Asymptotically efficient estimators of the drift function in ergodic diffusion processes

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Asymptotically efficient estimators of the drift function in ergodic diffusion processes

Jean-Yves Brua

To cite this version:

Jean-Yves Brua. Asymptotically efficient estimators of the drift function in ergodic diffusion processes.

2008. �hal-00437558�

Asymptotically efficient estimators of the drift function in ergodic diffusion

processes

J.-Y. BRUA

IRMA, 7 rue Ren´e-Descartes F67084, Strasbourg Cedex, France

Abstract

This paper deals with the pointwise estimation of the drift function of an ergodic diffusion using the absolute error loss. The optimal convergence rate and the sharp asymptotic lower bound are found for the minimax risk. An asymptotically efficient kernel estimator is constructed.

Key words: Asymptotic efficiency; Drift; Ergodic diffusion; Minimax;

Nonparametric estimation

Email address: brua@math.u-strasbg.fr (J.-Y. BRUA).

1 Introduction

This paper is devoted to the problem of estimating the drift coefficient of an ergodic diffusion, solution of the stochastic differential equation

dX _t = S(X _t )dt + dB _t , 0 ≤ t ≤ T (1) where (B _t ) _t≥0 is a scalar standard Brownian motion. Assuming that we ob- serve the continuous data (X _t , 0 ≤ t ≤ T ) and knowing the smoothness of S, we want to estimate the function S at a fixed point x 0 ∈ R.

Model (1) finds applications in numerous fields, namely financial mathemat- ics, econometrics, stochastic control, filtering and others (see for example A¨ıt- Sahalia (2002), Jiang and Knight (1997) and Liptser and Shiryaev (1978)).

There is a lot of papers studying the problem of nonparametric estimation of the drift of a diffusion. For our model (1) Banon (1978) proved the consistency whereas Pham (1981) considered the convergence rate of kernel estimators.

The reader is referred for instance to Pinsker (1980), Efro˘ımovich and Pinsker (1984), Tsybakov (1997) and Spokoiny (2000) in which time-continuous signal estimation problems are handled.

Our current purpose is to estimate the drift function at a single point using the absolute error loss to quantify the performance of an estimator through its corresponding risk. We are first interested in obtaining the exact asymptotic behavior of the minimax risk and particularly in finding an asymptotic lower bound of it. Secondly we aim at constructing an estimator of the drift coef- ficient for which the risk is asymptotically bounded from above by the same constant as for the minimax risk. Such an estimator will be called asymptoti- cally efficient (see Ibragimov and Has’minskii (1981)).

The problem of sharp estimation of the drift function in diffusion models has already been treated for some Sobolev classes: Dalalyan and Kutoyants (2002) with known smoothness then Dalalyan (2005) with an unknown one proposed asymptotically efficient estimators of the drift coefficient in model (1) for an L ² -type risk.

The drift estimation in some H¨older classes was handled as well. Galtchouk and Pergamenshchikov (2004) achieved the optimal convergence rate of the minimax risk for the L ² ([a, b], dx)-loss function as the regularity of the drift function is known but as it remains unknown too. The optimal convergence rate of the minimax risk is also obtained for the pointwise estimation of the drift function with unknown smoothness and for the absolute error loss in Galtchouk and Pergamenshchikov (2001). For any positive power of the ab- solute error loss and the same H¨older class, the sharp constant for the local minimax risk is given in Galtchouk and Pergamenshchikov (2005) as well as an asymptotically efficient estimator of the drift coefficient with known regu- larity.

We consider here the pointwise estimation of the drift function belonging to

a H¨older class with known smoothness using the absolute error loss. For this problem Galtchouk and Pergamenshchikov (2006) gave the sharp asymptotic lower bound for the local minimax risk and an asymptotically efficient kernel estimator. More precisely they assumed that the drift function belonged to a neighborhood centred on a Lipschitz function. The neighborhood consists in the centre plus a function satisfying a weak H¨older condition (involving a weak H¨older constant) and having a small norm. The asymptotic results were given with the time of observations tending to infinity, the weak H¨older constant and the diameter of the neighborhood to zero. We propose to find the sharp asymptotic lower bound for the minimax risk and an asymptotically efficient estimator without making the neighborhood tend to its centre but by keeping its diameter constant.

This paper takes the following organization. In the next section we describe the problem and define explicitly the function class, the neighborhoods and the risk of an estimator. The sequential procedure and the main results are given in section 3, their proofs in section 4.

2 Statement of the problem

In model (1) we are interested in the estimation of the unknown function S at a fixed point x ₀ ∈ R assuming that S lies in a neighborhood of a function S ₀ belonging to

Σ _L,M :=

(

f : |f(0)| ≤ L, −L ≤ f (x) − f (y)

x − y ≤ −M, ∀x, y ∈ R

)

, with 0 < M < L.

As mentioned in the introduction we need to construct neighborhoods of a function S 0 belonging to Σ L,M in order to define the risk of an estimator of S(x ₀ ) with S lying in this vicinity. Let S ₀ ∈ Σ _L,M and set

U _δ,β (S ₀ ) = ⁿ S : S = S ₀ + D, D ∈ H ^w _x

(δ, β) ^o where

H ^w _x

(δ, β) = { D differentiable : sup

x∈R

³ |D(x)| + | D(x)| ˙ ^´ ≤ B;

∀h > 0,

¯ ¯

¯ ¯ Z ₁

−1 (D(x ₀ + zh) − D(x ₀ ))dz

¯ ¯

¯ ¯ ≤ δh ^β

¾

, with β ∈]1; 2[, 0 < δ < 1 and 0 < B < M .

Remark 2.1 If S 0 ∈ Σ L,M one has U δ,β (S 0 ) ⊂ Σ L+B,M −B , so that for all

S ∈ U _δ,β (S ₀ ) the process (X _t ) _t≥0 is ergodic and there exists an ergodic density

(see Gihman and Skorohod (1968)) given by

q _S (x) = exp (2 ^R ₀ ^x S(z)dz)

Z _+∞

−∞ exp

µ

2

Z _y

0 S(z)dz

¶

dy .

Recall that the ergodicity of a process guarantees that it returns to any neigh- borhood of x ₀ infinitely many times.

The risk of an estimator ˜ S _T (x ₀ ) of S(x ₀ ) is defined by R _δ,β ( ˜ S _T , S ₀ ) = sup

S∈U

(S

)

q

2q _S (x ₀ )ϕ _T E _S | S ˜ _T (x ₀ ) − S(x ₀ )|, ϕ _T = T ^β/(2β+1) .

3 Main results

The asymptotic lower bound of the minimax risk is given in the following theorem.

Theorem 3.1 If S ₀ ∈ Σ _L,M we have for any δ ∈]0; 1[

lim inf

T →∞ inf

S ˜ R _δ,β ( ˜ S _T , S ₀ ) ≥ E|ξ|, ξ ∼ N (0, 1), where the infimum is taken over all estimators of S(x ₀ ).

In order to exhibit an asymptotically efficient estimator of S(x ₀ ) we begin with estimating the ergodic density at the point x 0 through the observations {X _t , t ≤ t ₀ }, where t ₀ = T ^2γ , γ _∗ < γ < 1/2 and γ _∗ = _2β+1 ^α . Let

ˆ

q _T (x ₀ ) = 1 2t ₀ l

Z _t

0 Q

µ X _t − x ₀ l

¶

dt, with Q = I _[−1;1] and l = l _T = o(1/ √

T ) as T → ∞.

Then for H > 0 we define the sequential procedure (τ _H , S _T ^∗ (x ₀ )) as

τ _H = inf{t ≥ t ₀ :

Z _t

t

Q

µ X t − x 0

h

¶

dt ≥ H}, S _T ^∗ (x 0 ) = 1

H

Z _τ

t

Q

µ X t − x 0

h

¶

dX t I _{τ

_{≤T }} . (2)

We choose the bandwidth h = h _T = T ^−1/(2β+1) and the level H = H _T =

(T − t 0 )(2˜ q T (x 0 ) − ε T )h, where ˜ q T (x 0 ) = max(ˆ q T (x 0 ), ν _T ^−1/2 ), ε T = 1/(ν T T ^γ

)

and ν _T = ln T .

The following lemmas proved in Galtchouk and Pergamenshchikov (2006) give some properties of ˆ q T (x 0 ), ˜ q T (x 0 ) and τ H .

Lemma 3.2 There exist two constants κ > 0 and T _∗ > 0 such that for all T ≥ T _∗ and all λ > 1/T , we have

sup

S∈Σ

P _S (|ˆ q _T (x ₀ ) − q _S (x ₀ )| > λ) ≤ 2e ^−κλ

^t

. Lemma 3.3 There exists a constant T ∗ > 0 such that

µ ^∗ = sup

T ≥T

sup

S∈Σ

√ t ₀ E _S | q ˜ _T (x ₀ ) − q _S (x ₀ )| < ∞.

Lemma 3.4 One has lim sup

T →∞ sup

S∈Σ

E _S

¯ ¯

¯ ¯

¯

1

˜

q T (x 0 ) − 1 q S (x 0 )

¯ ¯

¯ ¯

¯ = 0.

Lemma 3.5 For any p > 0,

T lim →∞ T ^p sup

S∈U

(S

)

P _S (τ _H > T ) = 0.

The centre of the considered neighborhood needs to verify an additional con- dition described by the following definition.

Definition 3.6 A function f satisfies the ”zero-constant” H¨older condition with an exponent ι > 0 at the point x ₀ if

y→x lim

f(y) − f (x ₀ )

|y − x ₀ | ^ι = 0.

An example of such a function can be found in Galtchouk and Pergamen- shchikov (2006).

We are now able to give the upper bound of the risk for the estimator (2).

Theorem 3.7 Let S 0 ∈ Σ L,M , β = 1 + α, α ∈]0; 1[ and assume that S ˙ 0

satisfies the ”zero-constant” H¨older condition with exponent α > 0 at the point x ₀ . Then one has

lim sup

δ→0 lim sup

T →∞ R δ,β (S _T ^∗ (x 0 ), S 0 ) ≤ E|ξ|, ξ ∼ N (0, 1).

4 Proofs

4.1 Proof of theorem 3.1

For u > 0, denote S _u (x) = S ₀ (x) + uD _ν (x), where

D _ν (x) = ϕ ⁻¹ _T V _ν

µ x − x 0

h

¶

, V _ν (x) = ν ⁻¹

Z _∞

−∞

Q ˜ _ν (u)g

µ u − x ν

¶

du, Q ˜ _ν (u) = I _{{|u|≤1−2ν}} + 2I {1−2ν≤|u|≤1−ν} ,

g(z) =

 



 

c exp(−(1 − z ² ) ⁻¹ ), |z| ≤ 1;

0, |z| > 1,

with 0 < ν < 1/4; the normalizing constant c > 0 is such that ^R ₋₁ ¹ g(z)dz = 1.

Now fix b > 0 and δ > 0. We can easily see that there exists T b,ν > 0 such that for any |u| ≤ b and T ≥ T _b,ν one has S _u ∈ U _δ,β (S ₀ ).

Furthermore we may write for all T ≥ T _b,ν

R δ,β ( ˜ S T , S 0 ) ≥ sup

|u|≤b

ϕ T

q

2q S

(x 0 )E S

| S ˜ T (x 0 ) − S u (x 0 )|

≥ 1 2b

Z _b

−b

q

2q S

(x 0 )E S

Ψ a,T ( ˜ S T , S u )du

=: I _T (a, b), with Ψ a,T ( ˜ S T , S) = v a

³ ϕ T ( ˜ S T (x 0 ) − S(x 0 ) ^´ ) and v a (x) = a ∧ |x|, a > 0.

Denoting P _S the distribution of the process (X _t ) in C([0; T ]) as the drift function is S, we have thanks to Galtchouk and Pergamenshchikov (2006, lemma 4.2)

ρ _T (u) := dP _S

dP _S

= exp{u∆ _T − 1

2 u ² σ _ν ² + r _T (u)}, ∀u > 0, where ∆ T =

Z _T

0 D ν (X t )dB t and σ _ν ² = q S

(x 0 )

Z ₁

−1 V _ν ² (z)dz.

Moreover

∆ _T −−−→ ^L

T →∞ ξ _ν , ξ _ν ∼ N (0, σ _ν ² ) et r _T (u) −−−→ ^P

T →∞ 0, (3)

and we can assume that ξ _ν and ∆ _T are independent.

Then write

I T (a, b) = 1 2b

Z _b

−b

q

2q S

(x 0 )E S

Ψ a,T ( ˜ S T , S u )ρ T (u)du

≥ 1 2b

Z _b

−b

q

2q _S

(x ₀ )E _S

I _B

Ψ _a,T ( ˜ S _T , S _u )ρ ⁰ _T (u)du + δ _T ⁰ (a, b)

=: J T (a, b) + δ ⁰ _T (a, b), where B _d = {|∆ _T | ≤ d}, d = σ ² _ν (b − √

b), ρ ⁰ _T (u) = exp(u∆ _T − u ² σ _ν ² /2) and δ _T ⁰ (a, b) = 1

2b

Z _b

−b

q

2q _S

(x ₀ )E _S

I _B

Ψ _a,T (( ˜ S _T , S _u )(ρ _T (u) − ρ ⁰ _T (u))du.

The family {ρ T (u), T > 0} is uniformly integrable. Actually for any T > 0, we have immediately E _S

ρ _T (u) = 1, Eη(u) = 1 and using (3) we can show that

ρ T (u) −−−→ ^L

T →∞ η(u) = exp(uξ ν − u ² σ _ν ² /2), ξ ν ∼ N (0, σ ² _ν ).

The uniform integrability criteria from Ibragimov and Has’minskii (1981, p.

364) allows us to conclude.

Since ρ ⁰ _T (u) is bounded on B _d and Ψ _a,T ( ˜ S _T , S _u ) is bounded from above by a, the family {I B

Ψ a,T ( ˜ S T , S u )(ρ T (u)−ρ ⁰ _T (u)), T > 0} is uniformly integrable too.

Rewrite

ρ T (u) − ρ ⁰ _T (u) = exp

µ

u∆ T − 1

2 u ² σ _ν ² + r T (u)

¶

− exp ^³ uξ ν − u ² σ _ν ² /2 ^´ . On the one hand we can prove via (3) that exp(r _T (u)) −−−→ ^P

T →∞ 1. On the other hand using in particular the independence between ∆ _T and ξ _ν , we can show that exp(u(ξ _ν − ∆ _T )) −−−→ ^P

T →∞ 1.

Then we get from the preceding results

³ ρ _T (u) − ρ ⁰ _T (u) ^´ Ψ _a,T ( ˜ S _T , S _u )I _B

−−−→ ^P

T →∞ 0.

Taking into account the uniform integrability of the corresponding random variables family, we obtain for all |u| ≤ b,

E _S

^{¯ ¯} ¯

³ ρ _T (u) − ρ ⁰ _T (u) ^´ Ψ _a,T ( ˜ S _T , S _u )I _B

^{¯ ¯} ¯ −−−→

T →∞ 0.

On the event B _d , one has ρ _T (u) ≤ exp(|u|d) and ρ ⁰ _T (u) ≤ exp(|u|d − u ² σ ² _ν /2).

Hence E S

(ρ T (u) − ρ ⁰ _T (u)) Ψ a,T ( ˜ S T , S u )I B

≤ 2a exp(|u|d), which is an inte- grable function on [−b; b]. Then bounded convergence yields

δ ⁰ _T (a, b) −−−→

T →∞ 0. (4)

Now we consider the quantity 1

2b

Z _b

−b E S

I B

Ψ a,T ( ˜ S T , S u )

µq

2q S

(x 0 ) −

q

2q S

(x 0 )

¶

ρ ⁰ _T (u)du.

Recall that S _u (z) = S ₀ (z) + uD _ν (z), S _u (x ₀ ) = S ₀ (x ₀ ) + uϕ ⁻¹ _T and specify that kD _ν k _∞ ≤ 3ϕ ⁻¹ _T . We set

c _T (y) := exp

µ

2

Z _y

0 uD _ν (z)dz

¶

, and rewrite

q _S

(x ₀ ) = exp (2 ^R ₀ ^x

S 0 (z)dz) c T (x 0 )

R _∞

−∞ exp (2 ^R ₀ ^y S ₀ (z)dz) c _T (y)dy . By bounded convergence one gets

exp

µ

2

Z _x

0 uD ν (z)dz

¶

−−−→ T →∞ 1 and c T (y) −−−→

T →∞ 1, ∀y ∈ R.

Note that if y > 0,

exp(−Ly ² + 2Ly) ≤ exp

µ

2

Z _y

0 S ₀ (z)dz

¶

≤ exp(−My ² − 2Ly);

and if y < 0,

exp(−Ly ² − 2Ly) ≤ exp

µ

2

Z _y

0 S ₀ (z)dz

¶

≤ exp(−My ² + 2Ly).

Moreover for all y ∈ R, we have |c _T (y)| ≤ exp(6|u||y|), so that the function y 7→ exp (2 ^R ₀ ^y S ₀ (z)dz) c _T (y) is bounded from above by an integrable function on R which does not depend on T . We conclude by bounded convergence that

q _S

(x ₀ ) −−−→

T →∞ q _S

(x ₀ ), (5)

and also that

¯ ¯

¯ ¯

¯

1 2b

Z _b

−b E _S

I _B

Ψ _a,T ( ˜ S _T , S _u )

µq

2q _S

(x ₀ ) −

q

2q _S

(x ₀ )

¶

ρ ⁰ _T (u)du

¯ ¯

¯ ¯

¯ −−−→

T →∞ 0.

Finally we have proved the following equality lim inf

T →∞

1 2b

Z _b

−b E _S

I _B

Ψ _a,T ( ˜ S _T , S _u ) ^q 2q _S

(x ₀ )ρ ⁰ _T (u)du = lim inf

T →∞ J _T (a, b).

At this point rewrite

ρ ⁰ _T (u) = ζ _T exp(−σ ² _ν (u − ∆ ˜ _T )2/2), ζ _T = exp( ˜ ∆ ² _T /2σ _ν ² ), ∆ ˜ _T = ∆ _T /σ ² _ν and put g _T = ϕ _T ( ˜ S _T (x ₀ ) − S ₀ (x ₀ )), ˜ g _T = g _T − ∆ ˜ _T . Then one successively has

1 2b

Z _b

−b E _S

I _B

Ψ _a,T ( ˜ S _T , S _u )

q

2q _S

(x ₀ )ρ ⁰ _T (u)du

= E S

I B

ζ T

1 2b

Z _b

−b v a (u − g T ) exp ^³ −σ ² _ν (u − ∆ ˜ T ) ² /2 ^{´ q} 2q S

(x 0 )du

= E _S

I _B

ζ _T 1 2b

Z _{b− ∆ ˜}

−b− ∆ ˜

v _a (u − g ˜ _T ) exp ^³ −σ _ν ² u ² /2 ^{´ q} 2q _S

(x ₀ )du

≥ E S

I B

ζ T

1 2b

Z ^√ _b

− √

b v a (u − g ˜ T ) exp ^³ −σ _ν ² u ² /2 ^{´ q} 2q S

(x 0 )du

≥ E _S

I _B

ζ _T 1 2b

Z ^√ _b

− √

b v _a (u) exp ^³ −σ _ν ² u ² /2 ^{´ q} 2q _S

(x ₀ )du,

the second inequality arising from Anderson’s lemma (see Ibragimov and Has’minskii, 1981, Lemma 10.2, p.157).

Thus 1 2b

Z _b

−b E _S

I _B

Ψ _a,T ( ˜ S _T , S _u ) ^q 2q _S

(x ₀ )ρ ⁰ _T (u)du

≥ E S

I B

ζ T 1 2b

Z ^√ _b

− √

b |u| exp ^³ −σ _ν ² u ² /2 ^{´ q} 2q S

(x 0 )du + δ _T ¹ (a, b), where

δ _T ¹ (a, b) = E S

I B

ζ T

1 2b

Z ^√ _b

− √

b (v a (u) − |u|) exp ^³ −σ _ν ² u ² /2 ^{´ q} 2q S

(x 0 )du.

Noticing that

T lim →∞ E _S

I _B

ζ _T = 2σ _ν (b − √ b)/ √

2π, one obtains for all b > 0,

a→∞ lim lim inf

T →∞ δ _T ¹ (a, b) = 0, and

lim inf

a→∞ lim inf

T →∞ I _T (a, b) ≥ b − √ b b

q 2q _S

(x ₀ )σ _ν

√ 2π

Z ^√ _b

− √

b |u| exp(−u ² σ _ν ² /2)du.

Remarking here that σ _ν → 2q _S

(x ₀ ) as ν → 0 and limiting b → ∞ before ν → 0 in the last inequality yield

lim inf

T →∞ R _δ,β ( ˜ S _T , S ₀ ) ≥ E|ξ|, ξ ∼ N (0, 1).

Eventually we have shown that lim inf

T →∞ inf

S ˜ R _δ,β ( ˜ S _T , S ₀ ) ≥ E|ξ|, ξ ∼ N (0, 1).

4.2 Proof of theorem 3.7

Let S ∈ U _δ,β (S ₀ ) and parse the error of estimation as S _T ^∗ (x 0 ) − S(x 0 ) =

Ã

B T − G T + ξ _T

√ H _T

!

I {τ

≤T } − S(x 0 )I {τ

>T } , where

B _T = 1 H _T

Z _T

t

Q

µ X _t − x ₀ h

¶

(S(X _t ) − S(x ₀ ))dt, G T = 1

H _T

Z _T

τ

Q

µ X _t − x ₀ h

¶

(S(X t ) − S(x 0 ))dt, ξ _T = 1

√ H _T

Z _T

t

Q

µ X _t − x ₀ h

¶

dB _t .

First we want to show that lim δ→0 lim sup

T →∞

sup

S∈U

(S

)

E _S ^q 2q _S (x ₀ )ϕ _T |B _T |I _{τ

_{≤T }} = 0. (6)

We begin with writing

B _T = T

(2˜ q T (x 0 ) − ε T )(T − t 0 )

à T − t ₀

T m(f _h ) + 1

√ T ∆ _t

_,T (f _h )

!

, (7)

where f _h (y) = φ _h (y)(S(y) −S(x ₀ )), φ _h (y) = _h ¹ Q ^{³ y−x} _h

^´ , m(f ) = ^R f (y)q _S (y)dy and ∆ _t

_,T (f) = ^{√ 1} _T ^R _t ^T

(f(X _t ) − m(f ))dt.

We can rewrite the term m(f _h ) as

m(f h ) = 1 h

Z _∞

−∞ Q

µ y − x 0

h

¶

(S(y) − S(x 0 ))q S (y)dy

=

Z ₁

−1 (S(x ₀ + hz) − S(x ₀ ))q _S (x ₀ + hz)dz

=

Z ₁

−1 (S(x ₀ + hz) − S(x ₀ ))(q _S (x ₀ + hz) − q _S (x ₀ ))dz + q _S (x ₀ )

Z ₁

−1 (S(x ₀ + hz) − S(x ₀ ))dz

=: m 1 (h) + q S (x 0 )m 0 (h).

Put r(y) := S ₀ (y) − S ₀ (x ₀ ) − S ˙ ₀ (x ₀ )(y − x ₀ ). Since ^R ₋₁ ¹ S ˙ ₀ (x ₀ )zhdz = 0 and S ∈ U _δ,β (S ₀ ), we get

|m 0 (h)| =

¯ ¯

¯ ¯ Z ₁

−1 (S(x 0 + zh) − S(x 0 ))dz

¯ ¯

¯ ¯

=

¯ ¯

¯ ¯ Z ₁

−1 r(x ₀ + zh)dz +

Z ₁

−1 (D(x ₀ + zh) − D(x ₀ ))dz

¯ ¯

¯ ¯

≤ 2 sup

|u|≤h

|r(x ₀ + u)|

|u| ^β h ^β + δh ^β =: (2r ^∗ (h) + δ)h ^β .

Using the ”zero-constant” H¨older condition one can show that lim

h→0 r ^∗ (h) = 0.

Consequently we obtain ϕ _T |m ₀ (h)| ≤ (2r ^∗ (h) + δ)ϕ _T h ^β = 2r ^∗ (h) + δ and lim δ→0 lim

T →∞ ϕ _T |m ₀ (h)| = 0. (8)

Moreover denoting C = C(L, M, B) a constant which depends on these pa- rameters but which is not necessarily the same each time it will appear, we have

ϕ _T |m ₁ (h)| ≤ ϕ _T

Z ₁

−1 |S(x ₀ + zh) − S(x ₀ )| |q _S (x ₀ + zh) − q _S (x ₀ )| dz

≤ Cϕ T h ² = Ch ^2−β −−−→

T →∞ 0, (9)

because |q _S (x ₀ +zh) −q _S (x ₀ )| = |zh q ˙ _S (c _z )| ≤ C|z|h and |S(x ₀ +zh) −S(x ₀ )| ≤ (L + B )|z|h.

Thanks to inequality (A.1) in Galtchouk and Pergamenshchikov (2006), there exists a constant κ > 0 such that for all λ > 0:

sup

T ≥1 sup

0≤t

≤T sup

S∈Σ

P _S (|∆ _t

_,T (f _h )| > λ) ≤ 2e ^−κλ

. (10)

Then it is easy to see that ϕ _T T ^−1/2 E _S |∆ _t

_,T (f _h )| tends to zero as T → ∞ uniformly in S ∈ U _δ,β (S ₀ ).

Set q _∗ (x ₀ ) := inf _S∈Σ

q _S (x ₀ ). Then q _∗ (x ₀ ) > 0 and we can write for sufficiently large T

1 2˜ q T (x 0 ) − ε T

≤ 1

˜

q T (x 0 ) ≤

¯ ¯

¯ ¯

¯

1

˜

q T (x 0 ) − 1 q S (x 0 )

¯ ¯

¯ ¯

¯ + 1 q ∗ (x 0 ) . Using lemma 3.4, that is the reason why we get

T lim →∞ sup

S∈U

(S

)

E _S ϕ _T

√ T (2˜ q T (x 0 ) − ε T ) |∆ _t

_,T (f _h )| = 0. (11)

Thanks to lemma 3.4 again, (9) shows that

T lim →∞ sup

S∈U

(S

)

E _S ϕ _T

2˜ q _T (x ₀ ) − ε _T |m ₁ (h)| = 0. (12) Finally since sup

S∈U

(S

)

q _S (x ₀ ) < ∞, (8) and lemma 3.4 prove that lim δ→0 lim

T →∞ sup

S∈U

(S

)

E _S ϕ _T

2˜ q _T (x ₀ ) − ε _T q _S (x ₀ )|m ₀ (h)| = 0. (13) Hence assertion (6) proceeds from (11), (12) and (13).

Now let us show that for all δ ∈ (0, 1)

T lim →∞ sup

S∈U

(S

)

E _S ^q 2q _S (x ₀ )ϕ _T |G _T |I _(τ

_{≤T )} = 0. (14)

First remark that

m(φ h ) − 2q S (x 0 ) =

Z ₁

−1 (q S (x 0 + uh) − q S (x 0 )) du.

By the Taylor formula, for all u ∈ [−1, 1] there exists θ ∈ (0, 1) such that q _S (x ₀ + uh) − q _S (x ₀ ) = ˙ q _S (x ₀ )uh + 1

2 u ² h ² q ¨ _S (x ₀ + θuh), therefore we obtain

sup

S∈U

(S

)

|m(φ _h ) − 2q _S (x ₀ )| = sup

S∈U

(S

)

1 2

¯ ¯

¯ ¯ Z ₁

−1 u ² h ² q ¨ _S (x ₀ + θuh)du

¯ ¯

¯ ¯ ≤ Ch ² ,

(15)

for a certain constant C = C(L, M, B).

Then we have successively

|G _T | ≤ 1 H T

Z _T

τ

hφ _h (X _t )|S(X _t ) − S(x ₀ )|dt

≤ 1 H _T

Z _T

τ

h ² (L + B)φ _h (X _t )dt ≤ (L + B )h ² H _T

ÃZ _T

t

φ _h (X _t )dt − H _T h

!

≤ (L + B)h ² H T

³ √

T ∆ _t

_,T (φ _h ) + (T − t ₀ )m(φ _h ) − (T − t ₀ )(2˜ q _T (x ₀ ) − ε _T ) ^´

≤ (L + B)h ² H _T

³ √

T ∆ t

,T (φ h ) + (T − t 0 )|2˜ q T (x 0 ) − m(φ h )| + ε T (T − t 0 ) ^´ .

So (15) and the definition of H _T yield

|G _T | ≤ (L + B )h

˜ q _T (x ₀ )

à √

T ∆ _t

_,T (φ _h )

T − t ₀ + 2|˜ q _T (x ₀ ) − q _S (x ₀ )| + Ch ² + ε _T

!

. As a consequence of lemma 3.3 and the definition of ˜ q _T (x ₀ ), we get E _S |G _T |I _(τ

_{≤T )} ≤ (L+B) √

ν _T

à h √ T T − t 0

E _S |∆ _t

_,T (φ _h )| + Ch ³ + hε _T + 2µ ^∗ h

√ t 0

!

. Since (10) implies that E _S |∆ _t

_,T (φ _h )| is uniformly bounded in S ∈ U _δ,β (S ₀ ) and T > 0, we can easily prove (14).

Eventually since ξ T is a Gaussian standard random variable, one has

¯ ¯

¯ ¯

¯ q

2q _S (x ₀ )ϕ _T E _S |ξ _T |

√ H _T − E _S |ξ|

¯ ¯

¯ ¯

¯ =

¯ ¯

¯ ¯

¯ ¯

q 2q _S (x ₀ )T ^β/(2β+1)

q

(T − t ₀ )(2˜ q _T (x ₀ ) − ε _T )h − 1

¯ ¯

¯ ¯

¯ ¯ E|ξ|

=

¯ ¯

¯ ¯

¯ ¯

q 2q _S (x ₀ )(1 − ^t _T

) ^−1/2

q 2˜ q _T (x ₀ ) − ε _T −

q 2q _S (x ₀ )

q 2˜ q _T (x ₀ ) +

q 2q _S (x ₀ )

q 2˜ q _T (x ₀ ) − 1

¯ ¯

¯ ¯

¯ ¯ E|ξ|

≤

¯ ¯

¯ ¯

¯ ¯ q

2q S (x 0 )(1 − ^t _T

) ^−1/2

q 2˜ q _T (x ₀ ) − ε _T −

q

2q S (x 0 )

q 2˜ q _T (x ₀ )

¯ ¯

¯ ¯

¯ ¯ E|ξ| +

¯ ¯

¯ ¯

¯ ¯ q

2q S (x 0 )

q 2˜ q _T (x ₀ ) − 1

¯ ¯

¯ ¯

¯ ¯ E|ξ|. (16)

It is not difficult to show that 2˜ q _T (x ₀ ) − ε _T ∼ 2˜ q _T (x ₀ ) and (1 − t ₀ /T ) ^1/2 ∼ 1

as T → ∞. Consequently we have (1 − ^t _T

) ^−1/2

q 2˜ q _T (x ₀ ) − ε _T − 1

q 2˜ q _T (x ₀ ) = o



 1

q q ˜ _T (x ₀ )



 = o((ln T ) ^−1/2 ) −−−→

T →∞ 0 uniformly on U _z

_,δ (S ₀ ).

Thanks to lemma 3.4, the second part of (16) tends to 0 uniformly on U _δ,β (S ₀ ) as T → ∞. Combining this with (6), (14) and lemma 3.5 finishes the proof of theorem 3.7.

References

A¨ıt-Sahalia, Y.: Maximum likelihood estimation of discretely sampled diffu- sions: a closed-form approximation approach, Econometrica 70(1) (2002), 223–262.

Banon, G.: Nonparametric identification for diffusion processes, SIAM J. Con- trol Optim. 16(3) (1978), 380–395.

Dalalyan, A.: Sharp adaptive estimation of the drift function for ergodic dif- fusions, Ann. Statist. 33(6) (2005), 2507–2528.

Dalalyan, A. S. and Kutoyants, Yu. A.: Asymptotically efficient trend coeffi- cient estimation for ergodic diffusion,Math. Methods Statist. 11(4) (2002), 402–427.

Efro˘ımovich, S. Yu. and Pinsker, M. S.: Learning algorithm for nonparametric filtering, Automat. Remote Control 11 (1984), 1434–1440.

Galtchouk, L. and Pergamenshchikov, S.: Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes, Math. Methods Statist. 10(3) (2001), 316–330.

Galtchouk, L.and Pergamenshchikov, S.: Nonparametric sequential estimation of the drift in diffusion processes via model selection, Math. Methods Statist.

13(1) (2004), 25–49.

Galtchouk, L. and Pergamenshchikov, S.: Nonparametric sequential minimax estimation of the drift coefficient in diffusion processes, Sequential Anal.

24(3) (2005), 303–330.

Galtchouk, L. and Pergamenshchikov, S.: Asymptotically efficient sequential kernel estimates of the drift coefficient in ergodic diffusion processes, Stat.

Inference Stoch. Process. 9(1) (2006), 1–16.

Gihman, I. I. and Skorohod, A. V.: Stochastic Differential Equations, Naukova Dumka, Kiev, 1968.

Ibragimov, I. A. and Has’minskii, R. Z.: Statistical Estimation: Asymptotic Theory, Springer, Berlin, New York, 1981.

Jiang, G. J. and Knight, J. L.: A nonparametric approach to the estimation of

diffusion processes with an application to a short-term interest rate model,

Econometric Theory 13(5) (1997), 615–645,

Liptser, R. S. and Shiryaev, A. N.: Statistics of random processes.I and II, Springer-Verlag, New York, 1978.

Pham D. T.: Nonparametric estimation of the drift coefficient in the diffusion equation, Math. Operationsforsch. Statist. Ser. Statist. 12(1) (1981), 61–73.

Pinsker, M. S.: Optimal filtration of square-integrable signals in Gaussian noise, Problems Inform. Transmission 16(2) (1980), 120–133.

Spokoiny, V. G.: Adaptive drift estimation for nonparametric diffusion model, Ann. Statist. 28(3) (2000), 815–836.

Tsybakov, A. B.: Asymptotically efficient estimation of a signal in L ₂ under

general loss functions, Problems Inform. Transmission 33(1) (1997), 78–88.