RATE OF ESTIMATION FOR THE STATIONARY DISTRIBUTION OF STOCHASTIC DAMPING HAMILTONIAN SYSTEMS WITH CONTINUOUS OBSERVATIONS

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RATE OF ESTIMATION FOR THE STATIONARY DISTRIBUTION OF STOCHASTIC DAMPING HAMILTONIAN SYSTEMS WITH CONTINUOUS

OBSERVATIONS

Sylvain Delattre, Arnaud Gloter, Nakahiro Yoshida

To cite this version:

Sylvain Delattre, Arnaud Gloter, Nakahiro Yoshida. RATE OF ESTIMATION FOR THE STATION-

ARY DISTRIBUTION OF STOCHASTIC DAMPING HAMILTONIAN SYSTEMS WITH CONTIN-

UOUS OBSERVATIONS. 2020. �hal-02455744�

DISTRIBUTION OF STOCHASTIC DAMPING HAMILTONIAN SYSTEMS WITH CONTINUOUS OBSERVATIONS

SYLVAIN DELATTRE, ARNAUD GLOTER, AND NAKAHIRO YOSHIDA

Abstract. We study the problem of the non-parametric estimation for the densityπof the stationary distribution of a stochastic two-dimensional damp- ing Hamiltonian system (Zt)t∈[0,T] = (Xt, Yt)t∈[0,T]. From the continuous observation of the sampling path on [0, T], we study the rate of estimation forπ(x0, y0) as T → ∞. We show that kernel based estimators can achieve the rateT^−vfor some explicit exponentv∈(0,1/2). One finding is that the rate of estimation depends on the smoothness ofπand is completely different with the rate appearing in the standard i.i.d. setting or in the case of two- dimensional non degenerate diffusion processes. Especially, this rate depends also ony0. Moreover, we obtain a minimax lower bound on the L²-risk for pointwise estimation, with the same rateT^−v, up to log(T) terms.

1. Introduction

The class of hypo-elliptic diffusion processes, for which the diffusion coefficient is degenerate, has been the subject of many recent works and is used for modeling in many fields, such as mathematical finance, biology, neuro-science, mechanics, ecology,... (see e.g. [8], [11], [7] and references therein). In this paper, we focus on the situation of a bi-dimensional hypo-elliptic process, describing the evolution in time of the couple position/velocity of some quantity. The velocityY is modeled by a non degenerate one-dimensional diffusion process, while the positionX is its integral, and the resulting bi-dimensional processZ = (X, Y) is hypo-elliptic. Our aim is to estimate the densityπof the stationary measure of this diffusion process, under the assumption of an ergodic setting.

The problem of non-parametric estimation of the stationary measure of a con- tinuous mixing process is a long-standing problem (see for instance N’Guyen [13], or Comte and Merlevede [3] and references therein). Based on sample of length T of the data Comte and Merlevede [3] find estimators of the stationary measure, converging at rate depending on the smoothness of the stationary measure and slower than√

T, as it is usual in non-parametric problems.

In the specific context where the continuous time process is a one-dimensional diffusion process, observed continuously on some interval [0, T], the finding is dif- ferent. It is shown that the rate of estimation of the stationary measure is√

T (see Kutoyants [10]). The rate of estimation is thus independent of the smoothness of the object that one estimates, in contrast to the typical non-parametric situation.

Date: 26, january, 2020.

2010Mathematics Subject Classification. Primary 62G07, 62G20; secondary 60J60.

Key words and phrases. hypo-elliptic diffusion, non-parametric estimation, stationary mea- sure, minimax rate.

1

The optimal estimator is very specific to the diffusive nature of the process as it relies on the local time of the process. Remark that if the process is a diffusion ob- served discretely on [0, T] with a sufficiently high frequency it is possible to estimate with rate√

T also (see [12], [4]).

The case of multi-dimensional non-degenerate diffusions is treated in Dalalyan and Reiß [6] and Srauss [14]. In that case, the local time process is not available, but it is shown that for non degenerate diffusion process of dimensiond= 2, there exists an estimator of the pointwise values of the stationary measure with rate

√

T /log(T)². This rate of estimation does not depend on the smoothness of the stationary measure for d= 2. In the situation of a non degenerate diffusion with dimensiond≥3, they find estimators whose rate is polynomial inT, depends on both the smoothness of the stationary measure and the dimensiond≥3. Ford≥3, the rate is strictly slower than√

T, but faster than the rate appearing in standard multivariate density estimation from T i.i.d. observations. Hence, for d≥ 3, the diffusive structure of the process, enables to get a faster estimation rate than for the i.i.d. case, as well.

In the case of hypo-elliptic processes, fewer results exist for the estimation of the stationary distribution. In [5], the authors consider the case of two-dimensional process Z = (X, Y), whereY is a velocity and X a position. Based on a discrete sampling of the path of size n, they propose an estimator of the stationary mea- sure which converges at a non parametric rate depending on the smoothness of the stationary measure. Assuming that the stationary measure has anisotropic regu- larity (k1, k2), the proposed estimator has a rate depending on the harmonic mean 2/(1/k1+ 1/k2) as it is for the optimal rate of estimation of distribution on in the case of i.i.d. sequence.

In this paper, we focus on the situation where the process is observed continu- ously on [0, T] and our goal is to determine what is the optimal rate of estimation ofπin this context. Assuming that the stationary density (x, y)7→π(x, y) has an anisotropic H¨older regularity with indexk1 with respect to the variablex, andk2

with respect to y, we construct an estimator of π(x₀, y₀) based on (X_t, Y_t)_t∈[0,T]. This estimator achieves some rateT^−v(k1,k2)and this rate of convergence depends on the smoothness of π in a very specific way. Indeed, the expression of the rate of estimation involves only k₁ or k₂, depending on the relative positions of these two smoothness indexes. This shows the specificity of the estimation problem for continuous observation of process Z = (X, Y) with a non degenerate diffusive ve- locity Y, and a degenerate componentX. It is noteworthy that we find a rate of estimation slower than in the case of continuous observation of a non-degenerate diffusion in dimension 2. Another interesting finding is that the rate of estimation ofπ(x0, y0) depends on the point where one estimates the stationary measure, and is slower for points corresponding to null velocityy0= 0. A crucial ingredient in the study of the rate of estimation is to derive the variance of a kernel estimator with a choice of bandwidth (h1, h2). We show that the variance of the kernel estimator depends in a completely unsymmetrical way onh1 and h2. As a consequence, we get that the optimal bandwidth choice for the estimator is such that one of the two bandwidth can go almost arbitrarily fast to 0, while the optimal choice for the other bandwidth depends sharply on the smoothness of the stationary measure.

Also, we show a lower bound for the minimax risk of estimation ofπ(x₀, y₀) on a class of hypo-elliptic diffusion models with stationary measure of H¨older regularity

(k1, k2). This proves that it is impossible to estimate uniformly on this class with a rate faster thanT^−v(k1,k2) (up to log(T) term).

The outline of the paper is the following. In Section 2, we present the model and give assumptions that are sufficient to get an ergodic system with stationary measure admitting a density π. In Section 3, we present the construction of the estimator and states the results on their rate of convergence (in Theorem 1 for y06= 0, and Theorem2fory0= 0). In Section4, we prove the upper bound on the variance of the kernel estimator. We also illustrate the very specific behaviour of the variance of the kernel estimator for hypo-elliptic diffusion by numerical simulations.

In Section5, we state and prove minimax lower bounds for the risk of estimation.

In the Appendix, we prove some technical results used in the proofs of Section4.

2. Hamiltonian system and mixing property

Let us consider (Ω,A,P), some probability space on which a standard one dimen- sional Brownian motion (Bt)_t≥0is defined. We assume that the process (Zt)_t≥0= (Xt, Yt)_t≥0is solution of the stochastic differential equation

dX_t=Y_tdt (2.1)

dY_t=a(X_t, Y_t)dB_t−[β(X_t, Y_t)Y_t+V⁰(X_t)]dt, (2.2)

where (X0, Y0) is a random variable independent of (Bt)t.

We introduce the following regularity assumption on the coefficients.

AssumptionHReg:

• The functionsa:R² →Ris a C^∞ function anda > a(x, y)≥a >0, and for some constantsa > a >0.

• The function β : R² → R is continuously differentiable, and such that

|β(x, y)| ≤β for all (x, y)∈ R², and for some constantβ > 0. Moreover, we have β(x, y) > β, ∀x ∈ [l,∞), y ∈ R, where l ≥ 0, β > 0 are two constants.

• The functionV :R→Ris lower bounded, and withC²regularity.

It is shown in [17], that under the assumptions HReg the S.D.E. (2.1)–(2.2) admits a weak solution, which satisfies the Markov property, and the associated semi group (Pt)_t≥0is strongly Feller [17], which in turn implies that the process is strongly Markovian. Let us stress that the sign condition onβ for largextogether with the existence of lower bound on V are crucial to insure that the solution of (2.1)–(2.2) does not explode in finite time. Of course, if we know that (x, y) 7→

a(x, y), (x, y)7→yβ(x, y) andx7→V⁰(x) are globally Lipschitz, the solutions of the S.D.E. exists in the strong sense.

We now introduce an assumption on the potentialV of the system that ensure that the process tends to some equilibrium.

AssumptionHErg: one has lim

|x|→∞V⁰(x)sign(x) = +∞.

Is is shown in [17] that underHRegandHErg, one can construct a Lyapounov function Ψ ≥ 1, and that a stationary probability π exists and is unique for the processZ = (X, Y), and satisfiesπ(Ψ)<∞. It is shown in [17] (see Theorem 2.4) that for someD >0 andρ >0,

(2.3) ∀t≥0, ∀z∈R², sup

|f|≤Ψ

Pt(f)(z)− Z

R²

f(z⁰)π(dz⁰)

≤DΨ(z)e^−ρt

for any function measurable function f such as f /Ψ is bounded on R and where (Pt)t≥0 is the semi group ofZ,

Pt(f)(z) =E[f(Xt, Yt)|(X0, Y0) =z].

Remark that under HReg and HErg, it is possible to construct a Lyapounov function such that Ψ(x, y)≥ _C¹ exp

1

C[|y|²+V(x)]

, for some constantC >0 (see (3.10) in [17]). As a consequence, (2.3) applies to functions f with exponential growth, and the invariant distribution admits finite exponential moments.

Hence, we can state the following proposition

Proposition 1. Assume that the coefficients of the equation (2.2) satisfy HReg andHErg, then there exists a stationary solutionZ = (X, Y)to the S.D.E.(2.1)–

(2.2), and the stationary distribution is unique and admits some densityπ. More- over, there exist constantsDerg>0andρ >0such that for any bounded measurable functionsf,g, we have

(2.4) ∀t≥0, |cov(f(Z0), g(Zt))| ≤Dergkfk_∞kgk_∞e^−ρt.

The Proposition (1) is a consequence of the results shown in [17]. Indeed, from the fact that the Lyapounov function Ψ is greater than 1 and integrable with respect to the stationary measure, one can check that (2.4) is a consequence of (2.3).

3. Estimator and upper bounds

In this section we introduce the expression for our estimator of the stationary measure π of the S.D.E. (2.1)–(2.2) and prove that the estimator achieves some rate of convergence, depending on the smoothness ofπ.

Letϕ:R→Ra bounded, compactly supported function. For convenience, we suppose that the support ofϕis [−1,1]. We assume that

(3.1) Z

R

ϕ(u)du= 1, Z

R

ϕ(u)u^ldu= 0, forl∈ {1, . . . , L} whereL≥1.

We let h₁(T) > 0, h₂(T) > 0 be two bandwidths which converge to zero as T → ∞, and we consider a kernel estimator ofπ at the point (x₀, y₀)∈R² as (3.2) πˆ_T(x₀, y₀) = 1

T Z T

0

ϕ_h1(T),h2(T)(X_s−x₀, Y_s−y₀)ds, where

(3.3) ϕ_h1,h2(x−x₀, y−y₀) = 1 h1h2

ϕ(x−x₀ h1

)ϕ(y−y₀ h2

).

We assume that the two bandwidths satisfy,

∃K >0, h1(T)⁻¹+h2(T)⁻¹≤K(1 +T^K), ∀T >0, (3.4)

ph1(T) +h2(T)≤K(log(T))^−3/2∧1, ∀T >1.

(3.5)

The two previous conditions insure that the bandwidths go faster to zero than the logarithmic rate by (3.5), but not faster than any polynomial rates by (3.4).

Actually these two bandwidths will be specified later (see equations (3.12), (3.13), (3.14), (3.15) in the proofs of Theorems1and2).

We introduce the class of H¨older functions.

Definition 1. For(k1, k2)∈(0,∞)², and R >0, we denote H^k1,k2(R) the set of functions f : R² → R such that x7→ f(x, y) and y 7→ f(x, y) are respectively of classC^bk1c andC^bk2c, and satisfy the control,∀x, y inRandh∈[−1,1],

|f(x, y)| ≤R,

∂^bk1cf

∂x^bk1c(x+h, y)−∂^bk1cf

∂x^bk1c(x, y)

≤R|h|^k1−bk1c,

∂^bk2cf

∂x^bk2c(x, y+h)−∂^bk2cf

∂x^bk2c(x, y)

≤R|h|^k2−bk2c.

We can state the main results on the asymptotic behaviour of the estimator.

This behaviour is different according to the fact that we estimate the value of the stationary measure on a point (x0, y0) corresponding to a null velocity or not.

Theorem 1. Assume that Z = (X, Y) is a stationary solution to (2.1)–(2.2)and that AssumptionsHReg,HErghold true. We assume that the stationary distribu- tionπbelongs to the set H^k1,k2(R)fork₁>0, k₂>0, R >0, withmax(k₁, k₂)≤L (recall (3.1)).

Assume that y₀6= 0. Then, there exist bandwidths(h₁(T))_T,(h₂(T))_T, depend- ing only onk₁ andk₂, such that the estimator satisfies :

if k1< k2/2, E

(ˆπT(x0, y0)−π(x0, y0))²

≤CT⁻

2k2 2k2 +1, (3.6)

if k₁≥k₂/2, E

(ˆπ_T(x₀, y₀)−π(x₀, y₀))²

≤CT⁻

2k1 2k1 +1/2, (3.7)

for some constant C independent ofT.

Theorem 2. Assume that Z = (X, Y) is a stationary solution to (2.1)–(2.2)and that AssumptionsHReg,HErghold true. We assume that the stationary distribu- tionπbelongs to the set H^k1,k2(R)fork1>0, k2>0, R >0, withmax(k1, k2)≤L (recall (3.1)).

Assume that y0= 0. Then, there exist bandwidths(h1(T))T,(h2(T))T, depend- ing only onk1 andk2, such that the estimator satisfies :

ifk1< k2/3, E

(ˆπT(x0, y0)−π(x0, y0))²

≤C( T logT)⁻

2k2 2k2 +2, (3.8)

ifk1≥k2/3, E

(ˆπT(x0, y0)−π(x0, y0))²

≤CT⁻

2k1 2k1 +2/3, (3.9)

for some constant C independent ofT.

Remark 1. The rates of estimation obtained in Theorems 1–2are completely dif- ferent with the usual one in several ways. First, they do not depend on the harmonic mean of the smoothness indexk₁,k₂, as it is usual in non-parametric setting. Sec- ond, the rate depends on the point(x₀, y₀)where the density is estimated. We state in Section5a minimax lower bound for the L² risk of estimation ofπ(x0, y0)with the same rates (up to logterms).

The asymptotic behaviour of the estimator relies on the standard bias variance decomposition. Hence, we need sharp evaluations for the variance of the estimator, that are stated below, and will be proved in Section4.

Proposition 2. Assume thatZ = (X, Y)is solution to (2.1)–(2.2), that Assump- tionsHReg,HErg, hold true and kπk_∞≤R for someR >0.

Assume thaty06= 0. Then, there exists some constantC, such that for allT >0, Var(ˆπT(x0, y0))≤C 1

T h₂(T)∧ 1 Tp

h₁(T)

!

+Cε(T, h1(T), h2(T)), where

(3.10) ε(T, h1(T), h2(T))≤|log(h1(T)h2(T))|^C

T .

Proposition 3. Assume that Z = (X, Y) is a solution to (2.1)–(2.2), that As- sumptionsHReg,HErg, hold true and thatkπk_∞≤R for someR >0.

Assume thaty0= 0. Then, there exists some constantC, such that for allT >0, Var(ˆπT(x0, y0))≤C

ln(T)

T h2(T)²∧ 1 T h1(T)^2/3

+Cε(T, h1(T), h2(T)), where

ε(T, h₁(T), h₂(T))≤|log(h₁(T)h₂(T))|^C

T .

We can now prove the that our estimator achieves the rates given in Theorems 1–2.

Proof[Proof of Theorem 1.] We write the usual bias-variance decomposition, (3.11) E[(ˆπT(x0, y0)−π(x0, y0))²]≤

|E(ˆπ_T(x₀, y₀))−π(x₀, y₀)|²+ Var(ˆπ_T(x₀, y₀)).

Using the stationarity of the process, we can upper bound the bias term as

|E(ˆπ_T(x₀, y₀))−π(x₀, y₀)|²

= Z

R²

ϕ(u)ϕ(v)[π(x0+uh1(T), y0+vh2(T))−π(x0, y0)]dudv

≤C(h1(T)^2k1+h2(T)^2k2), where in the last line we usedπ∈ H^k1,k2(R) with (3.1) (see e.g. [16], or Proposition 1 in [2] for details). We now use the results of Proposition2on the variance of the estimator and choose the optimal bandwidthsh1(T),h2(T).

•Case 1: k1< k2/2.

Using Proposition2

E[(ˆπT(x0, y0)−π(x0, y0))²]≤C(h1(T)^2k1+h2(T)^2k2)

+ C

T h2(T)+Cε(T, h₁(T), h₂(T)).

We now choose to balanceh₂(T)^2k2with the main contribution of the variance term and let the contribution ofh₁(T) on the bias be smaller. It yields us to set (3.12) h2(T) =T^−1/(2k2+1), h1(T) =T^−C1, whereC1≥ k2

k₁(2k₂+ 1). With these choices and recalling (3.10), we get (3.6).

•Case 2: k₁≥k₂/2.

We use again Proposition2:

E[(ˆπT(x0, y0)−π(x0, y0))²]≤C(h1(T)^2k1+h2(T)^2k2)

+ C

Tp

h1(T)+Cε(T, h₁(T), h₂(T)).

Balancing the variance and bias terms yields to

(3.13) h₁(T) =T^{−1/(2k1+1/2)}, h₂(T) =T^−C2, where C₂≥ k₁ k2(2k1+ 1/2),

and (3.7) follows.

Proof [Proof of Theorem2] We use again the bias/variance decomposition (3.11) and exploit now the results of Proposition3.

•Case 1: k₁< k₂/3.

We have that,

E[(ˆπ_T(x₀, y₀)−π(x₀, y₀))²]≤C(h₁(T)^2k1+h₂(T)^2k2) + Cln(T)

T h2(T)² +Cε(T, h1(T), h2(T)).

We set

(3.14) h2(T) = ( T

ln(T))^−1/(2k2+2), h1(T) =T^−C1, where C1≥ k2

k₁(2k₂+ 2), and (3.8) follows.

•Case 2: k₁≥k₂/3.

We have that,

E[(ˆπT(x0, y0)−π(x0, y0))²]≤C(h1(T)^2k1+h2(T)^2k2)

+ C

T h₁(T)^2/3 +Cε(T, h1(T), h2(T)).

The choice

(3.15) h1(T) =T^{−1/(2k1+2/3)}, h2(T) =T^−C2, where C2≥ k₁ k2(2k1+ 2/3)

gives (3.9).

Remark 2. • The optimal choices for the bandwidths are given in (3.12), (3.13),(3.14)and (3.15). In all the situations, we see that one of the two bandwidths h1(T) or h2(T) can be chosen “arbitrarily small” (as the con- stants C1 andC2 in (3.12),(3.13),(3.14), (3.15)can be arbitrarily large).

In means that the bias induced by the variation of π along one of the two variablesxory can be arbitrarily reduced by the choice of a very thin band- width. It explains why the expression of the rate of estimation depends only on one index of smoothnessk₁ ork₂.

• The fact that one of the two bandwidths can be chosen arbitrarily small is reminiscent to the situation of the estimation of the stationary measureπ(z) for a one dimensional diffusion process(Z_t)_t∈[0,T]observed continuously. In that case, the efficient estimator is based on the local time of the process (see [10]) and the rate is √

T independently of the smoothness of π. The

use of local time is a way to give a rigorous analysis of the quantityπ˜T(z) =

1 T

RT

0 δ_{z}(Z_s)ds, whereδ_{z} is the Dirac mass located atz. We see thatπ˜_T is essentially a kernel estimator with bandwidth h= 0, for which the bias is reduced to0.

4. Variance of the kernel estimator

In this section we prove the crucial upper bounds given in Propositions2–3. We first need to state two lemmas related to the behaviour of density and semi group of the process, and whose proofs are postponed to Section6.

Lemma 1 (Corollary 2.12 in [1]). Assume HReg. Then, the process admits a transition density (p_t)_t>0 which satisfies the following upper bound. For all K compact subset of R²,∀(x, y, x⁰, y⁰)∈K×K,∀t∈(0,1)

(4.1) pt((x, y); (x⁰, y⁰))≤p^G_t((x, y); (x⁰, y⁰)) +p^U_t((x, y); (x⁰, y⁰)) where

(4.2) p^G_t((x, y); (x⁰, y⁰)) = CG

t² exp − 1 C_G

"

(y−y⁰)²

t +(x⁰−x−^y+y₂ ⁰t)² t³

#!

,

for some CG >0 and p^U is a measurable non negative function such that for any compact K⊂R² and(x, y)∈K we have for all t∈(0,1),

(4.3)

Z

R²

p^U_t((x, y); (x⁰, y⁰))dx⁰dy⁰≤CUexp(−t⁻¹C_U⁻¹),

forC_U >0. The two constantsC_GandC_U are independent oft∈[0,1], but depend on the compact setK.

The Lemma 1 gives us a control on the short time behaviour of the transition density. For the sequel, we need a control valid for any time. This is the purpose of the following lemma about the semi group of the process.

Lemma 2. Assume HReg, and let K be a compact subset of R². Then, there exists a constant CeK such that for all 0 < D < 1, t ≥D, ∀z ∈ R², and any f measurable bounded function with support on K,

(4.4) |Pt(f)(z)| ≤Ce_K

"

kfk_L1(R²)

D² +kfk_∞e^−1/(CeKD)

# .

4.1. Proof or Proposition2. Throughout the proof we suppress in the notation the dependence uponT ofh₁(T) andh₂(T). The constantCmay change from line to line and is independent ofT. In the proof, we will use repeatedly Lemmas1–2.

To this end, we consider a compact setKthat contains a ball of radius√

2 centered at (x₀, y₀), and as a result the support of (x, y)7→ϕ(x−x₀, y−y₀) is included in this compact.

To prove the proposition, it is sufficient to prove that the following inequalities holds both, forT large enough,

Var(ˆπ_T(x₀, y₀))≤C 1 T h2

+Cε(T, h₁(T), h₂(T)), (4.5)

Var(ˆπT(x0, y0))≤C 1 T√

h1

+Cε(T, h1(T), h2(T)).

(4.6)

First step: we prove (4.5).

From (3.2), and the stationarity of the process we get that (4.7) Var (ˆπT(x0, y0)) = 1

T² Z T

0

Z T 0

κ(t−s)dtds, where

κ(u) = Cov (ϕh₁,h₂(X0−x0, Y0−y0), ϕh₁,h₂(Xu−x0, Yu−y0)). We deduce that

(4.8) Var (ˆπ_T(x₀, y₀))≤ 1 T

Z T 0

|κ(s)|ds.

We will find an upper bound for the integral on the right-hand side of the latter expression by splitting the time interval [0, T] into 4 pieces [0, T] = [0, δ)∪[δ, D₁)∪ [D1, D2)∪[D2, T], whereδ,D1,D2, will be chosen latter.

• For s∈[0, δ), we write from (4.7) and using Cauchy-Schwarz inequality and the stationarity of the process,

|κ(s)| ≤Var(ϕh₁,h₂(X0−x0, Y0−y0))^1/2Var(ϕh₁,h₂(Xs−x0, Ys−y0))^1/2

= Var(ϕh₁,h₂(X0−x0, Y0−y0)).

This variance is smaller than Z

R²

ϕh₁,h₂(x−x0, y−y0)²π(x, y)dxdy and using thatπis bounded and (3.3), we deduce

(4.9) |κs| ≤ C

h1h2

.

In turn, we have (4.10)

Z δ 0

|κ(s)|ds≤C δ h1h2

.

•Fors∈[δ, D1), whereδ < D1≤1. We write

|κ(s)| ≤E[|ϕh₁,h₂(X0−x0, Y0−y0)||ϕh₁,h₂(Xu−x0, Yu−y0)|] +

E[|ϕ_h1,h2(X₀−x₀, Y₀−y₀)|]E[|ϕ_h1,h2(X_s−x₀, Y_s−y₀)|]

Using that (Xt, Yt)tis stationary with marginal law having a bounded density we deduce that E[|ϕh1,h2(Xs−x0, Ys−y0)|] ≤ CR

R²|ϕh1,h2(x−x0, y−y0)|dxdy ≤ C from (3.3). This gives,

(4.11) |κ(s)| ≤ Z

R²

|ϕh₁,h₂(x−x0, y−y0)|

Z

R²

|ϕh₁,h₂(x⁰−x0, y⁰−y0)|ps(x, y;x⁰, y⁰)dx⁰dy⁰π(x, y)dxdy+C.

Using now equation (4.1) in Lemma1, we get that (4.12) |κ(s)| ≤κ¹(s) +κ²(s) +C

with

(4.13) κ¹(s) :=

Z

R²

|ϕ_h1,h2(x−x₀, y−y₀)|

Z

R²

|ϕ_h1,h2(x⁰−x₀, y⁰−y₀)|p^G_s(x, y;x⁰, y⁰)dx⁰dy⁰π(x, y)dxdy,

κ²(s) :=

Z

R²

|ϕh₁,h₂(x−x0, y−y0)|

Z

R²

|ϕh₁,h₂(x⁰−x0, y⁰−y0)|p^U_s(x, y;x, y)dx⁰dy⁰π(x, y)dxdy.

In order to upper boundκ¹(s) we show that the Gaussian kernel (4.2) appearing in the expression ofκ¹(s) takes small values fors∈[δ, D1) as soon asδis well cho- sen. Recall thaty₀6= 0, and for simplicity assume thaty₀>0. Then, using thatϕis

compactly supported on[−1,1] we know that|ϕh1,h2(x−x₀, y−y₀)ϕ_h1,h2(x⁰−x₀, y⁰−y₀)| 6=

0 implies that

(4.14) |x−x0| ≤h1,|x⁰−x0| ≤h1,|y−y0| ≤h2,|y⁰−y0| ≤h2.

Let us denoteK(h1, h2) the rectangle ofR⁴defined by the conditions (4.14). Then, (4.15) κ¹(s)≤C

Z

K(h1,h2)

|ϕ_h1,h2(x−x₀, y−y₀)|

|ϕh₁,h₂(x⁰−x0, y⁰−y0)|p^G_s(x, y;x⁰, y⁰)dxdydx⁰dy⁰, where we used that π is bounded. On K(h1, h2), we have ^y+y₂ ⁰ ≥ ^y₂⁰ > 0 if h2 is small enough, and |x−x⁰| ≤ 2h1. Hence, if we assume that s ≥ ^6h_y¹

0 we

have |x⁰−x| ≤ ^sy₃⁰. It entails, x⁰−x−^y+y₂ ⁰s ≤ ^sy₃⁰ −^y₂⁰s =−^sy₆⁰, and in turn

(x⁰−x−^y+y₂⁰s)²

s³ ≥^y₃₆⁰²¹_s. Plugging in (4.2) this yields top^G_s(x, y;x⁰;y⁰)≤ _s^C2exp(−_Cs¹ ) for some constantCindependent ofs. Using (4.15), we deduce

κ¹(s)≤ C

s²exp(− 1 Cs)

Z

K(h1,h2)

|ϕ_h1,h2(x−x₀, y−y₀)|

|ϕh1,h2(x⁰−x₀, y⁰−y₀)|dxdydx⁰dy⁰ and hence

(4.16) κ¹(s)≤ C

s²exp(− 1 Cs).

To controlκ²(s), we use (4.3) andkϕh₁,h₂(· −x0,· −y0)k_∞≤ _h^C

1h₂ to get that for allx, yin the compactKcontaining a ball of radius√

2 centered at (x0, y0), we have the upper boundR

R²|ϕh₁,h₂(x⁰−x0, y⁰−y0)|p^U_s(x, y;x⁰, y⁰)dx⁰dy⁰≤ _h^C

1h₂e^−Cs1 . As a consequence,

κ²(s)≤C Z

R²

|ϕh₁,h₂(x−x0, y−y0)| 1

h₁h₂e^−Cs1 π(x, y)dxdy

≤ C h₁h₂e^−Cs1 , (4.17)

where we used again thatπis bounded and that the support ofϕh₁,h₂(· −x0,· −y0) is included inK. From (4.12), (4.16)–(4.17), we deduce that for ^6h_y¹

0 ≤δ≤D1≤1, Z D₁

δ

|κ(s)|ds≤ Z D₁

δ

[C

s²exp(− 1

Cs) + C

h₁h₂exp(− 1

Cs) +C]ds

≤ Z D1

δ

[C

s²exp(− 1

Cs) + C

h1h2s²exp(− 1

Cs) +C]ds

≤Cexp(− 1 CD1

)[1 + 1 h1h2

] +CD1, (4.18)

where in the second line we usedD1≤1.

•Fors∈[D1, D2) withD1≤1≤D2< T, we start from the control (4.11) that we write

|κ(s)| ≤ Z

R²

|ϕ_h1,h2(x−x₀, y−y₀)|P_s(|ϕ_h1,h2(· −x₀,· −y₀)|)(x, y)π(x, y)dxdy+C.

Sinceϕ_h1,h2(·−x₀,·−y₀) vanishes outside the compact neighbourhoodKof (x₀, y₀) we can use Lemma2to upper bound the semi group term. Hence, we get for some constantC >0,

|κ(s)| ≤C Z

R²

|ϕh₁,h₂(x−x0, y−y0)|×

[kϕh₁,h₂k_L1(R²)

D²₁ + 1 h1h2

e^−1/(CD1)]π(x, y)dxdy+C, and we deduce|κ(s)| ≤C[_D¹2

1

+_h¹

1h₂e^−1/(CD1)+ 1]. This yields, (4.19)

Z D₂ D1

|κ(s)|ds≤C[D2

D₁² + D2

h1h2

e^−1/(CD1)+D2], for some constantC >0.

• For s ∈ [D₂, T], we use the covariance control (2.4), that allows us to write

|κ(s)| ≤Ckϕh₁,h₂(· −x0,· −y0)k²_∞e^−ρs≤C

1 h₁h₂

2

e^−ρs, forC >0 andρ >0. It entails the upper bound,

(4.20)

Z T D₂

|κ(s)|ds≤C e^−ρD2 (h1h2)².

Collecting together (4.8), (4.10), (4.18), (4.19), (4.20) we deduce, Var(ˆπT(x0, y0))≤ C

T δ

h1h2

+ exp(− 1 CD1

)[1 + D2

h1h2

]+

D1+D2+D2

D²₁ + e^−ρD2 (h₁h₂)²

,

forC >0 some constant. We choose δ= ^6h_y¹

0 ,D1= _C|log¹_h

1h₂|, D2= |^ln((h1h2)²)|

ρ .

By (3.4)–(3.5) we see that this choice is such that, ^6h_y¹

0 =δ < D1<1<D2< T for T large enough. And it yields,

Var(ˆπ_T(x₀, y₀))≤C T

6 y₀

1

h₂ +h₁h₂+D₁+D₂+D2

D²₁ + 1 .

Sinceh1→0,h2 →0, and as the value ofC may change from line to line, we can write that

Var(ˆπT(x0, y0))≤ C T

1 h2

+|ln(h1h2)|^C

and we have shown (4.5).

Second step: we prove (4.6).

We use the same decomposition ofRT

0 |κ(s)|dsin four terms as for the proof of (4.5), but we treat in a different way the contribution of the short time correlations Rδ

0 |κ(s)|ds.

• Let us find an upper bound forκ(s) fors∈(0, δ) withδ <1. We recall that, from Lemma1, the decomposition (4.12) holds true whereκ¹(s) is given by (4.13) andκ²(s) is upper bounded by (4.17). We now studyκ¹(s). To this end, we remark thatp^G_s(x, y;x⁰, y⁰)≤^√C_sqs(x⁰|x, y, y⁰) where

qs(x⁰|x, y, y⁰) = C

s^3/2exp −C⁻¹(x⁰−x−^y+y₂ ⁰s)² s³

! .

Let us stress that

(4.21) sup

s∈(0,1)

sup

(x,y,y⁰)∈R³

Z

R

qs(x⁰|x, y, y⁰)dx⁰≤C <∞.

Thus, using (4.13), we have κ¹(s)≤ C

√s Z

R²

|ϕ_h1,h2(x−x₀, y−y₀)|π(x, y) Z

R²

|ϕh1,h2(x⁰−x₀, y⁰−y₀)|qs(x⁰ |x, y, y⁰)dx⁰dy⁰ dxdy.

By (3.3), we have |ϕh₁,h₂(x⁰−x0, y⁰−y0)| ≤ _h^C

1

1 h₂

ϕ(^y0_h^−y0

2 )

, and thus, using (4.21), we get

Z

R²

|ϕh₁,h₂(x⁰−x0, y⁰−y0)|qs(x⁰|x, y, y⁰)dx⁰dy⁰

≤ C h1

Z

R

1 h2

ϕ(y⁰−y0

h2

) Z

R

qs(x⁰|x, y, y⁰)dx⁰dy⁰

≤ C h1

Z

R

1 h2

ϕ(y⁰−y0

h2

)

dy⁰≤ C h1

.

We deduce that κ¹(s)≤ ^√C_sh

1

R

R²|ϕh₁,h₂(x−x0, y−y0)|π(x, y)dxdy≤ ^√C_sh

1. Col- lecting the latter equation with (4.12) and (4.17), it yields

Z δ 0

|κ(s)|ds≤ Z δ

0

[ C

√sh1

+Ce^−Cs1 h1h2

+C]ds

≤ Z δ

0

[ C

√sh1

+C1 s²

e^−Cs1 h1h2

+C]ds

≤C[

√ δ h1

+e^−Cδ1 h1h2

+δ]

(4.22)

where we have used in the second line thatδ <1.

We now gather (4.8), (4.18), (4.19), (4.20), (4.22), to derive, Var(ˆπT(x0, y0))≤ C

T √

δ h1

+e^−Cδ1 h1h2

+δ+ exp(− 1 CD1

)[1 + D2

h1h2

]+

D1+D2+D1

D₂² + e^−ρD2 (h1h2)²

.

We choose the same thresholds as in the first step, δ = ^6h_y¹

0 , D1 = _C|log¹_h

1h₂|, D2 = |^ln((h1h₂)²)|

ρ . Recalling (3.4)–(3.5), we have e^{−(C6h1/y0)−1} =O(e^−εlog(T)3) = o(h1h2), with someε >0. We derive that

Var(ˆπ_T(x₀, y₀))≤ C T[ 1

√h₁ +|ln(h₁h₂)|^C],

for someC >0. The second step of the proposition is proved.

Remark 3. (1) The Proposition 2 consists actually in the two upper bounds (4.5)–(4.6)for the variance of the estimator. We see that one of these two bounds is smaller than the other, depending on the relative positions of h₂ or √

h₁. It explains why the expression for the rate of convergence of the estimator in Theorem 1 depends on the relative positions of k₁ and k₂/2, which determines which one of the two bounds (4.5)or (4.6)is used in the bias/variance decomposition of the estimation error (see proof of Theorem 1).

(2) The control ofκ(s) = Cov (ϕh₁,h₂(X0−x0, Y0−y0), ϕh₁,h₂(Xs−x0, Ys−y0)) fors∈[δ, D1], withδ≈h1 andD1≤1 depends on the fine structure of the main term (4.2)in the short time expansion of the transition density of the process and on the fact thaty06= 0. In the situationy06= 0, it is impossible to get such a refined result on the covariance, and eventually the bound on the variance of the estimator is larger (see Proposition3).

4.2. Proof of Proposition3. We need to prove that the following two inequalities hold true, forT large enough:

Var(ˆπ_T(x₀, y₀))≤C 1 T h^2/3₁

+Cε(T, h₁(T), h₂(T)), (4.23)

Var(ˆπT(x0, y0))≤Cln(T)

T h²₂ +Cε(T, h1(T), h2(T)).

(4.24)

Again we considerKa compact set ofR²that contains a ball of radius√

2 centered at (x0, y0).

First step : let us prove (4.23).

We recall the control (4.8) for the variance of ˆπ_T(x₀, y₀) and split the integral in (4.8) into four pieces corresponding to the partition [0, T] = [0, δ)∪[δ, D₁)∪ [D₁, D₂)∪[D₂, T], whereδ, D₁, D₂ will be specified latter. Let us stress that in the proof of Proposition 2, only the control of |κ(s)| fors ∈[δ, D₁), uses the fact thaty₀6= 0.

•Fors∈[0, δ) withδ <1, we recall the result obtained in (4.22) which states Z δ

0

|κ(s)|ds≤C[

√ δ h1

+e^−Cδ1 h1h2

+δ].

• For s ∈ [δ, D1] with 0 < δ < D1 < 1, exactly with the same proof as in Proposition 2, we have |κ(s)| ≤ κ¹(s) +κ²(s) +C where κ¹(s) is given by (4.13) andκ²(s) is upper-bounded as in (4.17). We need to find a control onκ¹(s) in the situation y0 6= 0. Using, from (4.2), that p^G_s(x, y;x⁰, y)≤ _s^C2, ∀(x, y, x⁰, y⁰)∈K², and the fact thatπis bounded, we get

κ¹(s)≤ C s²

Z

R⁴

|ϕh₁,h₂(x−x0, y−y0)|

|ϕh₁,h₂(x⁰−x0, y⁰−y0)|dxdydx⁰dy⁰ ≤ C s².

We deduce

Z D₁ δ

|κ(s)|ds≤ Z D₁

δ

C[1

s² +e^−Cs1 h1h2

+C]ds

≤C[1

δ + exp(− 1 CD₁) 1

h₁h₂ +D1]

≤C[1

δ + exp(− 1 CD1

) 1 h1h2

], (4.25)

where we usedD1≤1.

•Fors∈[D1, D2), withD1<1< D2< T, we use the control (4.19).

•Fors∈[D₂, T], withD₂< T, we use the control (4.20).

Collecting the four previous controls, we deduce

Var(ˆπ_T(x₀, y₀))≤ C T

√δ h1

+e^−Cδ1 h1h2

+1

δ +exp(−_CD¹

1) h1h2

(1 +D₂)+

D₁+D₂+D₂

D₁²+ e^−ρD2 (h1h2)²

whereC >0,ρ >0. We chooseδthat balances √

δ/h1 with 1/δ, namelyδ=h^2/3₁ which is smaller than 1 forT large enough, recalling (3.5). Next, we chooseD2=

|^ln((h1h₂)²)|

ρ , andD1= _|log^C_h

1h2|. We deduce Var(ˆπT(x0, y0))≤C

T[ 1 h^2/3₁

+|log(h1h2)|^C],

for someC >0, and where we have used that, from (3.4)–(3.5), exp(−1/(Ch^2/3₁ )) = O(exp(−εlog(T)²)) =o(h1h2), with someε >0. Hence (4.23) is proved.

Second step: we show (4.24).

Comparing with the first part of the proposition, we have to modify our upper bound onRδ

0|κ(s)|dswithδ <1. We split this integral into two parts,Rδ⁰

0 |κ(s)|ds+

Rδ

δ⁰|κ(s)|ds. On the first part, we use the control (4.9) and get (4.26)

Z δ⁰ 0

|κ(s)|ds≤C δ⁰ h1h2

.