Ergodic approximation of the distribution of a stationary diffusion : rate of convergence

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Ergodic approximation of the distribution of a stationary diffusion : rate of convergence

Gilles Pagès, Fabien Panloup

To cite this version:

Gilles Pagès, Fabien Panloup. Ergodic approximation of the distribution of a stationary diffusion : rate of convergence. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2012, 22 (3), 1059-1100 ; http://dx.doi.org/10.1214/11-AAP779. �10.1214/11-AAP779�. �hal-00512722�

Ergodic approximation of the distribution of a stationary diffusion : rate of convergence

Gilles Pag`es^∗, Fabien Panloup^† August 31, 2010

Abstract

We extend to Lipschitz continuous functionals either of the true paths or of the Euler scheme with decreasing step of a wide class of Brownian ergodic diffusions, the Central Limit Theorems formally established for their marginal empirical measure of these processes (which is classical for the diffusions and more recent as concerns their discretization schemes). We illustrate our results by simulations in connection with barrier option pricing.

Keywords: stochastic differential equation; stationary process; steady regime; ergodic diffusion; Central Limit Theorem; Euler scheme.

AMS classification (2000): 60G10, 60J60, 65C05, 65D15, 60F05.

1 Introduction

In a recent paper ([22]), we investigated weighted empirical measures based on some Euler schemes with decreasing step in order to approximate recursively the distributionP_ν of a stationary Feller Markov processX:= (X_t)_t≥0with invariant distributionν. To be precise, let ( ¯X_t) be such an Euler scheme, let (Γ_k)_k≥1denote its sequence of discretization times and let (η_k)_k≥1 be a sequence of weights. On the one hand we showed under some Lyapunov- type mean-reverting assumptions on the coefficients of the SDE and some conditions on the steps and on the weights that

¯

ν⁽ⁿ⁾(ω, F) = 1 η₁+. . .+η_n

Xn

k=1

η_kF( ¯XΓk+.)−−−−−→^n→+∞ P_ν(F) = Z

E[F(X^x)]ν(dx) a.s., (1.1) for a broad class of functionals F including bounded continuous functionals for the Sko- rokhod topology. On the other hand, in the marginal case,i.e.whenF(α) =f(α(0)), then the procedure converges to ν(f). When the Poisson equation related to the infinitesimal generator has a solution, this convergence is ruled by a Central Limit Theorem (CLT):

this has been extensively investigated in the literature (for continuous Markov processes, see [4], for the Euler scheme with decreasing step of Brownian diffusions, see [18, 21]). As concerns L´evy driven SDEs, see [24].

Our aim in this paper is to extend some of these rate results to functionals of the path process and its associated Euler scheme with decreasing step, i.e. to study the rate of

∗Laboratoire de Probabilit´es et Mod`eles Al´eatoires, UMR 7599, UPMC, Case 188, 4 pl. Jussieu, F-75252 Paris Cedex 5, France, E-mail: gilles.pages@upmc.fr

†Institut de Math´ematiques de Toulouse, Universit´e Paul Sabatier&INSA Toulouse, 135, av. de Rangueil, F-31077 Toulouse Cedex 4, France, E-mail: fabien.panloup@math.univ-toulouse.fr

convergence to P_ν(F) of (¹_tR_t

0F(X_s+.)ds)_t≥1 and (¯ν⁽ⁿ⁾(ω, F))_n≥1 respectively. Here, we choose to assume that (X_t) is an R^d-valued process solution to

dX_t=b(X_t)dt+σ(X_t)dW_t, (1.2) where (W_t)_t≥0 is a q-dimensional Brownian motion andb and σ are Lipschitz continuous functions with values in R^d and M_d,q respectively, where M_d,q denotes the set of d×q- matrices. Under these assumptions, strong existence and uniqueness hold and (X_t) is a Markov process whose semi-group is denoted by (Pt). We also assume that (Xt) has a unique invariant distribution ν and we denote by P_ν, the distribution of (X_t) when stationary.

Let us now focus on the discretization of (X_t). We are going to introduce some continuous-time Euler schemes with decreasing step: denoting by (Γ_n)_n≥1 theincreasing sequence of discretization times starting from Γ₀ = 0, we assume that the step sequence defined byγ_n := Γ_n−Γ_n−1,n≥1, isnonincreasing and satisfies

n→lim+∞γ_n= 0, Γ_n= Xn

k=1

γ_k −−−−−→^n→+∞ +∞, (1.3) First, we introduce the discrete time constant Euler scheme ( ¯X_Γn)_n≥0 recursively defined at the discretization times Γn by ¯X0 =x0 and

X¯_Γn+1 = ¯X_Γn+γ_n+1b( ¯X_Γn) +σ( ¯X_Γn+1)(W_Γn+1 −W_Γn). (1.4) There are several ways to extend this definition into a continuous time process. The simplest one is the stepwise constant Euler scheme ( ¯X_t)_t≥0 defined by

∀n∈N, ∀t∈[Γ_n,Γ_n+1), X¯_t= ¯X_Γn.

The stepwise constant Euler scheme is a right continuous left limited process (referred as c`adl`ag throughout the paper, following the French acronym). This scheme is easy to simulate provided one is able to compute the functions bandσ at a reasonable cost. One could also introduce the linearly interpolated process built on ( ¯X_Γn)_n≥0 but except the fact that it is a continuous process, it has no specific virtue in term of simulability or convergence rate.

The second possibility to extend the discrete time Euler scheme is what we will call thegenuine Euler scheme, denoted from now on by (ξ_t)_t≥0. It is defined by interpolating the two part of the discrete time scheme in its own scale (time, Brownian motion). It is defined by

∀n∈N, ∀t∈[Γ_n,Γ_n+1), ξ_t= ¯X_Γn+ (t−Γ_n)b( ¯X_Γn) +σ( ¯X_Γn)(W_t−W_Γn+1) (1.5) Such an approximation looks more accurate than the former one, especially in a func- tional setting, as it has been emphasized – in a constant step framework – in the literature on several problems related to the Monte Carlo estimation of (a.s. continuous) function- als of a diffusion (with a finite horizon) (see e.g. [7], Chapter 5). This follows from the classical fact that theL^p-convergence rate of this scheme for the sup norm is√γ instead of −√γlogγ for its stepwise constant counterpart (where γ stand for the step). On the other hand, the simulation of a functional of (ξ_t)_{t∈[τ,τ+T]} is deeply connected with the simulation of the Brownian bridge so that it is only possible for specific functionals (like running maxima, etc).

A convenient and synthetic form for the genuine Euler scheme is to write it as an Itˆo process satisfying the following pseudo-diffusion equation

ξt=x0+ Z t

0

b(ξs)ds+ Z t

0

σ(ξs)dWs (1.6)

where

t= Γ_N(t) with N(t) = min{n≥0,Γ_n+1> t}. (1.7) Taking advantage of this notation for the stepwise constant Euler scheme, one can also note that

∀t∈R₊, X¯t= ¯Xt.

When necessary, we will adopt the more precise notation ¯X^x,(hn) for a stepwise constant continuous-time Euler scheme to specify starting atx∈R^dat time 0 with a nonincreasing step sequence (h_n)_n≥1 satisfying (1.3).

Since we will deal with possibly c`adl`ag approximations of continuous processes we will introduce the spaces D_uc(I,R^d) ofR^d-valued c`adl`ag functions onI =R₊ or [0, T],T >0, endowed with the topology of the uniform convergence on compact sets, rather than the classical Skorokhod topology (see [6]). In fact, one must keep in mind that if α :I → R^d is a continuous function and (α_n) is a sequence of c`adl`ag functions, α_n ^Sk→ α iff α →^uc α (with obvious notations). Furthermore, usual Skorokhod distance d_Sk (so-called J₁ and J₂ topologies) onD([0, T],R^d) all satisfy

d_Sk(α, β)≤ kα−βkT := sup

t∈[0,T]|α(t)−β(t)|

so that any functional F : D([0, T],R^d) → R which is Lipschitz with respect to such a distance d_Sk will be Lipschitz continuous with respect to k.kT (hence measurable with respect to the Borel σ-field induced by the Skorokhod topology).

At this stage, we need to introduce further notations related to the long run behaviour of processes (or simply functions). Letδ_α(dβ) denotes the Dirac mass atα∈D(R₊,R^d) and α^(u):= (α_u+t)_t≥0 denotes theu-shift of α.

We will see below that our aim is to elucidate the asymptotic P(dω)-a.s. weak behaviour of the empirical measures 1

t Z t

0

δ_Y(s)(ω)(dβ)ds as t goes to infinity, where Y will be the diffusion X itself or one of its (simulatable) Euler time discretizations. This suggests to introduce a time dicretization at times Γ_n of the above time integral like we did to define the Euler scheme. This leads us to introduce, for anyα∈D(R₊,R^d), the following abstract

“Euler” empirical means

¯

ν⁽ⁿ⁾(α, dβ) = 1 Γn

Xn

k=1

γ_kδ_α(Γk−1)(dβ) = 1 Γn

Z Γn

0

δ_α(s)(dβ)ds.

Then, for a functional F defined onD(R₊,R^d) and α∈D(R₊,R^d),

¯

ν⁽ⁿ⁾(α, F) = Z

D(R₊,Rd)

F(β)¯ν⁽ⁿ⁾(α, dβ) = 1 Γ_n

Xn

k=1

γ_kF(α^(Γk−1)) = 1 Γ_n

Z Γn

0

F(α^(s))ds.

In the following, we will use this sequence of empirical measures for both stepwise con- stant and genuine Euler schemes. Compared to [22], this means that we assume that the sequence of weights (η_n) satisfies η_n=γ_n for everyn≥1.

Additional notations. hx, yi=P

ix_iy_i will denote the canonical inner product and

|x|=p

hx, xi will denote Euclidean norm of a vectorx∈R^d.

LetA= [a_ij]∈M_d,q be an R-valued matrix withdrows and q columns. A^∗ will denote the transpose of A, Tr(A) = P

ia_ii its trace and kAk := p

Tr(AA^∗) = (P

ija²_ij)¹². If d=q, one writes Ax^⊗2 forx^∗Ax.

2 Main results

2.1 Assumptions and background

We denote by (Ft)_t≥0 the usual augmentation ofσ(W_s,0≤s≤t) byP-negligible sets. Since b andσ are Lipschitz continuous functions, Equation (1.2) admits a unique (Ft)-adapted solution (X_t^x)_t≥0 starting from x ∈R^d. More generally, for every u ≥ 0 and every finite Fu-measurable random variable Ξ, we can consider (X_t^(u),Ξ)_t≥0, unique strong solution to theSDE:

dY_t=b(Y_t)dt+σ(Y_t)dW_t^(u), Y₀= Ξ, (2.8) where W_t^(u) =W_u+t−W_u,t≥0, is the u-shifted Brownian motion (independent of Fu).

Note that X_t^x =X_t^(0),x and that X_t^(u),Ξ can be also defined through the flow of (1.2) by setting

X_t^(u),Ξ = X_t^(u),x

|x=Ξ.

Throughout this paper, we consider a measurable functional F :D_uc([0, T],R^d)→R. We will denote byF_T the stopped functional defined onD_uc(R₊,R^d) by

∀α∈D_uc(R₊,R^d), F_T(α) =F(α^T) with α^T(t) =α(t∧T), t≥0. (2.9) Let us introduce the assumptions on F.

(C¹

F): F :D_uc([0, T],R^d)→Ris a bounded and Lipschitz continuous functional.

We set

f_F(x) =E[F_T(X^x)] =E[F(X_t^x,0≤t≤T)].

It is classical background (see e.g.[15]) that, under the Lipschitz assumption on band σ, E[sup_t∈[0,T]|X_t^x−X_t^y|]≤C_b,σ,T|x−y|so that f_F is in turn clearly Lipschitz continuous.

Additional regularity properties (like differentiability) can be transfered fromfF provided F, b and σ are themselves differentiable enough (see e.g. [15]). Furthermore, it follows from its very definition and the Markov property that

ν(f_F) =P_ν(F_T) = Z

E[F_T(X^x)]ν(dx).

(C²

F): There exists a boundedC²-functiong_F :R^d→Rwith bounded Lipschitz continuous derivatives such that

∀x∈R^d, f_F(x)−ν(f_F) =Ag_F

where A denotes the infinitesimal generator of the diffusion (1.2) defined for every C²- functionf on R^d by

Af(x) =h∇f, bi(x) +1

2Tr(σ^∗D²f σ(x)).

R_EMARK 2.1. In fact, we need in the sequel that f_F satisfies a CLT for the marginal occupation measures which follows (see [18, 24]) from Assumption (C²

F) combined with a Lyapunov stability assumption (such as (S_a,p) introduced below). Namely, we have for a class of regular functionsf satisfying f =Ag+C

√t 1

t Z t

0

f(X_s^x)ds−ν(f) L

−−−−→

t→+∞ N 0, σ²_f

(2.10)

and as soon as Xn

k=1

γ_k²

√Γ_k

n→+∞

−−−−−→0,

pΓ_n 1 Γ_n

Xn

k=1

γ_kf( ¯X_Γk−1)−ν(f)

!

−−−−−→L

n→+∞ N 0, σ_f²

, (2.11)

where

σ_f² = Z

Rd|σ^∗∇g(x)|²ν(dx) =−2 Z

g(x)Ag(x)ν(dx)

and L denotes the weak convergence of (real valued) random variables. For details on results in these directions, see [4] for the continuous case and [19, 21, 24] for the decreasing step Euler scheme.

Checking when Assumption (C²

F) is fulfilled is equivalent to solve the Poisson equation Au=f onR^d. Whenf has compact support, well-known results about the same equation in a bounded domain lead to Assumption (C²

F) when the diffusion is uniformly elliptic (see e.g. [16], Theorems III.1.1 and III.1.2). Such an assumption on f_F is clearly unrealistic.

In the general case, in [26], [27] and [28], the problem is solved under some ellipticity conditions in some Sobolev spaces and controls of the growth are given foru and its first derivatives. Finally, when the diffusion is an Ornstein-Uhlenbeck process, one can refer to [18] where the problem is solved in C²(R^d).

Let us now introduce the Lyapunov-type stability assumptions onSDE (1.2). LetEQ(R^d) denote the set of Essentially Quadratic functions, that is C²-functions V : R^d → (0,∞) such that

|x|→lim+∞V(x) = +∞, |∇V| ≤C√

V and D²V is bounded.

Note that since V is continuous, V attains its positive minimumv > 0 so that, for any A, r >0, there exists a real constant C_A,r such thatA+V^r≤C_A,rV^r.

Let us come to the mean-reverting assumption itself. First, for any symmetricd×dmatrix S, setλ⁺_S := max(0, λ₁, . . . , λ_d) whereλ₁, . . . , λ_ddenote the eigenvalues ofS. Leta∈(0,1]

and p∈[1,+∞). We introduce the following mean-reverting assumptionwith intensity a:

(S_a,p) : There exists a function V ∈ EQ(R^d) such that:

(i) ∃C_a>0 such that |b|²+ Tr(σσ^∗)≤C_aV^a.

(ii) There existβ ∈Rand ρ >0 such thath∇V, bi+λ_pTr(σσ^∗)≤β−ρV^a, whereλp := 1

2 sup

x∈Rd

λ⁺

D²V(x)+(p−1)^∇V_V^⊗∇V. The functionV is then called a Lyapunov func- tion for the diffusion (X_t)_t≥0.

In Theorem 3 of [19], it is shown that this assumption leads to an a.s. marginal weak convergence result to the set of invariant distributions of the diffusion. When p≥2 and the invariant distribution is unique, this result reads as follows.

P_ROPOSITION 2.1. Let a∈(0,1]and p≥2 such that (S_a,p) holds. Then, sup

n≥1

1 Γ_n

Xn

k=1

γ_kV^p2+a−1( ¯XΓ_k−1)<+∞ a.s. (2.12) Let ν denote the unique invariant distribution of (1.2). Then, a.s.,

1 Γ_n

Xn

k=1

γ_kf( ¯XΓ_k−1)−−−−−→^n→+∞ ν(f)

for every continuous function f satisfying f(x) =o(V ^p2+a−1(x)) as |x| →+∞.

R_EMARK 2.2. In the case V(x) = 1 +|x|², one checks for instance that for a given a ∈ (0,1], Assumption (S_a,p) is fulfilled for every p ≥ 1 if Tr(σσ^∗)(x) = o(1 +|x|^2a) as

|x| →+∞and

b(x) =−ρ(x) x

|x| +T(x) whereC₁|x|^2a−1 ≤ρ(x)≤C₂|x|^2a−1, and T satisfies for every x∈R^d hT(x), xi= 0 and|T(x)| ≤C(1 +|x|^a).

As concerns the uniqueness of the invariant distributionν, we need an additional assump- tion related to the transition P_T. Namely, we assume that:

(S^ν

T): ν is an invariant distribution for (P_t)_t≥0 and the unique one for P_T.

Then, ν is in particular the unique invariant distribution for (P_t)_t≥0. In fact, checking uniqueness of the invariant distribution for P_T at a given time T > 0 is a standard way to establish uniqueness for the whole semi-group (P_t)_t≥0. To this end, one may use the following two typical criterions:

•Irreducibility based on ellipticity: for everyx∈R^d,P_T(x, dy) has a density (p_T(x, y))_y∈Rd

w.r.t. the Lebesgue measure λ_d and λ_d(dy)−a.s.,p_T(x, y)>0 for every x∈R^d.

•Asymptotic confluence: for every bounded Lipschitz continuous functionf, for every compact subset K of R^d,

sup

(x1,x2)∈K|P_kTf(x₁)−P_kTf(x₂)|−−−−→^k→+∞ 0 (seee.g.[3, 21]).

2.2 Main results

We are now in position to state our main results.

T_HEOREM 2.1. Let T >0. Assumebandσ are Lipschitz continuous functions satisfying (S_a,p) with an essentially quadratic Lyapunov function V :R^d→(0,+∞) and parameters a∈(0,1] and p >2. Assume furthermore that V satisfies the growth assumption:

lim inf

|x|→+∞

V^p+a−1(x)

|x| >0. (2.13)

Assume that the uniqueness assumption(S^ν

T)holds. Finally, assume that the step sequence (γ_n)_n≥1 satisfies (1.3) and

X

k≥1

γ^3/2_k

√Γ_k <+∞. (2.14)

Let F :D_uc([0, T],R^d)→R be a functional satisfying (C¹

F) and (C²

F).

(a) Genuine Euler scheme: Then pΓ_n

¯

ν⁽ⁿ⁾(ξ(ω), F_T)−P_ν(F_T) _L

−−−−−→_n

→+∞ N 0, σ_F²

, (2.15)

where σ_F² = 1

T Z

E

E

A^x_2T| F2T

−E

A^x_T| FT

− Z 2T

T

σ^∗∇gF(X_u^x)dWu

2 ν(dx)

(2.16) and A^x_t :=R_t

0 F_T(X_u+.^x )−f_F(X_u^x)

du, t≥0, (is Ft+T-adapted).

(b) Stepwise constant Euler scheme: furthermore, if there exists δ >0 such that X

k≥1

γ

3 2−δ

√k

Γ_k <+∞, (2.17)

then,

pΓ_n

¯

ν⁽ⁿ⁾( ¯X(ω), F_T)−P_ν(F_T) _L

−−−−−→

n→+∞ N 0, σ²_F

. (2.18)

R_EMARK 2.3. By a series of computations, we can obtain other expressions for σ_F². In particular, we check in the Appendix A that σ_F² reads

σ_F² = 2 Z _T

0

(1− v

T)C_F(v)dv−2E_ν F_T(X)

Z _T

0

σ^∗∇g_F(X_u)dW_u +

Z

R^d

|σ^∗∇g_F(x)|²ν(dx), (2.19) where E_ν denotes the expectation under the stationary regime and C_F is the covariance function defined by

C_F(u) =E_ν F_T(X_u+.)−f_F(X_u))(F_T(X)−f_F(X₀))

. (2.20)

This expression is not clearly positive but has the advantage to separate the “marginal part” that is represented by the last term from the “functional part” which corresponds to the first two ones.

For instance, whenF(α) =φ(α(0)),φbeing bounded and such thatφ−ν(φ) =Ahwhere h is a boundedC²-function with bounded derivatives, thenf_F =φand one observes that the first two terms of (2.19) are equal to 0 so thatσ_F² =

Z

R^d

|σ^∗∇g_F(x)|²ν(dx). This means that we retrieve the marginalCLT given by (2.11) (under a slightly more condition on the step sequence which is adapted to the more general functionals we are dealing with, thus, more constraining than that of the original paper; see below for more detailed comments on the steps conditions).

If we now consider F_T definedF_T(α) =φ(α(T)),φsatisfying the same assumptions as before, one can straightforwardly deduce from a simple change of variable that the limiting variance is still R

R^d|σ^∗∇h(x)|²ν(dx). In the appendix (Part B), we show that retrieving this limiting variance using (2.16) is possible but requires some non trivial computations.

In particular, this calculus emphasizes the intricate nature of the structure of the functional variance.

Given the form of ¯ν⁽ⁿ⁾, it seems natural to introduce the (non-simulatable) sequence 1

Γn

Z Γn

0

F_T(ξ^(u))du

which in fact appears naturally as a tool in the proof of the above theorem.

T_HEOREM 2.2. Assume the assumptions of Theorem 2.1(a). Then,

√t 1

t Z _t

0

F_T(ξ^(s))ds−P_ν(F_T) L

−−−−−→_n

→+∞ N 0, σ_F²

. (2.21)

Finally, we also state the Central Limit Theorem for the stochastic process (X_t)_t≥0 itself. This result can be viewed as a (partial) extension to functionals of Bhattacharya’s CLT established in [4] for a class of ergodic Markov processes.

T_HEOREM 2.3. Let T >0. Assumebandσ are Lipschitz continuous functions satisfying (S_a,p) with an essentially quadratic Lyapunov function V and parameters a ∈ (0,1] and p > 2. Assume (S^ν

T) holds. Let F :D_uc([0, T],R^d) → R be a functional satisfying (C¹

F) and (C²

F). Then, for every x∈R^d,

√t 1

t Z t

0

F(X_u^(s),x,0≤u≤T)ds−P_ν(F_T) L

−−−−−→_n

→+∞ N 0, σ_F²

. (2.22) This means that our approach (averaging decreasing step schemes) induces no loss of weak rate of convergence with respect to that of the empirical mean of the process itself towards its steady regime. If we look at the problem from an algorithmic point of view, the situation becomes quite different. First, we will no longer discuss the recursive aspects as well as the possible storing problems induced by the use of decreasing steps: it has already been done in [22] and we showed that they can easily be encompassed in practice, especially for additive functionals or functions of running extrema (seee.g.simulations in Section 7).

Our aim here is to discuss the rate of convergence in terms of complexity. It is clear from its design that the complexity of the algorithm grows linearly with the number of iterations. Thus, if γ_n ∝ n^−ρ, 0 < ρ < 1, then Γ_n ∼ ⁿ₁^1−ρ_−ρ so that the effective rate of convergence as a function of the complexity is essentially proportional ton^1−ρ2 . However, the choice ofρis constrained by conditions (2.14) or (2.17) that are required for the control of the discretization error. These conditions imply that ρmust be taken greater than 1/2 and lead to an “optimal” rate proportional to n^14−ε for every ε >0. This means that we are not able to recover the optimal rate of the marginal case that is proportional ton^−1/3 and obtained for ρ = 1/3 (see [19] for details). Indeed, in this functional framework, the weak discretization error is generally smaller and thus, is negligible compared to the long time error under a more constraining step condition (2.14) instead of P

γ_k²/√

Γ_n < +∞ in the marginal case).

The paper is organized as follows. In Sections 3, 4 and 5, we will focus on the proof of Theorem 2.1(a) and Theorem 2.3 about the rate of convergence of the two considered occupation measures of the genuine Euler scheme. Then, in Section 6, we will summarize the results of the previous sections and will give the main arguments of the proof of Theorems 2.1(a) and 2.3. Finally, Section 7 is devoted to numerical tests in a financial framework: the pricing of a barrier option when the underlying asset price dynamics is a stationary stochastic volatility model.

3 Preliminaries

As for the marginal rate of convergence (see [18]), the first idea is to find a good decom- position of the error (see Lemma 3.1). In particular, we have to exhibit a main martingale component. Here, sinceF depends on the trajectory of the process between 0 andT, the idea is that the “good” filtration for the main martingale component is (FkT)_k≥0. That

is why, in the main part of the proof of these theorems, we will introduce and study the sequence of random probabilities (P^(n,T)(ω, dα))n≥1 defined by:

P^(n,T)(ω, dβ) = 1 nT

Z nT 0

δξ^(u∼)(dβ)du= 1 nT

Xn

k=1

Z kT (k−1)T

δξ^(∼u)(dβ)du whereu_∼ is a deterministic real number lying in [u, u].

To alleviate the notations, we will denote from now on, Gk = FkT and E_k[.] = E[.| Gk], k≥0.

At this stage, the reader can observe on the one hand that for a bounded functional F,P^(n,T)(ω, F_T) is Gn+1 =F(n+1)T-adapted for every n≥0 and on the other hand that P^(n,T)(ω, F_T) is very close to the random measures ¯ν⁽ⁿ⁾(ξ(ω), dβ) of Theorem 2.1(b) by takingu_∼=u∨[u] and exactly equal to its continuous time counterpart in Theorem 2.2 if one sets u_∼=u. (This fact will be made more precise in Section 6).

Hence, the main step of the proof of the above theorems will be to study the rate of convergence of the sequence (P^(n,T)(ω, F_T))_n≥0 to P_ν(F_T) for which the main result is given in Section 6 (see Proposition 6.3). In this way, we state in this section a series of preliminary lemmas. In Lemma 3.1, we decompose the error between this new sequence (P^(n,T)(ω, F_T))_n≥1 and the target P_ν(F_T). In Lemma 3.2, we recall a series of results on the stability of diffusion processes and their genuine Euler scheme in finite horizon.

Finally, in Lemma 3.3, we recall and extend results of [19] about the long-time behavior of the marginal Euler scheme.

For every k∈N, we define theGk-measurable random variableφ_F(k) by φ_F(1) = 0, φ_F(k) =

Z

Ik−1

F_T(ξ^(u∼))du if k≥2. (3.23) whereI_k= [(k−1)T, kT). Please note that φ_F(k) isFkT-measurable.

L_EMMA 3.1. For every F satisfying (C¹

F) and (C²

F), we have P^(n,T)(ω, F_T)−P_ν(F_T) = M_n

nT + Θ_n,1+ Θ_n,2+ Θ_n+1,3 nT

where

(Mn)n≥1 is a(Gn)-martingale decomposed as follows : Mn= X4

i=1

Mn,i with

∆M_k,1=φ_F(k)−E_k−1[φ_F(k)], ∆M_k,2=E_k[φ_F(k+ 1)]−E_k−1[φ_F(k+ 1)],

∆M_k,3= Z

Ik

E_k−1[F_T(X^{(u∼),ξ∼u})]−f_F(ξ_u

∼

)du, ∆M_k,4 =− Z

Ik

h∇g_F(ξ_u

∼

), σ(ξ_u

∼

)dW_ui and (Θn,1), (Θn,2) and (Θn,3) are (Gn)-adapted sequences defined for every n≥1, by:

Θ_n,1= Xn

k=1

Z

Ik

E_k−1

F_T(ξ^(u∼))−F_T(X^{(u∼),ξ∼u})

du,

Θ_n,2= Xn

k=1

Z

Ik

Ag_F(ξ_u

∼

)du−∆M_k,4

, Θ_n,3= (φ_n(F)−E_n(φ_n(F))).

Proof. With our newly defined notations, we have, for every n≥1, P^(n,T)(ω, F_T) = 1

nT Xn

k=1

φ_F(k+ 1).

Now, for every k ≥ 1, going twice backward through martingale increments, one checks that

φ_F(k+ 1) = ∆M_k+1¹ + ∆M_k²+E_k−1(φ_F(k+ 1)).

Then, noting thatE_k−1(φ_F(k+ 1)) = Z

Ik

E_k−1(F_T(ξ^(u∼)))du, we introduce the approxima- tion term ∆Θ_n,1 between the genuine Euler scheme ξ and the true diffusion X so that

φ_F(k+ 1) = ∆M_k+1¹ + ∆M_k²+ ∆Θ_k,1+ Z

Ik

E_k−1(F_T(X^{(u∼),ξ∼u}))du.

At this stage the Markov property applied to the original diffusion process yields E_k(F_T(X^{(u∼,ξ∼u)})) =E_k E_u

∼

F_T(X^{(u∼),ξ∼u})

=E_kf_F(ξ_u

∼

) =f_F(ξ_u

∼

) sinceu_∼≤u≤kT. As a consequence, ∆M_k³ is a trueGk-martingale increment and

φ_F(k+ 1) = ∆M_k+1¹ + ∆M_k²+ ∆Θ_k,1+ ∆M_k³+ Z

Ik

f_F(ξ_u

∼

)du On the other handf_F =Ag_F +P_ν(F), so that

Z

Ik

f_F(ξ_u

∼

)du−P_ν(F) = Z

Ik

Ag_F(ξ_u

∼

)du = ∆Θ_n,2+ ∆M_k⁴. Finally, summing up all these terms yields

P^(n,T)(ω, F_T)−P_ν(F) = 1

nT M_n+1¹ + X4

i=2

M_nⁱ + X2

i=1

Θ_n,i

!

= 1

nT X4

i=1

M_nⁱ + X3

i=1

Θ_n,i

!

since Θ_n+1,3 =M_n+1¹ −M_n¹.

R_EMARK 3.4. The term Θ_n,1 sums up the error resulting from the approximation of X^{(u∼),ξ∼u} by its Euler scheme (with decreasing step) ξ_u

∼+.. The term Θ_n,2 is a residual approximation term as well: indeed, if we replacemutatis mutandisξ_u

∼

byX_u, Itˆo’s formula implies that

g_F(X_(k+1)T)−g_F(X_kT) = Z

Ik

Ag_F(X_u)du+ Z

Ik

h∇g_F(X_u), σ(X_u)dW_ui, so that the resulting term would be, instead of Θ_n,2, ^gF(X(n+1)T_nT^)−gF(XnT) =O(1/n).

L_EMMA 3.2. Let p > 0 and T > 0. Assume that b and σ are Lipschitz continuous functions and that there existsφ∈ EQ(R^d)such that |b|²+kσk² ≤C_b,σφfor a positive real constant C_b,σ. Then,

(i) There exists a real constant C_p,T,b,σ > 0, such that for every u ≥ 0 and every finite Fu-measurable random vector Ξ

E[ sup

t∈[0,T]

φ^p(X_t^(u),Ξ)| Fu]≤C_p,T,b,σφ^p(Ξ) and E[ sup

t∈[0,T]

φ^p(ξ_u+t)| Fu]≤C_p,T,b,σφ^p(ξ_u).

(ii) There exists a real constant C_p,T >0 such that, for every u≥0, E[ sup

t∈[0,T]|ξ_u+t−X_t^(u),ξu|^p| Fu]≤C_p,T(1 +|ξ_u|^p)γ

p 2

N(u)+1. (iii) There exists a real constant C_p>0 such that, for every n≥0,

E[ sup

u∈[Γn,Γn+1)|ξ_u−ξ_Γn|^p|FΓn]≤C_pφ^p2(ξ_Γn)γ

p

n+12 .

(iv) Let p >2. Then, there exists C_p,T,δ>0 such that, for every u≥0, E[ sup

t∈[0,T]|ξu+t−ξu+t|^p| Fu]≤Cp,Tφ^p2(ξu)γ

p 2−1 N(u)+1.

Proof. The proofs follow the lines of their classical counterpart for the constant step Euler scheme of a diffusion (seee.g. [7], Theorem B.1.4 p.276 and the remark that follows). In particular, as concerns (ii), the only thing to be checked is that (ξ_u+t)_t≥0 is the Euler scheme with decreasing stepγ^(u) of X^(u),ξu where the step sequenceγ^(u) is defined by

γ^(u)₁ = Γ_N(u)+1−u, γ_k^(u)=γ_N(u)+k, k ≥2. (3.24)

L_EMMA 3.3. Let p >2 and a∈(0,1]such that (S_a,p) holds and assume thatb andσ are Lipschitz continuous functions.

(i) Let g:R₊→R₊ be a nonincreasing function such that R_∞

0 g(u)du <+∞. Let(δ_k) be a nonincreasing sequence of positive numbers such that P

k≥1δ_k<+∞. Then, a.s., Z +∞

0

E[V^p+a−1(ξ_u)]g(u)du <+∞ and X

k≥1

δ_kE[V^p+a−1(ξ_(k−1)T)]<+∞. (3.25) (ii) We have:

sup

t≥Γ1

1 t

Z t 0

V ^p2+a−1(ξ_s)ds <+∞ a.s. (3.26) and

sup

n≥1

1 n

Xn

k=1

V^p2+a−1(ξ_(k−1)T)<+∞ a.s. (3.27) In particular, the families of empirical measures ¹_tRt

0δ_ξsds

t≥1 and _n¹Pn

k=1δ_ξ(k−1)T

n≥1

are a.s. tight.

(iii)Assume(S^ν

T). Then,a.s., for every continuous functionf such thatf(x) =o(V^p2+a−1(x)) as |x| →+∞,

1 t

Z t 0

f(ξ_s)ds−−−−→^t→+∞ ν(f) and 1 n

Xn

k=1

f(ξ_(k−1)T)−−−−→^t→+∞ ν(f).

Proof. (i) First, note that Z _∞

0

V^p+a−1(ξ_u)g(u)du=X

n≥1

θ_nγ_nV^p+a−1(ξ_Γn−1),

where θn = γ⁻_n¹RΓn

Γn−1g(u)du. Consequently, the first statement is simply a rewriting with continuous time notations of Lemma 4 of [19]. As concerns the second one, using Lemma 3.2(i) withφ=V and the exponent^p+a−1 yields for everyk≥1 and everyu∈I_k:

E[V^p+a−1(ξ_kT)]≤C_p,a,TE[V^p+a−1(ξ_u)].

As a consequence, considering the integrable, nonincreasing, nonnegative function g = P

k≥11_I

k−1δ_k leads to X

k≥2

δ_kE[V^p+a−1(ξ_(k−1)T)]≤Cp,a,T

X

k≥2

Z

Ik−1

E[V^p+a−1(ξu)]g(u)du <+∞ owing to the previous statement.

(ii) Set r = ^p₂ +a−1 > 0 since p > 2 and a > 0. First, for every n ≥ 1 and every t∈[Γn,Γn+1),

1 t

Z t 0

V^r(ξ_s)ds≤ Γ_n+1 Γ_n

1 Γ_n+1

n+1X

k=1

γ_kV^r(ξ_Γ

k−1)≤ 2 Γ_n+1

n+1X

k=1

γ_kV^r(ξ_Γ

k−1), sinceγ_nis nonincreasing. Now, owing to Proposition 2.1,

sup

n≥1

1 Γ_n

Xn

k=1

γ_kV^r(ξ_Γ

k−1)<+∞ a.s.

and (3.26) follows.

Let us deal now with (3.27). Given (3.26), it is clear that (3.27) is equivalent to showing that for an increasing sequence (t_k) such that t0 = 0, sup_k≥1(t_k −t_k−1) < +∞ and t_k→+∞,

sup

n≥1

1 n

Xn

k=1

(t_k−t_k−1)V^r(ξ_kT)− Z tk

tk−1

V^r(ξ_u)du

!

<+∞ a.s. (3.28) Setting t_k = Γ_N(kT)+1 for every k≥1, this suggests to introduce the martingale defined by N0 = 0 and for every n≥1,

N_n= Xn

k=1

1 k

Z tk

t_k−1

V^r(ξ_kT)−V^r(ξ_u)du−E_t

k−1

"Z tk

t_k−1

V^r(ξ_kT)−V^r(ξ_u)du

#!

, whereE_t

k[.] :=E[.|Ftk].Setε= _2r^p so that (1 +ε)r =p+a−1. Using that sup_k≥1(t_k− t_k−1)<+∞ and the elementary inequality|u+v|^1+ε≤2^ε(u^1+ε+v^1+ε) foru,v≥0,

X

k≥1

1 k^1+εE

Z tk

t_k−1

V^r(ξ_kT)−V^r(ξ_u)du

1+ε

≤CX

k≥1

δ_kE[V^r(1+ε)(ξ_kT)] +C Z +∞

0

E[V^r(1+ε)(ξ_u)]g(u)du,

where δ_k = k^−(1+ε) and g is the nonincreasing function defined by g(u) = k^−(1+ε) on [t_k−1, t_k). Thus, we deduce from (3.25) that,

X

k≥1

1 k^1+εE

Z tk

t_k−1

V^r(ξ_kT)−V^r(ξ_u)du

1+ε <+∞.

It follows from the Chow Theorem (seee.g. [10]) that (N_n) a.s. converges toward a finite random variableN_∞ which in turn implies by the Kronecker Lemma that

1 n

Xn

k=1

Z tk

tk−1

V^r(ξ_kT)−V^r(ξ_u)du−E_t

k−1

"Z tk

tk−1

V^r(ξ_kT)−V^r(ξ_u)du

#!

n→+∞

−−−−−→0 a.s.

Then, (3.28) will follow from sup

n≥1

1 n

Xn

k=1

Z tk

t_k−1

E_t

k−1

V^r(ξ_kT)−V^r(ξ_u)

du <+∞ a.s. (3.29) In order to prove (3.29), we need to inspect two cases for r:

Case r ≥1. We decompose the incrementV^r(ξ_kT)−V^r(ξ_u) into elementary increments, namely

V^r(ξ_kT)−V^r(ξ_u) =V^r(ξ_kT)−V^r(ξ_kT) +

N(kT)

X

ℓ=N(u)+1

V^r(ξ_Γ

ℓ)−V^r(ξ_Γ

ℓ−1).

Owing to the second order Taylor formula, we have for every ℓ∈ {N(u) + 1, . . . , N(kT)}: V^r(ξ_Γ

ℓ)−V^r(ξ_Γ

ℓ−1) = γ_lh∇V^r, bi(ξ_Γ

ℓ−1) +h∇V^r(ξ_Γ

ℓ−1), σ(ξ_Γ

ℓ−1)(W_Γℓ−W_Γℓ−1)i +1

2D²V^r(θ_l)(ξ_Γ

ℓ−ξ_Γ

ℓ−1)^⊗2 whereθ_l∈(ξ_Γ

ℓ−1, ξ_Γ

ℓ).

Note that a similar development holds for V^r(ξ_kT)−V^r(ξ_kT). Now, one checks that the fact that V ∈ EQ(R^d) implies that kD²V^rk ≤ C_VV^r−1 and that √

V is a Lipschitz continuous function with Lipschitz constant [√

V]₁. Consequently

|D²V(θ_ℓ)(ξ_Γ

ℓ−ξ_Γ

ℓ−1)^⊗2| ≤ C_V q V(ξ_Γ

ℓ−1) + [√ V]₁|ξ_Γ

ℓ −ξ_Γ

ℓ−1|2(r−1)

|ξ_Γ

ℓ−ξ_Γ

ℓ−1|²

≤ C_r,VV^r−1(ξ_Γ

ℓ−1)|ξ_Γ

ℓ−ξ_Γ

ℓ−1|²+C|ξ_Γ

ℓ−ξ_Γ

ℓ−1|^2r,

where we used in the second inequality the standard control |u+v|^s ≤2^s−1(|u|^s+|v|^s).

Then, summing overℓ and using thath∇V, bi ≤β owing to (S_a,p)(ii), we deduce that V^r(ξ_kT)−V^r(ξ_u) ≤ β(kT −u) +

Z kT

u h∇V^r(ξ_v), σ(ξ_v)dW_vi +C_V

Z kT u

V^r−1(ξ_v)|ξ_v¯∧kT −ξ_v|²+|ξ_v¯∧kT −ξ_v|^2r dv γ_N(v)+1 where ¯v = Γ_N(v)+1. By (S_a,p)(i), we can use Lemma 3.2(iii) with φ=V^a and p =s to obtain for everys >0,

E_v[|ξ_v¯∧kT −ξ_v|^s]≤C_sV^as2 (ξ_v)γ_N(v)+1^s/2 . (3.30)