Simulation and approximation of Lévy-driven stochastic differential equations

DOI:10.1051/ps/2009017 www.esaim-ps.org

SIMULATION AND APPROXIMATION OF L´ EVY-DRIVEN STOCHASTIC DIFFERENTIAL EQUATIONS

Nicolas Fournier

Abstract. We consider the approximate Euler scheme for L´evy-driven stochastic differential equa- tions. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them.

For example, when the L´evy measure of the driving process behaves like|z|^−1−αdz near 0, for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of ordern^α. For a similar error when neglecting the small jumps, see [S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a L´evy process. Stochastic Process. Appl. 103 (2003) 311–349], the computational cost is of ordern^α/(2−α), which is huge whenαis close to 2. In the same spirit, we study the problem of the approximation of a L´evy-driven S.D.E. by a Brownian S.D.E. when the L´evy process has no large jumps. Our results rely on some results of [E. Rio, Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincar´e Probab. Stat. 45(2009) 802–817] about the central limit theorem, in the spirit of the famous paper by Koml´os-Major-Tsun´ady [J. Koml´os, P. Major and G. Tusn´ady, An approximation of partial sums of independent rvs and the sample df I.Z. Wahrsch. verw. Gebiete32(1975) 111–131].

Mathematics Subject Classification. 60H35, 60H10, 60J75.

Received April 10, 2009. Revised September 11, 2009.

1. Introduction

Let (Z_t)_t≥0 be a one-dimensional square integrable L´evy process. Then for some a∈R, b∈ R+ and some measureν onR∗:=R\{0}satisfying

R∗z²ν(dz)<∞, Z_t=at+bB_t+

_t

0

R∗

zN˜(ds,dz), (1.1)

where (B_t)_t≥0is a standard Brownian motion, independent of a Poisson measureN(ds,dz) on [0,∞)×R∗with intensity measure dsν(dz) and where ˜N is its compensated Poisson measure, see Jacod-Shiryaev [10].

We consider, for some x∈Rand some functionσ:R→R, the S.D.E.

X_t=x+ _t

0 σ(X_s−)dZ_s. (1.2)

Keywords and phrases.L´evy processes, stochastic differential equations, Monte-Carlo methods, simulation, Wasserstein distance.

1 LAMA UMR 8050, Facult´e de Sciences et Technologies, Universit´e Paris Est, 61 avenue du G´en´eral de Gaulle, 94010 Cr´eteil Cedex, France;nicolas.fournier@univ-paris12.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2011

Using some classical results (seee.g. Ikeda-Watanabe [5]), there is strong existence and uniqueness for (1.2) as soon asσis Lipschitz continuous: for any given couple (B, N), there exists an unique c`adl`ag adapted solution (X_t)_t≥0 to (1.2). Byadapted, we mean adapted to the filtration (Ft)_t≥0generated by (B, N).

We consider two related problems in this paper. The first one deals with the numerical approximation of the solution (X_t)_t≥0. It is discussed in the next section in the case whereσ is bounded and Lipschitz continuous.

We extend our study in Section7to the case whereσis locally Lipschitz continuous with at most linear growth, and where the L´evy measureν satisfies only

R∗min(z²,1)ν(dz)<∞.

The second problem concerns the approximation of (X_t)_t≥0 by the solution to a Brownian S.D.E., whenZ has only small jumps, and is discussed in Section3.

Our results are based on a recent work of Rio [13] that concerns the rate of convergence in the central limit theorem, when using the quadratic Wasserstein distance. This result and its application to L´evy processes are discussed in Section4.

The proofs are handled in Sections5and6. We give some numerical illustrations in Section8.

2. Numerical simulation

The first goal of this paper is to study a numerical scheme to solve (1.2). The first idea is to perform an Euler scheme (X_i/nⁿ )_n≥0 with time-step 1/n, see Jacod [6], Jacod-Protter [9], Protter-Talay [12] for rates of convergence. However, this is generally not a good scheme in practise, unless one knows how to simulate the increments of the underlying L´evy process, which is the casee.g. whenZ is a stable process.

We assume here that the L´evy measureν is known explicitly: one can thus simulate random variables with lawν(dz)1_A(z)/ν(A), for anyA such thatν(A)<∞.

The first idea is to approximate the increments ofZ byΔ^n,_i =Z_i/n −Z_(i−1)/n , whereZ_t is the same L´evy process as Z without its (compensated) jumps smaller than . However, Asmussen-Rosinski [1] have shown that for a L´evy process with many small jumps, it is more convenient to approximate small jumps by some Gaussian variables than to neglect them. We thus introduce Δ^n,_i =Δ^n,_i +U_i^n,, whereU_i^n, is Gaussian with same mean and variance as the neglected jumps and is independent ofΔ^n,_i . The arguments of [1] concern only L´evy processes and rely on explicit computations.

Let us write (X_[nt]/n^n, )_t≥0(resp. (X_[nt]/n^n, )_t≥0) for the Euler scheme using the approximate increments (Δ^n,_i )_i≥1 (resp. (Δ^n,_i )_i≥1). They of course have a similar computational cost.

Jacod-Kurtz-M´el´eard-Protter [8] have computed systematically the weak error for the approximate Euler scheme. In particular, they prove some very precise estimates ofE[g(X_[nt]/n^n, )]−E[g(X_t)] forg smooth enough.

The obtained rate of convergence is very satisfying.

Assume now that the goal is to approximate some functional of the path of the solution (e.g. sup_t∈[0,T]|Xt|).

Then we have to estimate the error between the laws of the paths of the processes (not only between the laws of the time marginals). A common way to perform such an analysis is to introduce a suitable coupling between the numerical scheme (X_[nt]/n^n, )_t≥0 and the true solution (X_t)_t≥0 and to estimate the (discretized) strongerror E[sup_t∈[0,T]|X_[nt]/n^n, −X_[nt]/n|²]. We refer to Jacod-Jakubowski-M´emin [7] for the speed of convergence of the discretized process (X_[nt]/n)_t≥0 to the whole process (X_t)_t≥0.

Rubenthaler [14] has studied the strong error when neglecting small jumps. He obtains roughlyE[sup_t∈[0,T]| X_[nt]/n^n, −X_[nt]/n|²]C_T(n⁻¹+

|z|≤z²ν(dz)) (ifb= 0). Forν very singular near 0, the obtained precision is very low. Let us mention that in some particular cases, a more efficient method was introduced in Rubenthaler- Wiktorsson [15].

Our aim here is to prove that the strong error is much lower when usingX_[nt]/n^n, , see Sections2.4and8below.

The main difficulty is to find a suitable coupling between the true increments (Z_i/n−Z_(i−1)/n)_i≥1 and the

approximate increments (Δ^n,_i )_i≥1: clearly, one considersZ, then one erases its jumps smaller than, but how to build the additional Gaussian variable in such a way that it is a.s. close to the erased jumps? We will use a recent result of Rio [13], which gives some very precise rate of convergence for the standard central limit theorem in Wasserstein distance, in the spirit of Koml´os-Major-Tsun´ady [11].

2.1. Notation

We introduce, for∈(0,1),k∈N, F(ν) =

|z|>ν(dz), m_k(ν) =

R∗

|z|^kν(dz), (2.1)

m_k,(ν) =

|z|≤|z|^kν(dz), β(ν) = m_4,(ν) m_2,(ν)· Observe that we always haveβ(ν)≤² andF(ν)≤⁻²m₂(ν).

Forn∈Nandt≥0, we setρ_n(t) = [nt]/n, where [x] is the integer part ofx.

2.2. Numerical scheme

Letn∈Nand∈(0,1) be fixed. We introduce an i.i.d. sequence (Δ^n,_i )_i≥1of random variables, with

Δ^n,₁ =a_n,+b_n,G+

Nn,

i=1

Y_i, (2.2)

where a_n, = (a−

|z|>zν(dz))/n, where b²_n, = (b² +m_2,(ν))/n, where G is Gaussian with mean 0 and variance 1, where N_n, is Poisson distributed with mean F(ν)/n and where Y₁, Y₂, ... are i.i.d. with law ν(dz)1_|z|>/F(ν). All these random variables are assumed to be independent. Then we introduce the scheme

X₀^n,=x, X_(i+1)/n^n, =X_i/n^n,+σ(X_i/n^n,)Δ^n,_i+1 (i≥0). (2.3) Remark 2.1.

(i) The cost of simulation of Δ^n,₁ is of order 1 +E[Nn,] = 1 +F(ν)/n, whence that of (X_ρ^n,_n(t))_t∈[0,T] is of orderT n(1 +F(ν)/n) =T(n+F(ν)), as in [14].

(ii) Δ^n,_i+1has the same law asZ_(i+1)/n −Z_i/n +U_i^n,, whereU_i^n,is Gaussian with same mean and variance as _(i+1)/n

i/n

|z|≤zN(ds,˜ dz) and whereZ_t=at+bB_t+_t

0

|z|>zN(ds,˜ dz).

(iii) The key argument of the paper is to use a suitable coupling betweenU_i^n,and_(i+1)/n

i/n

|z|≤zN(ds,˜ dz).

As shown in Lemma5.2, there exists such a coupling satisfyingE[|U_i^n,−_(i+1)/n

i/n

|z|≤zN˜(ds,dz)|²]≤ Cβ(ν). Then we will choose Δ^n,_i+1=Z_(i+1)/n −Z_i/n +U_i^n,, which thus satisfiesE[|Δ^n,_i+1−(Z_(i+1)/n− Z_i/n)|²]≤Cβ(ν).

2.3. Main result

We may now state our main result.

Theorem 2.2. Assume thatσ:R→R is bounded and Lipschitz continuous. Let ∈(0,1)and n∈N. There is a coupling between a solution (X_t)_t≥0to (1.2) and an approximated solution(X_ρ^n,

n(t))_t≥0as in Subsection2.2 such that for all T >0,

E

t∈[0,T]sup |X_ρn(t)−X_ρ^n,

n(t)|²

≤C_T

n⁻¹+nβ(ν) , where the constant C_T depends only on T, σ, a, b, m₂(ν).

The first bound n⁻¹ is due to the time discretization (Euler scheme) and the second bound nβ(ν) is due to the approximation of the increments of the L´evy process. As noted by Jacod [6], the first bound may be improved if there is no Brownian motionb= 0 (but we have to work with some weaker norm).

2.4. Optimization

Choose= 1/n. Then recalling thatβ(ν)≤², we get E

t∈[0,Tsup]|X_ρn(t)−X_ρ^n,1/n

n(t)|²

≤C_T/n, for a mean cost to simulate (X_ρ^n,1/n

n(t))_t∈[0,T] of orderT(n+F_1/n(ν)).

• We always haveF(ν)≤m₂(ν)⁻², so that the cost is always smaller thanCT n².

• Ifν(dz)^z→0 |z|^−1−αdz for someα∈(0,2), thenF(ν)^−α, so that the cost is of orderT(n+n^α).

When neglecting the small jumps, one gets, for a mean cost of orderT(n+F(ν)), E

sup

t∈[0,T]|X_ρn(t)−Xˆ_ρ^n,

n(t)|²

≤C_T(1/n+m₂()),

see [14]. In the case where ν(dz)^z→0 |z|^−1−αdz for some α∈(0,2), we have m₂()^2−α and F(ν)^−α. Thus to get an mean squared error of order 1/n, one has to choose=n^{−1/(2−α)}, which yields a cost of order T(n+n^α/(2−α)). This is huge whenαis close to 2.

2.5. Discussion

The computational cost to get a given precision does not explode when the L´evy measure becomes very singular near 0. The more ν is singular at 0, the more there are jumps greater than , which costs many simulations. But the more it is singular, the more the jumps smaller thanare well-approximated by Gaussian random variables. These two phenomena are in competition and we prove that the second one compensates (partly) the first one.

Our result involves a suitable coupling between the solution (X_t)_t≥0 and its approximation (X_t^n,)_t≥0. This might seem uninteresting in practise, since by assumption, (X_t)_t≥0 is completely unknown. But this is always the case when dealing withstrong errors. In our opinion, this is just an artificial way to estimate the rate of convergence of the pathsin law, using a Wasserstein type distance. For example, our result allows us to estimate the error when approximatingE[F((X_ρn(t))_t∈[0,T])] byE[F((X_ρ^n,_n(t))_t∈[0,T])], for any Lipschitz functionalF.

Recall that Theorem2.2is extended in Section7to the case whereσis locally Lipschitz with at most linear growth and wherem₂(ν) might be infinite.

Let us finally mention that the simulation algorithm can easily be adapted to the case of dimensiond≥2.

We believe that the result still holds. However, the result of Rio [13] is not known in the multidimensional setting. We could use instead the results of Einmahl [3] or Zaitsev [19]. This would be much more technical.

3. Brownian approximation

It is classical in applied sciences to approximate discontinuous phenomena by continuous ones. Here, we examine how far the solution to (1.2) is from the solution of a continuous Brownian SDE. Let us mention some more complicated models where such a problem occurs.

(a) The Boltzmann equation is a P.D.E. that can be related to a Poisson-driven S.D.E. (see Tanaka [16]), while the Landau equation can be related to a Brownian S.D.E. (see Gu´erin [4]). In the grazing collision limit, the Boltzmann equation is known to converge to the Landau equation (see Villani [17]). However, no convergence rate is known.

(b) In [18], Walsh approximates a Poisson-driven stochastic heat equation by a white-noise driven S.P.D.E.

He proves some convergence results, without rate.

Here, we consider only the simple case of one-dimensional L´evy-driven S.D.E.s, but we hope that the main ideas of our proof might apply to more complicated models.

Consider the L´evy process introduced in (1.1), consider x∈ R, σ : R → R Lipschitz continuous, and the unique solution (X_t)_t≥0 to (1.2). Recall (2.1), consider a Brownian motion (W_t)_t≥0 and set

Z˜_t=at+

b²+m₂(ν)W_t, (3.1)

which has the same mean and variance asZ_t. Let ( ˜X_t)_t≥0be the unique solution to X˜_t=x+

_t

0 σ( ˜X_s−)d ˜Z_s. (3.2)

Theorem 3.1. Assume thatσ is Lipschitz continuous and bounded. Then it is possible to couple the solutions (X_t)_t≥0 to (1.2) and ( ˜X_t)_t≥0 to (3.2) in such a way that for all p≥4, all T >0, alln≥1,

E

sup

t∈[0,T]|X_t−X˜_t|²

≤C_T,p

n^2/p−1+m_p(ν)^2/p+nm₄(ν)

,

whereC_T,p depends only onp, T, σ, a, b, m₂(ν).

If we only know thatm₄(ν)<∞, we choosep= 4 andn= [m₄(ν)^−2/3] + 1 to getE

sup_t∈[0,T]|X_t−X˜_t|²

≤ C_T(m₄(ν)^1/3+m₄(ν)).

Consider a sequence of L´evy processes (Z_t)_t≥0 with drift a, diffusion coefficient b and L´evy measure ν, such that z²ν(dz) tends weakly to the Dirac massδ₀(dz). Then lim_→0m₂(ν) = 1, while in almost all cases, lim_→0m_p(ν) = 0 for some (or all)p >2.

Consider the solution toX_t=x+_t

0σ(X_s− )dZ_s. Then it is well-known and easy to show that (X_t)_t≥0tends in law to the solution of a Brownian S.D.E. Theorem 3.1provides a rate of convergence (for some Wasserstein distance). To our knowledge, it is the first result in that direction. For example, we will immediately deduce the following corollary.

Corollary 3.2. Assume that σ is Lipschitz continuous and bounded. Assume that ν({|z| > }) = 0 for some ∈(0,1]. Then it is possible to couple the solutions(X_t)_t≥0 to (1.2) and( ˜X_t)_t≥0 to (3.2) in such a way that for allη ∈(0,1), allT >0,

E

sup

t∈[0,T]|X_t−X˜_t|²

≤C_T,η^1−η whereC_T,η depends only on η, T, σ, a, b, m₂(ν).

4. Coupling results

Consider two lawsP, QonRwith finite variance. The Wasserstein distanceW2is defined by W₂²(P, Q) = inf

E

|X−Y|²

, L(X) =P,L(Y) =Q .

With an abuse of notation, we also writeW₂(X, Y) =W₂(X, Q) =W₂(P, Q) ifL(X) =P andL(Y) =Q. We recall the following result of Rio [13], Theorem 4.1.

Theorem 4.1. There is an universal constantC such that for any sequence of i.i.d. random variables (Y_i)_i≥1 with mean0 and variance θ², for any n≥1,

W₂²

√1 n

n i=1

Y_i,N(0, θ²)

≤CE[Y₁⁴] nθ² ·

HereN(0, θ²) is the Gaussian distribution with mean 0 and varianceθ². Recall now (2.1).

Corollary 4.2. Consider a pure jump centered L´evy process (Y_t)_t≥0 with L´evy measure μ. In other words Y_t = _t

0

R∗zM˜(ds,dz), where M˜ is a compensated Poisson measure with intensity dsμ(dz). There is an universal constant C such that

∀t≥0, W₂²(Y_t,N(0, tm₂(μ)))≤Cm₄(μ) m₂(μ)· Proof. Lett >0. Forn≥1,i≥1, writeY_iⁿ=n^1/2_it/n

(i−1)t/n

R∗zM˜(ds,dz), whenceY_t=n^−1/2_n

i=1Y_iⁿ. The Y_iⁿ are i.i.d., centered,E[(Y₁ⁿ)²] =tm₂(μ), and

E[(Y₁ⁿ)⁴] =n²E

⎡

⎣ _t/n

0

R∗

z²M(ds,dz) ₂⎤

⎦

=n²E

⎡

⎣ _t/n

0

R∗

z²M˜(ds,dz) + (t/n)m₂(μ) ₂⎤

⎦

=n²

tm₄(μ)/n+ (tm₂(μ)/n)²

=ntm₄(μ) +t²m₂(μ).

Using Theorem4.1, we get

W₂²(Y_t,N(0, tm₂(μ)))≤Cntm₄(μ) +t²m₂(μ) ntm₂(μ)

n→∞−→ Cm₄(μ) m₂(μ),

which concludes the proof.

This result is quite surprising at first glance: since the variances of the involved variables aretm₂(μ), it would be natural to get a bound that decreases to 0 astdecreases to 0 (and that explodes for larget). Of course, we deduce the boundW₂²(Y_t,N(0, tm₂(μ)))≤Cmin(m₄(μ)/m₂(μ), tm₂(μ)), but this is now optimal, as shown in the following example.

Example. Consider, for >0,μ= (2²)⁻¹(δ+δ₋) and the corresponding pure jump (centered) L´evy process (Y_t)_t≥0. It takes its values inZ. Observe thatm₂(μ) = 1 andm₄(μ) =². There isc >0 such that for all t≥0, all >0,W₂²(Y_t,N(0, t))≥cmin(², t) =cmin(m₄(μ)/m₂(μ), tm₂(μ)). Indeed,

• ift≤², thenP(Y_t = 0)≥e^−tμ(R)= e^−t/2 ≥1/e, from which the lower-boundW₂²(Y_t,N(0, t))≥ct= cmin(t, ²) is easily deduced;

• if t ≥ ², use thatW₂²(Y_t,N(0, t)) ≥ E[min_n∈Z|t^1/2G−n|²] = tE[min_n∈Z|G−nt^−1/2|²], where G is Gaussian with mean 0 and variance 1. Tedious computations show that there is c > 0 such that for any a∈(0,1],E[minn∈Z|G−na|²]≥(a/4)²P(G∈ ∪n∈Z[(n+ 1/4)a,(n+ 3/4)a])≥ca². Hence W₂²(Y_t,N(0, t))≥ ct(t^−1/2)²=c²=cmin(t, ²).

5. Proof of Theorem 2.2

We recall elementary results about the Euler scheme for (1.2) in Section 5.1. We introduce our coupling in Section 5.2, which allows us to compare our scheme with the Euler scheme in Section 5.3. We conclude in Section 5.4. We assume in the whole section thatσis bounded and Lipschitz continuous.

5.1. Euler scheme

We introduce the Euler scheme with step 1/nassociated to (1.2). Let

Δⁿ_i =Z_i/n−Z_(i−1)/n (i≥1), (5.1)

X₀ⁿ=x, X_(i+1)/nⁿ =X_i/nⁿ +σ

X_i/nⁿ

Δⁿ_i+1 (i≥0). (5.2)

The following result is classical.

Proposition 5.1. Consider a L´evy process (Z_t)_t≥0 as in (1.1). For (X_t)_t≥0 the solution to (1.2) and for (X_i/nⁿ )_i≥0 defined in (5.1)–(5.2),

E

t∈[0,T]sup |X_ρn(t)−X_ρⁿ_n(t)|²

≤C_T/n, whereC_T depends only onT, a, b, m₂(ν) andσ.

We give a proof for the sake of completeness.

Proof. Using the Doob and Cauchy-Schwarz inequalities, we get, for 0≤s≤t≤T, E

sup

u∈[s,t]|X_u−X_s|²

≤CE a _t

s |σ(X_u)|du 2

+ sup

u∈[s,t]

b

_u

s σ(X_v)dB_{v 2}

(5.3)

+ sup

u∈[s,t]

_u

s

R∗

σ(X_v−)zN(ds,˜ dv) ₂

≤C_{T t}

s

(a²+b²+m₂(ν))||σ||²_∞dv≤C_T(t−s).

Observe now that X_ρⁿ_n(t) = x+_ρn(t)

0 σ(X_ρⁿ_n(s)−)dZ_s. Setting Aⁿ_t = sup_[0,t]|X_ρn(s)−X_ρⁿ_n(s)|², we thus get Aⁿ_t = sup_[0,t]|_ρn(s)

0 (σ(X_u−)−σ(X_ρⁿ

n(u)−))dZ_u|². Using the same arguments as in (5.3), then the Lipschitz property ofσand (5.3), we get

E[Aⁿ_t]≤C_{T ρn(t)}

0 (a²+b²+m₂(ν))E[(σ(Xs)−σ(X_ρⁿ_n(s)))²]ds

≤C_{T t}

0 E[(Xs−X_ρn(s))²+ (X_ρn(s)−X_ρⁿ_n(s))²]ds

≤C_{T t}

0 (|s−ρ_n(s)|+E[Aⁿ_s]) ds.

We conclude using that|s−ρ_n(s)| ≤1/nand the Gronwall lemma.

5.2. Coupling

We now introduce a suitable coupling between the Euler scheme (see Sect. 5.1) and our numerical scheme (see Sect. 2.2). Recall (2.1).

Lemma 5.2. Let n∈ Nand > 0. There exist two coupled families of i.i.d. random variables (Δⁿ_i)_i≥1 and (Δ^n,_i )_i≥1, distributed respectively as in (5.1) and (2.2) in such a way that for eachi≥1,

E[(Δⁿ_i −Δ^n,_i )²]≤Cβ(ν),

whereC is an universal constant. Furthermore, for all >0, all n∈N, all i≥1, E[Δⁿ_i] =E[Δ^n,_i ] = a

n, Var[Δⁿ_i] =Var[Δ^n,_i ] =b²+m₂(ν)

n ·

Proof. It of course suffices to build (Δⁿ₁,Δ^n,₁ ) and then to take independent copies. Consider a Poisson measure N(ds,dz) with intensity measure dsν(dz)1_{|z|≤} on [0,∞)× {|z| ≤ }. Observe that _t

0

|z|≤zN(ds,˜ dz) is a centered pure jump L´evy process with L´evy measure ν(dz) =1_|z|≤ν(dz). Then we use Corollary4.2 and enlarge the underlying probability space if necessary: there is a Gaussian random variableG^n,₁ with mean 0 and variancem₂(ν)/n=m_2,(ν)/nsuch thatE

|_1/n

0

|z|≤zN(ds,˜ dz)−G^n,₁ |²

≤Cm₄(ν)/m₂(ν) =Cβ(ν).

We consider a Brownian motion (B_t)_t≥0 and a Poisson measure N with intensity measure dsν(dz)1_{|z|>}

on [0,∞)× {|z|> }, independent of the couple (G^n,₁ ,_1/n

0

|z|≤zN˜(ds,dz)) and we set

• Δⁿ₁ :=a/n+bB_1/n+_1/n

0

|z|≤zN˜(ds,dz) +_1/n

0

|z|>zN(ds,˜ dz);

• Δ^n,₁ :=a/n+bB_1/n+G^n,₁ +_1/n

0

|z|>zN˜(ds,dz).

Then Δⁿ₁ has obviously the same law as Z_1/n−Z₀ (see (1.1) and (5.1)), while Δ^n,₁ has also the desired law (see (2.2)). Indeed, bB_1/n+G^n,₁ has a centered Gaussian law with variance b²/n+m_2,(ν)/n = b²_n, and a/n+_1/n

0

|z|>zN(ds,˜ dz) = a_n,+_1/n

0

|z|>zN(ds,dz). This last integral can be represented as in (2.2).

FinallyE[(Δⁿ₁ −Δ^n,₁ )²]≤E

|_1/n

0

|z|≤zN(ds,˜ dz)−G^n,₁ |²

≤Cβ(ν) and the mean and variance estimates

are obvious.

5.3. Estimates

We now compare our scheme with the Euler scheme. To this end, we introduce some notation. First, we consider the sequence (Δⁿ_i,Δ^n,_i )_i≥1 introduced in Lemma 5.2. Then we consider (X_i/nⁿ )_i≥0 and (X_i/n^n,)_i≥0 defined in (5.2) and (2.3). We introduce the filtration F_i^n, =σ(Δⁿ_k,Δ^n,_k , k ≤i) and the processes, fori ≥0 (withV₀^n,= 0)

Y_i^n,=X_i/nⁿ −X_i/n^n,, V_i^n,= a n

i−1 k=0

[σ(X_k/nⁿ )−σ(X_k/n^n,)], M_i^n,=Y_i^n,−V_i^n,. Lemma 5.3. There is a constant C, depending only on σ, a, b, m₂(ν)such that for all N ≥1,

E

i=0,...,Nsup |Y_i^n,|²

≤Cnβ(ν)(1 +C/n)^N(1 +N²/n²).

Proof. We divide the proof into four steps.

Step 1. We prove that for alli≥0,E

|Y_i^n,|²

≤Cnβ(ν)(1 +C/n)ⁱ. First, E[|Y_i+1^n,|²] =E[|Y_i^n,|²] +E[(σ(X_i/nⁿ )Δⁿ_i+1−σ(X_i/n^n,)Δ^n,_i+1)²]

+ 2E

Y_i^n,(σ(X_i/nⁿ )Δⁿ_i+1−σ(X_i/n^n,)Δ^n,_i+1)

=E[|Y_i^n,|²] +I_i^n,+J_i^n,. Now, using Lemma5.2and that (Δⁿ_i+1,Δ^n,_i+1) is independent ofF_i^n,, we deduce that

J_i^n,=2a nE

Y_i^n,(σ(X_i/nⁿ )−σ(X_i/n^n,)) ≤C

nE[|Y_i^n,|²],

sinceσ is Lipschitz continuous. Using now the Lipschitz continuity and the boundedness of σ, together with Lemma5.2 and the independence of (Δⁿ_i+1,Δ^n,_i+1) with respect toF_i^n,, we get

I_i^n,≤CE[|Y_i^n,|²(Δ^n,_i+1)²] +CE[(Δ^n,_i+1−Δⁿ_i+1)²]≤C

nE[|Y_i^n,|²] +Cβ(ν).

Finally, we get

E[|Y_i+1^n,|²]≤(1 +C/n)E[|Y_i^n,|²] +Cβ(ν).

SinceY₀^n,= 0, this entails thatE[|Y_i^n,|²]≤Cβ(ν)[1 + (1 +C/n) +...+ (1 +C/n)ⁱ⁻¹]≤Cnβ(ν)(1 +C/n)ⁱ. Step 2. We check that for N ≥ 1, E[sup_0,...,N|V_i^n,|²] ≤ Cnβ(ν)(1 +C/n)^NN²/n². It suffices to use the Lipschitz property ofσ, the Cauchy-Schwarz inequality and then Step 1:

E

1,...,Nsup |V_i^n,|²

≤CE

⎡

⎣

1 n

N−1

i=0

|Y_i^n,| 2⎤

⎦≤CN n²

N−1 i=0

E[|Y_i^n,|²]

≤CN²

n²nβ(ν)(1 +C/n)^N.

Step 3. We now verify that (M_i^n,)_i≥0 is a (F_i^n,)_i≥0-martingale. We have M_i+1^n, −M_i^n, = σ(X_i/nⁿ )[Δⁿ_i+1− a/n]−σ(X_i/n^n,)[Δ^n,_i+1−a/n]. The step is finished, since the variables Δⁿ_i+1−a/nand Δ^n,_i+1−a/n are centered and independent ofF_i^n,.

Step 4. Using the Doob inequality and then Steps 1 and 2, we get E

i=0,...,Nsup |M_i^n,|²

≤C sup

i=0,...,NE

|M_i^n,|²

≤C sup

i=0,...,NE

|Y_i^n,|²

+C sup

i=0,...,NE

|V_i^n,|²

≤Cnβ(ν)(1 +C/n)^N(1 +N²/n²).

But now, sinceY_i^n,=M_i^n,+V_i^n,, E

i=0,...,Nsup |Y_i^n,|²

≤CE

i=0,...,Nsup |M_i^n,|²

+CE

i=0,...,Nsup |V_i^n,|²

,

which allows us to conclude.

Let us rewrite these estimates in terms ofXⁿand X^n,.

Lemma 5.4. Consider the sequence(Δⁿ_i,Δ^n,_i )_i≥1 introduced in Lemma5.2and then (X_i/nⁿ )_i≥0 and(X_i/n^n,)_i≥0 defined in (5.2) and (2.3). For allT ≥0,

E

sup

t∈[0,T]|X_ρⁿ_n(t)−X_ρ^n,

n(t)|²

≤C_Tnβ(ν),

whereC_T depends only onT, a, b, m₂(ν), σ.

Proof. With the previous notation, sup_t∈[0,T]|X_ρⁿ

n(t)−X_ρ^n,

n(t)| = supi=0,...,[nT]|Y_i^n,|. Thus using Lemma 5.3, we get the boundCnβ(ν)(1 +C/n)^[nT](1 + [nT]²/n²)≤Cnβ(ν)e^CT(1 +T²), which ends the proof.

5.4. Conclusion

Proof of Theorem2.2. Fixn∈Nand >0. Denote byQ(du,dv) the joint law of (Δⁿ₁,Δ^n,₁ ) built in Lemma5.2 and write Q(du,dv) =Q₁(du)R(u,dv), whereQ₁(du) is the law of Δⁿ₁ and whereR(u,dv) is the law of Δ^n,₁ conditionally to Δⁿ₁ =u.

Consider a L´evy process (Z_t)_t≥0 as in (1.1) and (X_t)_t≥0 the corresponding solution to (1.2). Set, fori≥0, Δⁿ_i =Z_i/n−Z_(i−1)/nand consider the Euler scheme (X_i/nⁿ )_i≥0as in (5.2). For eachi≥1, let Δ^n,_i be distributed according toR(Δⁿ_i,dv), in such a way that (Δ^n,_i )_i≥1 is an i.i.d. sequence. Finally, let (X_i/n^n,)_i≥0 as in (2.3).

By this way, the processes (X_t)_t≥0, (X_i/nⁿ )_i≥0 and (X_i/n^n,)_i≥0 are coupled in such a way that we may apply Proposition5.1and Lemma5.4. We get

E

t∈[0,T]sup |X_ρn(t)−X_ρ^n,_n(t)|²

≤2E

t∈[0,T]sup |X_ρn(t)−X_ρⁿ_n(t)|²+ sup

t∈[0,T]|X_ρⁿ_n(t)−X_ρ^n,_n(t)|²

≤C_T[n⁻¹+nβ(ν)].

This concludes the proof.

6. Proofs of Theorem 3.1 and Corollary 3.2

We assume in the whole section that σ is bounded and Lipschtiz continuous. We start with a technical lemma.

Lemma 6.1. Let (X_t)_t≥0 and ( ˜X_t)_t≥0 be solutions to (1.2) and (3.2). Then for p ≥ 2, for all t₀ ≥ 0, all h∈(0,1],

E

sup

t∈[t0,t0+h]|X_t−X_t0|^p

≤C_p(h^p/2+hm_p(ν)),

E

sup

t∈[t0,t0+h]|X˜_t−X˜_t0|^p

≤C_ph^p/2,

whereC_p depends only on p, σ, a, b, m₂(ν).

Proof. It clearly suffices to treat the case of (X_t)_t≥0, because ( ˜X_t)_t≥0solves the same equation (withν replaced by 0 andb replaced byb+m₂(ν)). Let thusp≥2.

Using the Burkholder-Davies-Gundy inequality and the boundedness ofσ, we get

E

sup

t∈[t0,t0+h]|X_t−X_t0|^p

≤C_pE _t0+h

t0

|aσ(X_s)|ds _p

+C_pE

⎡

⎣ _t0+h

t0

b²σ²(X_s)ds _p/2⎤

⎦

+C_pE

⎡

⎣ _t0+h

t0

R∗

σ²(X_s)z²N(ds,dz) _p/2⎤

⎦

≤C_ph^p+C_ph^p/2+C_pE

⎡

⎣ _t0+h

t0

R∗

z²N(ds,dz) _p/2⎤

⎦

≤C_ph^p/2+C_pE[U_h^p/2], where U_t =_t

0

R∗z²N(ds,dz). It remains to check that for t ≥0, E[U_t^p/2] ≤ C_p(t^p/2+tm_p(ν)). But, with C_p depending onm₂(ν),

E[U_t^p/2] = _t

0

ds

R∗

ν(dz)E[(U_s+z²)^p/2−U_s^p/2]

≤C_{p t}

0 ds

R∗

ν(dz)E[z²U_s^p/2−1+|z|^p]≤C_{p t}

0 E[U_s^p/2−1]ds+C_pm_p(ν)t

≤C_{p t}

0 E[U_s^p/2]⁻¹ds+C_p(^p/2−1+m_p(ν))t,

for any >0. HenceE[U_t^p/2]≤C_p(^p/2−1t+m_p(ν)t)e^Cpt/by the Gronwall lemma. Choosing=t, we conclude

that E[U_t^p/2]≤C_p(t^p/2+m_p(ν)t).

Proof of Theorem 3.1. We fixn≥1,T >0 andp≥4.

Step 1. Using Lemma 4.2 (see also Lem. 5.2) we deduce that we may couple two i.i.d. families (Δⁿ_i)_i≥1 and ( ˜Δⁿ_i)_i≥1, in such a way that:

• (Δⁿ_i)_i≥1 has the same law as the increments (Z_i/n−Z_(i−1)/n)_i≥1 of the L´evy process (1.1);

• ( ˜Δⁿ_i)_i≥1 has the same law as the increments ( ˜Z_i/n−Z˜_(i−1)/n)_i≥1 of the L´evy process (3.1);

• for alli≥1,E[(Δⁿ_i −Δ˜ⁿ_i)²]≤Cm₄(ν) (we allow constants to depend onm₂(ν)).

Step 2. We then setX₀ⁿ = ˜X₀ⁿ =x and fori≥1,X_i/nⁿ =X_(i−1)/nⁿ +σ(X_(i−1)/nⁿ )Δⁿ_i and ˜X_i/nⁿ = ˜X_(i−1)/nⁿ + σ( ˜X_(i−1)/nⁿ ) ˜Δⁿ_i. Using exactly the same arguments as in Lemmas5.3and5.4, we deduce thatE

sup_t∈[0,T]|X_ρⁿ

n(t)

−X˜_ρⁿ_n(t)|²

≤C_Tnm₄(ν), whereC_T depends only onT, σ, a, b, m₂(ν).

Step 3. But (X_ρⁿ

n(t))_t≥0 is the Euler discretization of (1.2), while ( ˜X_ρⁿ

n(t))_t≥0 is the Euler discretization of (3.2). Hence using Step 2, Proposition 5.1 and a suitable coupling as in the final proof of Theorem 2.2, E

sup_t∈[0,T]|X_ρn(t)−X˜_ρn(t)|²

≤C_T(1/n+nm₄(ν)).

Step 4. We now prove thatE

sup_t∈[0,T]|X_t−X_ρn(t)|²

≤C_T,p(n^2/p−1+m_p(ν)^2/p).