME ONE-DIMENSIONAL LÉVY-DRIVEN NON-LINEAR SDES

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WELL-POSEDNESS AND APPROXIMATION OF SOME ONE-DIMENSIONAL LÉVY-DRIVEN

NON-LINEAR SDES

Noufel Frikha, Libo Li

To cite this version:

Noufel Frikha, Libo Li. WELL-POSEDNESS AND APPROXIMATION OF SOME ONE- DIMENSIONAL LÉVY-DRIVEN NON-LINEAR SDES. 2020. �hal-02446337�

WELL-POSEDNESS AND APPROXIMATION OF SOME ONE-DIMENSIONAL L ´EVY-DRIVEN NON-LINEAR SDES.

NOUFEL FRIKHA AND LIBO LI

Abstract. In this article, we are interested in the strong well-posedness together with the numerical approximation of some one-dimensional stochastic differential equations with a non-linear drift, in the sense of McKean-Vlasov, driven by a spectrally-positive L´evy process and a Brownian motion. We provide criteria for the existence of strong solutions under non-Lipschitz conditions of Yamada-Watanabe type without non-degeneracy assumption.

The strong convergence rate of the propagation of chaos for the associated particle system and of the corresponding Euler-Maruyama scheme are also investigated. In particular, the strong convergence rate of the Euler-Maruyama scheme exhibits an interplay between the regularity of the coefficients and the order of singularity of the L´evy measure around zero.

1. Introduction

In this paper, we are interested in the strong well-posedness as well as the numerical approximation of the following one-dimensional non-linear stochastic differential equation (SDE for short) with positive jumps and with dynamics

X_t=ξ+ Z t

0

b(s, X_s,[X_s])ds+ Z t

0

σ(s, X_s)dW_s+ Z t

0

h(s, X_s−)dZ_s, (1)

Zt= Z t

0

Z ∞ 0

zNe(ds, dz) (2)

whereW = (W_t)_t≥0is a standard one-dimensional Brownian motion,Ne is a compensated Poisson random measure with intensity measure dsν(dz) satisfying the condition R∞

0 (z∧z²)ν(dz) < ∞ and with starting point ξ (with law µ). We here assume that the random variables ξ, W and Ne are defined on some filtered probability space (Ω,F,(Ft)_t≥0,P) satisfying the usual conditions and that they are mutually independent. Throughout the article, we will denote byP(R) the space of probability measures onR, Pq(R) the subset of probability measures inP(R) that have finite moment of orderq >0, equipped with the corresponding Wasserstein distanceWq, and by [θ] the law of a random variableθ.

The SDE (1) generally appears as the mean-field limit of an individual particle evolving within a system of particles, which interact with each other only through the empirical measure of the whole system, when the number of particles goes to infinity. This property is commonly referred in the literature as the propagation of chaos phenomenon. The weak and strong well-posedness in the diffusive setting, i.e. when h ≡ 0, have attracted considerable attention of the research community from the last decades to the present day. We refer the reader to the works of Funaki [11], Oelschlaeger [31], G¨artner [12], Sznitman [33], Jourdain [18] and more recently, Li and Min [26], Chaudru de Raynal [2], Mishura and Veretennikov [30], Lacker [25], Chaudru de Raynal and Frikha [3] for a small and incomplete sample. When Z is a square integrable L´evy process, the strong well-posedness of multi-dimensional non-linear SDEs with time-homogeneous Lipschitz coefficients R^d × P₂(R^d) 3 (x, µ) 7→ b(x, µ), σ(x, µ), h(x, µ) together with the strong convergence rate of propagation of chaos have been investigated by Jourdain, M´el´eard and Woyczynski [19]. The well-posedness in the weak and strong sense of some multi-dimensional non-linear SDEs driven by non-degenerate symmetricα-stable L´evy processes,α∈(0,2), under some mild H¨older regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables have been recently established by Frikha, Konakov and Menozzi [9].

In the absence of the Lipschitz regularity of coefficients and/or non-degeneracy of the underlying noise, it ac- tually turns out to be challenging to establish the well-posedness of such non-linear dynamics and to obtain some quantitative rates of convergence for the propagation of chaos by the related system of particles and for the time

2010Mathematics Subject Classification. Primary 60H10, 60G46; Secondary 60H30, 35R09.

Key words and phrases. McKean-Vlasov SDEs; L´evy driven SDEs; strong uniqueness; propagation of chaos.

The financial support from the Europlace Institute of Finance is gratefully acknowledged.

1

discretization by the so-called Euler-Maruyama scheme. Concerning the well-posedness, we again mention the works [2, 3, 9, 18, 25]. For some quantitative rates of propagation of chaos in the diffusive setting, we mention the recent work of Holding [16] who studies systems of interacting particles with a constant diffusion coefficient and a drift coefficient given by an H¨older continuous interacting kernel of convolution type are established. Still in the non- degenerate diffusive setting, some new quantitative estimates for propagation of chaos have been recently obtained by Chaudru de Raynal and Frikha [4] for McKean-Vlasov SDEs under some mild H¨older regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables.

Our main objective here is twofold as it consists specifically in investigating the strong well-posedness and the numerical approximation of the one-dimensional non-linear SDE (1) without any non-degeneracy condition on the underlying noise and only assuming thatR× P1(R)3(x, µ)7→b(t, x, µ) is one-sided Lipschitz, uniformly in time, and that both coefficientsσand hare H¨older continuous in space, uniformly in time.

Motivated by the recent developments of continuous-state branching processes, the well-posedness of such dy- namics in the linear setting, i.e. when there is no dependence with respect to the measure argument in the drift coefficient, was first addressed by Fu and Li [10] and then later extended by Li and Mytnik [27]. Equations of such type have recently found applications in financial mathematics due to the positivity of the solution and the ability to capture self-exciting effects through the inclusion of the L´evy componentZ. We refer to the works of Jiao et al.

[22, 23, 24] for the modelling of sovereign interest rate, electricity prices and stochastic volatility. For multi-curve term structure models we refer to Fontana et al. [5]. Our results thus allow to consider and to study from both theoretical and numerical point of view some mean-field extension of the models proposed in the aforementioned references. In this direction, it is worth mentioning for instance the recent works of Bo and Capponi [1] and of Fouque and Ichiba [6] for the modelling of systemic risk of a banking system through the lens of interacting particle system driven by independent Brownian motions.

Concerning the study of the time discretization schemes in the linear framework, we can mention the two recent works of Li and Taguchi [28, 29] concerning the strong rate of convergence of the corresponding Euler-Maruyama approximation scheme and of a positivity preserving time discretization scheme. In the spirit of the aforementioned works, we employ the Yamada-Watanabe approximation technique to derive both theoretical and numerical results.

Let us however mention that compared to the two recent works [28, 29], we here improve the strong convergence rate of the Euler-Maruyama scheme and we also remove the restrictive boundedness assumption on the diffusion and jump coefficients at the price of some integrability constraints. Let us also mention that such boundedness assumption on the jump coefficienthhas appeared persistently in the literature on the numerical approximation of α-stable driven SDEs by the Euler-Maruyama scheme, see for example the works of Hashimoto [14] and Hashimoto and Tsuchiya [15].

The organization of the paper is as follows. The main theorems together with their proofs are presented in Section 2. In Section 3, we mention briefly conditions for which positivity of the solution holds and some potential applications. Section 4 is an appendix and contains some important but auxiliary results. In particular, a general weak existence result as well as as some moment estimates for some non-linear SDEs of jump-type with continuous coefficients having at most linear growth are presented.

2. Assumptions and main results

2.1. Strong solutions to the mean-field SDE. Let (Ω,F,(Ft)_t≥0,P) be a filtered probability space satisfying the usual conditions. On this probability space, let us consider a standard one-dimensional (F_t)_t≥0-Brownian motionW and a compensated (Ft)t≥0-Poisson random measureNe with L´evy measureν on (R+,B(R+)) such that R

R+z∧z²ν(dz) <∞. Suppose that W and Ne are independent. Given a real-valued random variable ξ defined on (Ω,F,(Ft)_t≥0,P) and independent of the pair (W,Ne), we will say that strong existence holds for the non-linear SDE (1) starting from ξwith law µat time 0 if there exists a real-valued and c`adl`ag process X = (Xt)_t≥0, with marginal law [Xt] at timet, adapted to the augmented (and completed) natural filtration (Gt)_t≥0 generated byW andN, that satisfies equation (1) almost surely for alle t≥0. We will say that weak existence holds for (1) starting fromξ with lawµat time 0 if there exists a filtered probability space (Ω,F,(Ft)_t≥0,P) and a triplet (X, W,Ne) of (Ft)_t≥0-adapted process such that X is c`adl`ag with [X0] = µ, W is a Brownian motion, Ne is a Poisson random measure independent ofW with L´evy measureν and satisfying (1). We will say that pathwise uniqueness holds for the same equation if for any two solutionsX¹ andX² of (1) (possibly defined on two different filtered probability spaces) satisfying X₀¹ =X₀² =ξ, W¹ =W² and Ne¹ =Ne², we have X_t¹ =X_t² almost surely for everyt ≥ 0. In particular, note that strong uniqueness implies uniqueness of marginal laws, i.e. [X_t¹] = [X_t²] for allt≥0, which in turn implies that the non-linear SDE (1) may be regarded as a linear SDE with time-inhomogeneous coefficients.

In the spirit of Li and Mytnik [27], we introduce the quantity α_ν := inf{β >1 : lim

x→0+x^β−1 Z ∞

x

z ν(dz) = 0}

which represents the order of singularity of the L´evy measure at zero. For instance, if ν is the L´evy measure of a spectrally positive α-stable like process, that is, if ν(dz) = 1_(0,∞)(z)g(z)/z^1+αdz, with α ∈ [1,2], g being a non-negative bounded and continuous function on R+, one has α_ν =αand the infimum is actually achieved. We recall from Lemma 2.1 of [27] thatα_ν∈[1,2]. Moreover, for anyα > α_ν,

(3) lim

x→0+x^α−2 Z x

0

z²ν(dz) = 0 and lim

x→0+x^α−1 Z ∞

x

zν(dz) = 0.

Note that in our current setting, both limits in (3) are also valid forα= 2. We now introduce our assumptions on the coefficientsb,σandhin order to tackle the strong well-posedness of the SDE (1).

Assumption 2.1.

(i) The coefficientsR⁺×R× P1(R)3(t, x, µ)7→b(t, x, µ), σ(t, x), h(t, x)are continuous functions,P1(R)being equipped with the Wasserstein metricW1, and have at most linear growth inxandµlocally uniformly with respect tot, that is, for anyT >0, there exists some positive constantCT such that for any(x, µ)∈R×P(R)

(4) sup

0≤t≤T

{|b(t, x, µ)|+|σ(t, x)|+|h(t, x)|} ≤C_T(1 +|x|+W₁(µ, δ₀)).

(ii) There exists some positive constant denoted by [b]L such that for all(t, x, y, µ, ν)∈[0,∞)×R²× P1(R)², sign(x−y)(b(t, x, µ)−b(t, y, ν))≤[b]L(|x−y|+W1(µ, ν)).

(iii) For anyt≥0, the diffusion coefficientx7→σ(t, x)isγ-H¨older continuous with γ∈[1/2,1]andx7→h(t, x) is η-H¨older continuous with η ∈(1−1/α_ν,1], uniformly with respect to t. We denote respectively by[σ]_γ and[h]_η the (uniform) H¨older modulus ofσand h.

(iv) There exists a positive constant denoted by[h]_L such that for allt∈[0,∞), for allx, y∈R, sign(y−x)(h(t, x)−h(t, y))≤[h]L|x−y|.

The assumption (i) is required in order to prove weak existence of solutions to the SDE (1) by employing a compactness argument together with some moment estimates. Though the previous result seems natural, to the best of our knowledge, it is new. The precise statement together with its proof are postponed to the Appendix, see Lemma 4.1 in Section 4.1. The regularity assumptions (ii), (iii) and (iv) allow to prove pathwise uniqueness as well as to study the propagation of chaos for the corresponding system of interacting particles. As already mentioned in the introduction, we will rely on the Yamada-Watanabe approximation technique which is briefly exposed in Section 4.2.

Before proceeding to the statement of our first main result, let us make some additional comments on Assumption 2.1. Observe that the condition on b holds as soon asb(t, x, µ) = b1(t, x) +b2(t, x, µ), with b1 non-increasing in space, uniformly in time, and b2(t, ., .) Lipschitz-continuous onR× P1(R), uniformly with respect to timet. Our assumption on h holds if h(t, x) = h1(t, x) +h2(t, x) with h1(t, .) Lipschitz continuous (uniformly in time) and h₂(t, .) η-H¨older continuous (uniformly in time) for someη ∈(1−1/α_ν,1] and non-decreasing in space. This last assumption is reminiscent of those introduced in [27].

We now present two results concerning weak existence and pathwise uniqueness of solutions for the mean-field SDE (1).

Lemma 2.2. Suppose that assumption 2.1 (i) holds and thatR

R|x|^βµ(dx) +R

z≥1z^βν(dz) <∞ for some β > 1.

Then, weak existence holds for the SDE (1) starting at time0 from the initial pointξwith lawµ.

Proof. Weak existence follows from a compactness argument stated in a more general form in Lemma 4.1 of the

Appendix in Section 4.1.

Lemma 2.3. Suppose that assumption 2.1 holds. Then, pathwise uniqueness holds for the SDE (1)starting at time 0 from the initial pointξwith law µ∈ P1(R).

Proof. Consider two solutions X and Y to (1) with the same input data (ξ, W, Z). We will use the notation

∆_t:=X_t−Y_tand γ_t:=σ(t, X_t)−σ(t, Y_t),λ_t:=h(t, X_t)−h(t, Y_t). Let ε∈(0,1) andδ∈(0,1). Using (38) and then applying Itˆo’s formula, we get

(5) |∆t| ≤ε+φδ,ε(∆t) =ε+M_t^δ,ε+I_t^δ,ε+J_t^δ,ε+K_t^δ,ε,

where we set

M_t^δ,ε:=

Z t 0

φ⁰_δ,ε(∆s)γsdWs+ Z t

0

Z ∞ 0

{φδ,ε(∆_s−+λ_s−z)−φδ,ε(∆_s−)}Ne(ds, dz), I_t^δ,ε:=

Z t 0

φ⁰_δ,ε(∆s){b(s, Xs,[Xs])−b(s, Ys,[Ys])}ds, J_t^δ,ε:=1

2 Z t

0

φ⁰⁰_δ,ε(∆_s)|γ_s|²ds, K_t^δ,ε:=

Z t 0

Z ∞ 0

n

φδ,ε(∆_s−+λ_s−z)−φδ,ε(∆_s−)−λ_s−zφ⁰_δ,ε(∆_s−)o

ν(dz)ds.

By a standard localization argument, we can define a sequence (τm)_m≥1 satisfyingτm↑ ∞ asm↑ ∞ and such that (M_t∧τ^δ,ε_m)_t≥0is anL¹(P)-martingale. After taking expectation and performing the above computations, one can finally pass to the limit as m↑ ∞ using Fatou’s lemma and the right-continuity of (∆t)_t≥0. Since this procedure is standard, for sake of simplicity, we omit it here and refer the reader to the proof of Theorem 2.2 in [27] for a similar argument. We now quantify the contribution ofI_t^δ,ε, J_t^δ,ε andK_t^δ,ε.

In order to deal withI_t^δ,ε we make use of Assumption 2.1 (iii)

φ⁰_δ,ε(∆s){b(s, Xs,[Xs])−b(s, Ys,[Ys])}=|φ⁰_δ,ε(∆s)|sign(∆s){b(s, Xs,[Xs])−b(s, Ys,[Ys])}

≤[b]L(|∆s|+W1([Xs],[Ys])) so that

I_t^δ,ε≤[b]L

Z t 0

(|∆s|+W1([Xs],[Ys]))ds.

From Assumption 2.1 (iii) and (41), we directly get J_t^δ,ε≤[σ]²_γ

Z t 0

|∆s|^2γ

|∆s|log(δ)1[ε/δ,ε](|∆s|)ds≤[σ]²_γε^2γ−1 log(δ)t.

In order to deal withK_t^δ,ε, we write it as the sum of K_t^1,δ,ε andK_t^2,δ,εwith K_t^1,δ,ε:=

Z t 0

Z ∞ 0

n

φ_δ,ε(∆_s−+λ_s−z)−φ_δ,ε(∆_s−)−λ_s−zφ⁰_δ,ε(∆_s−)o

1_{{∆s−λs−>0}}ν(dz)ds, K_t^2,δ,ε:=

Z t 0

Z ∞ 0

n

φδ,ε(∆_s−+λ_s−z)−φδ,ε(∆_s−)−λ_s−zφ⁰_δ,ε(∆_s−)o

1_{{∆s−λs−<0}}ν(dz)ds.

To deal withK_t^1,δ,ε, we apply Lemma 4.2 withy= ∆_s−,x=λ_s− andu >0 to be chosen later. From Assumption 2.1 (iii), we obtain

Z ∞ 0

n

φδ,ε(∆_s−+λ_s−z)−φδ,ε(∆_s−)−λ_s−zφ⁰_δ,ε(∆_s−)o

1_{{∆s−λs−>0}}ν(dz)

≤21_{{∆s−λs−>0}}1_(0,ε](|∆_s−|)

|λ_s−|²

|∆_s−|logδ Z u

0

z²ν(dz) +|λ_s−| Z ∞

u

zν(dz)

≤ 2[h]²_η log(δ)ε^2η−1

Z u 0

z²ν(dz) + [h]ηε^η Z ∞

u

zν(dz).

(6)

To proceed, we aim at selecting a sufficiently smallu, which equilibrates the two terms that appear in the above upper-estimate. Having in mind (3), we observe that the upper-bound (6) can be further bounded as follows

2[h]²_η log(δ)ε^2η−1

Z u 0

z²ν(dz) + [h]ηε^η Z ∞

u

zν(dz)

≤ 2[h]²_η log(δ)

ε^2η−1 u^α−2u^α−2

Z u 0

z²ν(dz) + [h]η

ε^η u^α−1u^α−1

Z ∞ u

zν(dz)

≤ 2[h]²_η

log(δ) ε^2η−1

u^α−2 + [h]_η ε^η u^α−1

(I_ε^α+J_ε^α) = (2[h]²_η+ [h]_η)ε^{1−α(1−η)}

log(δ)^α−1(I_ε^α+J_ε^α)

where in the last equality we have chosen uso that _log(δ)u^ε2η−1α−2 = _uα−1^εη , that is, u= log(δ)ε^1−η and introduced the two quantities

I_ε^α:= (log(δ)ε^1−η)^α−2

Z log(δ)ε^1−η 0

z²ν(dz) and J_ε^α:= (log(δ)ε^1−η)^α−1 Z ∞

log(δ)ε^1−η

zν(dz)

for a fixed α∈ (α_ν,1/(1−η)). Note that this choice of α is admissible since η ∈ (1−1/α_ν,1]. If η < 1, from (3), there exists ε₀:=ε₀(α, δ)>0 such thatI_ε^α+J_ε^α ≤1 for any ε∈(0, ε₀). If η = 1, we select δ= 2 and from Assumption 2.1, one may bound the quantityI_ε^α+J_ε^αby 2R∞

0 z∧z²ν(dz). From the above computation, for any α∈(α_ν,1/(1−η)), there exists a positiveε₀such that for any ε∈(0, ε₀)

K_t^1,δ,ε≤Cε^{1−α(1−η)} log(δ)^α−1.

In order to deal with K_t^2,δ,ε, we remark that on the set {∆s−λs− <0}, from assumption 2.1 (iv), one has 0<−sign(∆_s−)λ_s−=|λ_s−| ≤[h]L|∆_s−|so that separating theν(dz)-integral into the two domainsz∈(0,_2|λ^|∆s−|

s−|∧1]

andz∈(_2|λ^|∆s−|

s−|∧1,∞), we get

∀z∈

0, |∆s−| 2|λ_s−|∧1

, φδ,ε(∆s−+λs−z)−φδ,ε(∆s−)−λs−zφ⁰_δ,ε(∆s−)≤C1(0,ε](|∆s−|) |λs−|²

|∆_s−|logδz² and

∀z∈

|∆s−| 2|λ_s−|∧1,∞

, |φδ,ε(∆s−+λs−z)−φδ,ε(∆s−)|+|λs−zφ⁰_δ,ε(∆s−)| ≤C|λs−|z≤C|∆s−|z.

Combining the two previous bounds, we obtain Z ∞

0

n

φδ,ε(∆_s−+λ_s−z)−φδ,ε(∆_s−)−λ_s−zφ⁰_δ,ε(∆_s−)o

1_{{∆s−λs−<0}}ν(dz)

≤C1_(0,ε](|∆s−|) |λs−|²

|∆_s−|logδ Z

|∆s− | 2|λs− |∧1 0

z²ν(dz) +C|∆s−| Z ∞

|∆s− | 2|λs− |∧1

zν(dz)

≤C ε

log(δ)+|∆s−|

. (7)

Hence,

K_t^2,δ,ε≤C ε

log(δ)t+ Z t

0

|∆s|ds

. Taking expectation in (5) and gathering the above estimates we obtain (8) E[|∆_t|]≤C

ε

1 + 1

log(δ)

+ Z t

0

(E[|∆_s|] +W₁([X_s],[Y_s]))ds+ ε^2γ−1

log(δ)+ε^{1−α(1−η)} log(δ)^α−1

for any α∈(α_ν,1/(1−η)) and for any ε∈(0, ε₀). Hence, passing to the limit asε↓0 and then using Gronwall’s lemma yield

E[|∆t|]≤C Z t

0

W1([Xs],[Ys]))ds

By the very definition of the Wasserstein metric of order one, one has W₁([X_t],[Y_t])≤E[|∆t|] which combined

with the previous inequality allows us to conclude the proof.

Combining the two previous lemmas with Yamada-Watanabe theorem, we thus obtain our first main result.

Theorem 2.4. Suppose that assumption 2.1 holds and that R

R|x|^βµ(dx) +R

z≥1z^βν(dz) < ∞ for some β > 1.

Then, there exists a unique strong solution to the SDE (1)starting at time 0from the initial pointξ with lawµ.

Remark 2.5. We recall that in the case of α-stable like L´evy measure, that is, ν(dz) = 1_(0,∞)(z)g(z)/z^1+αdz, withα∈(1,2],gbeing a non-negative bounded and continuous function onR+, one hasαν =α. In this case, one may slightly weaken assumption 2.1 (iii) by letting η ∈[1−1/αν,1], ifα∈ (1,2]. Indeed, ifη = 1−1/αν with αν∈(1,2], then the last term appearing in the right-hand side of (8) becomes log(δ)^1−αν so that, after passing to the limit asε↓0, one may conclude by lettingδ↑ ∞.

2.2. Approximation of the mean-field limit dynamics by a system of interacting particles. In this section we consider the approximation of the dynamics (1) by the corresponding system of particles. For a positive integer N, let us introduce a sequence (ξⁱ, Wⁱ, Zⁱ)_1≤i≤N of i.i.d. copies of (ξ, W, Z) which is assumed to be defined on the same probability space (Ω,F,P) for sake of simplicity. The system of interacting particlesn

X_t^i,N,1≤i≤N, t≥0o is defined by the followingN-dimensional SDE with dynamics

(9) X_t^i,N =ξⁱ+ Z t

0

b(s, X_s^i,N, µ^N_s )ds+ Z t

0

σ(s, X_s^i,N)dW_sⁱ+ Z t

0

h(s, X_s−^i,N)dZ_sⁱ, 1≤i≤N whereµ^N_t :=N⁻¹PN

i=1δ_Xi,N

t ,t≥0, is the empirical measure associated to (9) taken at timet.

Let us point out that even if the above system of particles appears as the natural candidate for the approximation of the mean-field SDE (1), at the moment it is not clear if it is well-defined. Our first aim here is to investigate the well-posedness in the strong sense of (9). We then provide an error bound for theL¹-distance between X^i,N and the dynamics ¯X^i,N constructed as i.i.d. copies of the limit equation (1) with the same input (ξⁱ, Wⁱ, Zⁱ)1≤i≤N as the system (9), namely

(10) X¯_t^i,N =ξⁱ+ Z t

0

b(s,X¯_s^i,N, µⁱ_s)ds+ Z t

0

σ(s,X¯_s^i,N)dW_sⁱ+ Z t

0

h(s,X¯_s−^i,N)dZ_sⁱ, 1≤i≤N

where µⁱ_t = [ ¯X_t^i,N], t ≥0. Note that by weak uniqueness of solutions to the SDE (10) and interchangeability in law of theξⁱ satisfying [ξⁱ] =µ one hasµⁱ_t=µt, for any t≥0 and for anyi∈ {1,· · ·, N}, so that the system of particles ( ¯X^i,N)_1≤i≤N indeed corresponds to i.i.d. copies of the SDE (1).

Theorem 2.6. Under assumption 2.1, the SDE (9) admits a unique strong solution for any initial distribution µ∈ P1(R).

Assume additionally that R

z≥1z^βν(dz)<∞for some β >1,β 6= 2, and that the initial distributionµ∈ P1(R) of the SDE (1) has a finite β-moment, that is, Mβ(µ) =R

R|x|^βµ(dx)<∞. Then, for any positive integer N, for any T >0, one has

(11) max

1≤i≤N sup

0≤t≤TE[|X_t^i,N −X¯_t^i,N|] + sup

0≤t≤TE[W1(µ^N_t , µt)]≤C(N^−1/2+N^{−(β−1)/β}) for some positive constant C depending only onT,β andMβ(µ).

Proof. Step 1: Weak existence. First let us note that under assumption 2.1 the maps R+ ×R^N 3 (t, x) 7→

b(t, xi, µ^N_x), σ(t, xi), h(t, xi), with µ^N_x := N⁻¹PN

i=1δx_i, are continuous with at most linear growth for any 1 ≤ i ≤ N. Indeed, for any fixed N ≥ 1, if (tⁿ, xⁿ)_n≥1 converges to (t, x) ∈ R+×R^N, then lim_nW₁(µ^N_xn, µ^N_x) ≤ limnN⁻¹PN

i=1|xⁿ_i −xi| = 0. Therefore, the triple (b(tn, xⁿ_i, µ_x^Nn), σ(tn, xⁿ_i), h(tn, xⁿ_i))n≥1 converges to the triple (b(t, x_i, µ^N_x), σ(t, x_i), h(t, x_i)). In the spirit of step 2 in the proof of Lemma 4.1, we then introduce the sequence (X^(m)= (X^i,(m))1≤i≤N)m≥1of SDEs with dynamics

(12) X_t^i,(m+1)=ξ+ Z t

0

b(s, X_s^i,(m), µ^N

X_s^(m))ds+ Z t

0

σ(s, X_s^i,(m))dWs+ Z t

0

h(s, X_s−^i,(m))dZ_s^i,m, 1≤i≤N whereZ_t^i,m=Rt

0

R

R\{0}zNe_mⁱ (ds, dz),Ne_mⁱ , 1≤i≤N, beingN-independent compensated Poisson random measures on [0,∞)×R\{0}with intensity measuredt1_|z|≤mν(dz). Following similar lines of reasonings as those employed in the first step of Lemma 4.1, one may prove that for any weak solution to (12) with starting distributionµ∈ P1(R), for anyT >0, one has

sup

m≥1

1≤i≤Nmax E[ sup

0≤t≤T

|X_t^i,(m)|]<∞.

Similarly, relabelling the indices if necessary, one may assume that the sequence (X^i,(m), Z^i,m,1 ≤i≤N)m≥1

converges in law to (Xⁱ, Zⁱ,1≤i≤N) in D([0,∞),R^N×R^N). The sequence (Z^i,(m),1≤i≤N)m≥1 also satisfies the P-UT property since

max

1≤i≤NE[ sup

s∈[0,t]

|∆Z_s^i,m|]≤1 +Ct with C :=R

z≥1zν(dz) <∞. Finally, in a completely analogous manner as in step 2 of the proof of Lemma 4.1, since the maps [0,∞)×R^N 3(t, x)7→b(t, xi, µ^N_x), σ(t, xi), h(t, xi) are continuous, from the continuous mapping theorem, the family

(X_t^i,(m+1), b(t, X_t^i,(m+1), µ^N

X_t^(m)), σ(t, X_t^i,(m+1)), h(t, X_t^i,(m+1)), W_tⁱ, Z_t^i,m,1≤i≤N)_t≥0, m≥0

converges in law to (X_tⁱ, b(t, X_tⁱ, µ^N_Xt), σ(t, X_tⁱ), h(t, X_tⁱ), W_tⁱ, Z_tⁱ,1 ≤i≤N)t≥0 in D([0,∞),(R⁶)^N). Thus, passing to the limit in the dynamics (12), we deduce that there exists a weak solution to the SDE (9). It thus suffices to prove pathwise uniqueness.

Step 2: Pathwise uniqueness. Let us consider two weak solutionsX^N := (X^i,N,1≤i≤N) and (Y^i,N,1≤i≤N) of (9) with the same input (ξⁱ, Wⁱ, Zⁱ)1≤i≤N. Following exactly the same lines of reasonings as those employed in the proof of Theorem 2.4, introducing similarly the quantities ∆ⁱ_t:= X_t^i,N −Y_t^i,N for i = 1,· · ·, N and ν_t^N = N⁻¹PN

i=1δ_Yi,N

t , instead of (8) we get E[|∆ⁱ_t|]≤C

ε+

Z t 0

E[|∆ⁱ_s|] +E

W1(µ^N_s, ν_s^N)

ds+ε^2γ−1+ε^{1−α(1−η)}

for anyα∈(1−1/α_ν,1/(1−η)). Passing to the limit asε↓0, then summing overiand finally using the standard inequalityW₁(µ^N_t , ν_t^N)≤N⁻¹PN

i=1|∆ⁱ_t|, we get 1

N

N

X

i=1

E[|∆ⁱ_t|]≤C Z t

0

1 N

N

X

i=1

E[|∆ⁱ_s|]ds

so that, by Gronwall’s lemma, we deduce thatE[W1(µ^N_t , ν_t^N)] = 0 for all t≥0. Hence, pathwise uniqueness holds for the SDE (9) so that, by the Yamada-Watanabe theorem, it has a unique strong solution.

Step 3: Propagation of chaos. In order to prove (11), we again follow the lines of reasoning of the proof of Theorem 2.4. Namely, introducing the quantity ¯∆ⁱ_t:=X_t^i,N −X¯_t^i,N fori= 1,· · · , N and ¯µ^N_t =N⁻¹PN

i=1δ_X¯i,N

t , instead of (8) we get

E[|∆¯ⁱ_t|]≤C

ε+ Z t

0

E[|∆¯ⁱ_s|] +E

W₁(µ^N_s, µ_s)

ds+ε^2γ−1+ε^{1−α(1−η)}

. By the triangle inequality

(13) W₁(µ^N_s, µ_s)≤W₁(µ^N_s,µ¯^N_s) +W₁(¯µ^N_s, µ_s)≤ 1 N

N

X

i=1

|∆¯ⁱ_s|+W₁(¯µ^N_s, µ_s) and noticing that the processes ((X^i,N,X¯^i,N))_1≤i≤N are identically distributed yield

E[|∆¯¹_t|]≤C

ε+ Z t

0

E[|∆¯¹_s|] +E

W1(¯µ^N_s, µs)

ds+ε^2γ−1+ε^{1−α(1−η)}

. Applying Gronwall’s lemma and then lettingε↓0 we finally get

(14) E[|∆¯¹_t|]≤C

Z t 0

E

W₁(¯µ^N_s, µ_s) ds.

We now discuss the rate of convergence stated in (11) under the additional assumption that the L´evy measure satisfiesR

z≥1z^βν(dz)<∞and that the initial distributionµhas a finite moment of orderβ, for someβ >1. Now, from Lemma 4.1 (i), it holds

1≤i≤Nmax E[ sup

0≤t≤T

|X¯_t^i,N|^β]<∞.

It then follows from Theorem 4 in Fournier and Guillin [8] that there exists some positive constant C only depending onβ such that

E[W₁(¯µ^N_t , µ_t)]≤CE[|Xt|^q]^1/q(N^−1/2+N^{−(β−1)/β}), β 6= 2.

Taking the supremum over t∈ [0, T] and plugging the above bound into (14) we firstly get the desired upper- bound for the quantity max_1≤i≤Nsup_0≤t≤TE[|∆¯ⁱ_t|]. Then, from (13), we derive the similar estimate for the quantity

sup_0≤t≤TE[W1(µ^N_t , µt)]. The proof is now complete.

2.3. Euler-Maruyama time-dscretization scheme for the system of particles. In the previous section, we established a strong rate of convergence of propagation of chaos for the system of particles associated to the McKean-Vlasov SDE (1). From a numerical perspective, the system of particles is not tractable and one usually has to approximate the dynamics of the system (9) by considering the so-called Euler-Maruayama time discretization scheme that we now introduce and analyze. To facilitate our computations, we here only consider the time- homogeneous setting and we claim that, given appropriate regularity assumptions on the maps t 7→ b(t, x, µ), t7→σ(t, x) andt7→h(t, x), the time non-homogeneous case could be dealt with from similar lines of reasonnings.

For a given finite time horizonT >0 and a positive integern, let us introduce the equally spaced time grid on the interval [0, T], given by 0 = t0 < t1 < t2 <· · · < tn =T, where tk =kδ andδ =T /n. We define η(t) =tk

fort ∈(tk, tk+1], for k= 0,· · ·, n−1,η(0) =−∞and η(t) =T for t > T. The Euler-Maruyama approximation scheme (X^n,i,N)_t∈[0,T],i= 1, . . . , N, of the system of particles (9) is given by the following dynamics

(15) X_t^n,i,N =ξⁱ+

Z t 0

b(X_η(s)^n,i,N, µ^n,N_η(s))ds+ Z t

0

σ(X_η(s)^n,i,N)dW_sⁱ+ Z t

0

h(X_η(s)^n,i,N)dZ_sⁱ where we setµ^n,N_s :=N⁻¹PN

i=1δ_Xn,i,N

s . We also introduce the integrability index of the tail of the L´evy measure βν = sup{β≤2 :

Z ∞ 1

z^βν(dz)<∞}.

Note that the above supremum does exist and satisfiesβ_ν ≥1 sinceR∞

1 z ν(dz)<∞. In the case of α-stable like L´evy measure, with index α∈[1,2], we have β_ν =α_ν=αand in the case of tempered α-stable L´evy measure, we have β_ν = 2 andα_ν=α. In order to derive the strongL¹(P) convergence rate of the Euler-Maruyama scheme, we introduce the following additional assumptions on the coefficients and the L´evy measure:

Assumption 2.7.

(i) For any µ ∈ P1(R), the map x 7→ b(x, µ) is ρ-H¨older continuous, uniformly in µ, for some ρ ∈ (0,1].

Namely, there exists some positive constant κsuch that for anyµ∈ P1(R):

|b(x, µ)−b(y, µ)| ≤κ|x−y|^ρ. (ii) γ∈[1/2, βν/2).

(iii) βν > ηαν.

The assumptions (ii) and (iii) stem from some integrability constraints when one investigates the convergence rate of the Euler-Maruyama approximation schemes. Let us note that for α-stable like L´evy measure with index α∈(1,2], the assumption (ii) imposesγ∈[1/2, α/2) and (iii) imposesη <1 while for temperedα-stable like L´evy measure with index α∈ [1,2], the assumption (ii) imposes γ ∈ [1/2,1) and (iii) is always satisfied if α6= 2 and imposesη <1 forα= 2.

Before stating the main result of this section, we start with the following preparatory lemmas.

Lemma 2.8. Let S be a F-stopping time taking values in the interval[0, T]then

(i) the random timeτ(S) = inf{s≥0 :η(s)≥S} is aF-stopping time and {τ(S)≥t}={η(t)< S}, (ii) for any F-adapted processX, we haveX_η(S)and η(S)are both FS− measurable.

Proof. (i) Using the fact thatη is left-continuous, for any it holds {τ(S)≥t}={η(t)< S}=

n−1

[

i=0

{t_i < t≤t_i+1} ∩ {t_i< S}

!

∪

{T < t} ∩ {T < S}

∈ F_t−

(ii) For anyc∈R, using the fact thatS is a F-stopping time taking value in [0, T] we have {Xη(S)< c}=

n−1

[

i=0

{ti< S≤t_i+1} ∩ {Xti < c} ∈ FS−.

Similarly, we see thatη(S) isF_S− measurable.

Lemma 2.9. Under the same assumptions as in the statement of Lemma 4.1 (i), for any T > 0, there exists a positive constantCT such that for all positive integern

1≤i≤Nmax E h

sup

t∈[0,T]

|X_t^n,i,N|^βi

≤CT and max

1≤i≤N sup

0≤t≤TE

h|X_t^n,i,N−X_η(t)^n,i,N|^βi

≤CTn^−β2 where we recall thatη(t) =tk for any t∈(tk, tk+1] and for anyk= 0,· · ·, n−1.

Proof. The proof of the first moment estimate follows from similar lines of reasonings as those employed to prove (31) of Lemma 4.1. We focus below on the particularities which are induced by having piecewise constant coefficients.

Step 1: For notational convenience, in the proof of the first moment estimate, we write Xⁱ := (X_t^n,i,N)_0≤t≤T, µt=µ^n,N_t , t ∈[0, T], Wⁱ = (W_tⁱ)_0≤t≤T, Zeⁱ = (Ze_tⁱ)_0≤t≤T andZbⁱ = (Zbⁱ)_0≤t≤T where we introduced the notations Ze_tⁱ = Rt

0

R1

0 zdNeⁱ(ds, dz) and Zb_tⁱ = Rt 0

R∞

1 zdNeⁱ(ds, dz). From the Yamada-Watanabe theorem, there exists a measurable map

ΦN : (R^d)^N ×

C([0, T];R^d) N

×

D([0, T];R^d) N

×

D([0, T];R^d) N

→

C([0, T];R^d) N

such that

(X¹,· · ·, X^N) = ΦN((ξ¹,· · ·, ξ^N),(W¹,· · · , W^N),(Ze¹,· · ·,Ze^N),(Zb¹,· · · ,Zb^N)) Moreover, by symmetry of the dynamics (15), for any permutationζof{1,· · ·, N}, we get

(X^ζ(1),· · · , X^ζ(N)) = Φ_N((ξ^ζ(1),· · ·, ξ^ζ(N)),(W^ζ(1),· · · , W^ζ(N)),(Ze^ζ(1),· · ·,Ze^ζ(N)),(Zb^ζ(1),· · · ,Zb^ζ(N))).

Now, combining the fact that (Zb¹,· · ·,Zb^N) and (ξ¹,· · · , ξ^N),(W¹,· · ·, W^N),(Ze¹,· · · ,Ze^N)

are independent together with the fact that (ξⁱ, Wⁱ,Zeⁱ)_1≤i≤N are i.i.d, we deduce that conditionally on (Zb¹,· · ·,Zb^N) the processes (Xⁱ)_1≤i≤N are identically distributed.

Step 2: The integral againstZbⁱ generates jumps at discrete instants, that is, one may write the restriction ofNeⁱ to the set [0,∞)× {z≥1}asP

n≥1δ_(Ti

n,Zⁱ_n)where (T_nⁱ)_n≥1 are the jump times of a Poisson processJⁱ with intensity λ=R∞

1 ν(dz), the random variables (Z_nⁱ)_n≥1 being i.i.d. with lawλ⁻¹1_z≥1ν(dz). We denote by Gⁱ:=σ(T_nⁱ, n≥1) theσ-algebra generated by all the jump times of the compound Poisson processZbⁱ.

We now prove some conditional moment estimate forXⁱ. As the computations are similar between two successive instantsT_nⁱ, T_n+1ⁱ , we only give the estimate on the interval [T₁ⁱ∧T, T₂ⁱ∧T]. The dynamics afterT₁ⁱ∧T and strictly beforeT₂ⁱ∧T is given by

X_tⁱ=X_Tⁱi 1∧T +

Z

(T₁ⁱ∧T ,t]

eb(X_η(s)ⁱ , µ_η(s))ds+ Z

(T₁ⁱ∧T ,t]

σ(X_η(s)ⁱ )dW_sⁱ+ Z

(T₁ⁱ∧T ,t]

h(X_η(s)ⁱ )dZe_sⁱ (16)

witheb(x, µ) :=b(x, µ)−h(x)R

z≥1zν(dz).

By using Lemma 2.8 and the linear growth condition onb andh, for anyt∈(T₁ⁱ∧T, T), the absolute value of the drift can bounded as follows

Z

(T₁ⁱ∧T ,t]

eb(X_η(s)ⁱ , µ_η(s))ds

≤C_T Z t

T₁ⁱ∧T

(1 +|X_η(Tⁱ i

1∧T)|)1_{s≤τ(Ti

1∧T)}ds+C_T Z t

T₁ⁱ∧T

(1 +|X_η(s)ⁱ |)1_{s>τ(Ti

1∧T)}ds+C_T Z t

T₁ⁱ∧T

1 N

N

X

j=1

|X_η(s)^j |ds

≤CT(1 +|X_η(Tⁱ i

1∧T)|)T n⁻¹+CT

Z t T₁ⁱ∧T

1_{η(s)≥Ti

1∧T}|X_η(s)ⁱ |ds+CT

Z t T₁ⁱ∧T

1 N

N

X

j=1

|X_η(s)^j |ds

≤C_T



1 +|X_η(Tⁱ i

1∧T)|+ 1 N

N

X

j=1

|X_η(T^j i 1∧T)|+

Z t T₁ⁱ∧T

1_{η(s)≥Ti

1∧T}|X_η(s))ⁱ |ds+ Z t

T₁ⁱ∧T

1_{η(s)≥Ti 1∧T}

1 N

N

X

j=1

|X_η(s)^j |ds



. where we have used the fact that|τ(s)−s| ≤T n⁻¹ for alls∈[0, T].

On the other hand, again by using Lemma 2.8, the stochastic integrals against theL²-martingalesWⁱ and Zeⁱ can be similarly decomposed into

Z

(T₁ⁱ∧T ,t]

σ(X_η(s)ⁱ )dW_sⁱ=σ(X_η(Tⁱ i

1∧T))(W_τ(Tⁱ i

1∧T)−W_Tⁱi 1∧T) +

Z

(T₁ⁱ∧T ,t]

σ(X_η(s)ⁱ )1_{η(s)≥Ti

1∧T}dW_sⁱ. By using the Burkholder-Davis-Gundy inequality and then the Jensen inequality, we obtain

E

sup

T₁ⁱ∧T≤t<T₂ⁱ∧T

|X_tⁱ|^β F_Tⁱ

1∧T ∨σ(Zb¹,· · ·,Zb^N)

≤C_T 1 +|X_Tⁱi

1∧T|^β∨ |X_η(Tⁱ i

1∧T)|^β+ 1 N

N

X

j=1

|X_η(T^j i 1∧T)|^β +

Z T₂ⁱ∧T T₁ⁱ∧T

1_{η(s)≥Ti 1∧T}E

|X_η(s)ⁱ |^β F_Ti

1∧T ∨σ(Zb¹,· · ·,Zb^N) ds