Penalized nonparametric drift estimation in a continuous time one-dimensional diffusion process

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Penalized nonparametric drift estimation in a continuous time one-dimensional diffusion process

Eva Loecherbach, Dasha Loukianova, Oleg Loukianov

To cite this version:

Eva Loecherbach, Dasha Loukianova, Oleg Loukianov. Penalized nonparametric drift estimation in a

continuous time one-dimensional diffusion process. ESAIM: Probability and Statistics, EDP Sciences,

2011, 15, pp.197–216. �10.1051/ps/2009016�. �hal-00367993v3�

Penalized nonparametric drift estimation for a continuously observed one-dimensional

diffusion process.

Eva L¨ ocherbach ^∗ , Dasha Loukianova ^† , Oleg Loukianov ^‡ September 17, 2009

Abstract

Let X be a one dimensional positive recurrent diffusion continu- ously observed on [0, t]. We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained us- ing a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data.

The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in stationary regime neither to be exponentially β - mixing. This is possible thanks to the use of a new polynomial inequality in the ergodic theorem [16].

Key words : diffusion process, adaptive estimation, regeneration method, mean square estimator, model selection, deviation inequalities.

MSC 2000 : 60 F 99, 60 J 35, 60 J 55, 60 J 60, 62 G 99, 62 M 05

∗

Centre de Math´ ematiques, Facult´ e de Sciences et Technologie, Universit´ e Paris- Est Val-de-Marne, 61 avenue du G´ en´ eral de Gaulle, 94010 Cr´ eteil, France. E-mail:

locherbach@univ-paris12.fr

†

D´ epartement de Math´ ematiques, Universit´ e d’Evry-Val d’Essonne, Bd Fran¸ cois Mit- terrand, 91025 Evry, France. E-mail: dasha.loukianova@univ-evry.fr

‡

D´ epartement Informatique, IUT de Fontainebleau, Universit´ e Paris-Est, route Hur-

tault, 77300 Fontainebleau, France. E-mail: oleg@iut-fbleau.fr

1 Introduction

Let X

t

be a one-dimensional diffusion process given by dX

_t

= b(X

_t

) dt + σ(X

_t

) dW

_t

, X

₀

= x,

where W is a standard Brownian motion. Assuming that the process is pos- itive recurrent but not necessarily in the stationary regime (i.e. not starting from the invariant measure) and not necessarily exponentially β-mixing, we want to estimate the unknown drift function b on a fixed interval K from observations of X during the time interval [0, t], for fixed t. We do not require any knowledge about smoothness of the drift function : b is not supposed to belong to some known Besov or Sobolev ball. Hence we aim at studying nonparametric adaptive estimators for the unknown drift b.

Nonparametric estimation in continuous time of the drift coefficient of diffusion processes has been widely studied over the last decades. To mention just a few, let us cite Banon [1], Prakasa Rao [18], Pham [17], Galtchouk and Pergamenschikov [9], Dalalyan and Kutoyants [7], Delattre, Hoffmann and Kessler [8], Loukianova and Loukianov [14], L¨ ocherbach and Loukianova [15]

and the extensive book of Kutoyants [11].

The adaptive estimation for the drift at a fixed point has been studied by Spokoiny [20], who uses Lepskii’s method (see [12]) in order to construct an adaptive procedure. Dalalyan [6], uses kernel-type estimators and considers a weighted L

²

-risk, where the weight is given by the invariant density. He has to work under quite strong ergodicity assumptions.

Our aim in this paper is twofold. Firstly, we aim at introducing a non- parametric estimation procedure based on model selection. Our estimator is obtained by minimizing a contrast function within a fixed finite-dimensional linear sub-space of L

²

(K, dx) – quite in the spirit of mean square estimation and following ideas presented by Comte et al. [5], for discretely observed diffusions. These finite-dimensional sub-spaces include spaces such as piece- wise polynomials or compactly supported wavelets. The risk we consider for a given estimator ˆ b of b is the expectation of an empirical L

²

−norm defined by

E

x

k ˆ b − bk

²_t

, where k ˆ b − bk

²_t

= 1 t

Z

t 0

(ˆ b − b)

²

(X

_s

)ds.

The dimension of the space is chosen by a data-driven method using a pe-

nalization.

Secondly, we aim at working under the less restrictive assumptions on the ergodicity properties of the process that seem to be possible. We do not impose the diffusion to be exponentially β-mixing and do not assume the existence of exponential moments for the invariant measure, though we do have to impose the existence of a certain number of moments. Finally, note that we do not work in the stationary regime : the process starts from a fixed point x ∈ K, and is not yet in equilibrium. Note also that our approach is non-asymptotic in time. But we have to suppose that t ≥ t

₀

for some fixed explicitly given time horizon t

0

that is needed for theoretical reasons and defined precisely later in the text (see Proposition 3.4). A main ingredient of the proofs is a new polynomial inequality ensuring that empirical norm and theoretical L

²

-norm are not too far away. This inequality is given in Loukianova et al. [16].

The paper is organized as follows. In section 2 we describe our frame- work and give the main results : in section 2.1 we give precise assumptions on the diffusion model, explain these assumptions and give some examples for models satisfying them. In section 2.2 we introduce both the non-adaptive and adaptive estimator, section 2.3 gives assumptions on the approximation spaces and section 2.4 provides some examples of approximation spaces veri- fying these assumptions. The main results (rate of convergence of estimators) are given in section 2.5. Section 3 presents probabilistic tools and auxiliary results necessary for the proof of the main results. Section 4 is devoted to the proofs of the main results : section 4.1 deals with non-adaptive and section 4.2 with adaptive drift estimation. Finally, section 5 is an appendix, where we give the proof of one technical result (Lemma 4.2).

Acknowledgments. The subject of this paper has been proposed to the authors by Fabienne Comte and Valentine Genon-Catalot during their re- search period at the University Paris-Descartes in spring 2008. The authors thank both of them for their kindness, their patience, and last, but not least for all the time spent on discussions and explanations on the paper.

The authors are also grateful to the referees for comments that helped to significantly improve the paper.

Eva L¨ ocherbach has been partially supported by an ANR project : Ce

travail a b´ en´ efici´ e d’une aide de l’Agence Nationale de la Recherche portant

la r´ ef´ erence ANR-08-BLAN-0220-01.

2 Framework, assumptions and main results.

2.1 Assumptions on the diffusion.

Let X

_t

be a one-dimensional diffusion process given by

dX

_t

= b(X

_t

) dt + σ(X

_t

) dW

_t

, X

₀

= x. (2.1) We would like to estimate the drift function b on a fixed interval K, say K = [0, 1]. To insure the existence and the unicity of a strong non exploding solution of (2.1) we suppose

Assumption 2.1 1. b and σ are locally Lipschitz and b is at most of linear growth.

2. There exist 0 < σ

²₀

≤ σ

₁²

< ∞ such that for all x, σ

₀²

≤ σ

²

(x) ≤ σ

₁²

. A more particular assumption is needed for the drift function to guarantee some “speed” of ergodicity of X.

Assumption 2.2 1. There are two known constants M

₀

and b

₀

such that K ⊂ [−M

₀

, M

₀

] and for all x with |x| ≤ M

₀

, |b(x)| ≤ b

₀

.

2. We suppose that there is a positive constant γ such that for all x with

|x| ≥ M

₀

,

xb(x) ≤ −γ.

3. The constant γ satisfy 2γ > 31σ

₁²

.

To clarify the meaning of Assumptions 2.2 let us recall some well-known facts about linear diffusions. We refer the reader to the book of Revuz and Yor [19]. The scale density of X is given by

s(x) = exp

−2 Z

x

0

b(u) σ

²

(u) du

, and the scale function by S(x) = R

x

0

s(t)dt. X is recurrent if and only if lim

x→±∞

S(x) = ±∞. In the case of recurrence the diffusion admits a unique up to a constant multiple invariant measure m(dx), given by m(dx) = 1/(s(x)σ

²

(x))dx. Denote M = R

+∞

−∞

m(dx). The diffusion is positively recur- rent if and only if M < ∞. In this case put

µ(dx) = p(x)dx, where p(x) = 1

M s(x)σ

²

(x) .

The probability µ is called invariant or stationary probability of X.

Using Assumptions 2.1.2 and 2.2.1, 2.2.2 we see that for any x such that

|x| ≤ M

0

,

s

⁻¹

(x) ≤ e

2M0b0 σ2

0

, and for |x| ≥ M

₀

,

s

⁻¹

(x) ≤ e

2M0b0 σ2

0

M

₀

|x|

^2γ

σ2 1

.

This shows that S(x) → ±∞, when x → ±∞. Hence X is recurrent. The same estimation gives M < ∞ (and X is positively recurrent) as soon as 2γ > σ

²₁

.

Actually Assumption 2.2.3: 2γ > 31σ

²₁

guarantees more than positive recurrence. It is well known that the positive recurrence of X is equivalent to E

^x

T

_a

< ∞ for all a ∈ R , x ∈ R , where T

_a

is the hitting time of level a. Under Assumptions 2.1.2 and 2.2.1, 2.2.2 the moments of hitting times of X satisfy E

x

T

_aⁿ

< ∞ for n < γ/σ

₁²

+ 1/2, for all x ∈ R , a ∈ R , see Loukianova et al. [16], Theorem 5.5. Thus under Assumption 2.2.3 we have E

x

T

_aⁿ

< ∞ for n ≤ 16. This means that the “speed of recurrence” of X is polynomial of order 16 and will be used to bound the speed of convergence of our estimator. Though we do not use the mixing coefficient, note that Assumption 2.2 guarantees that the diffusion is polynomially β-mixing (see Veretennikov [21]).

It follows from the above assumptions that the invariant density p is con- tinuous and hence bounded from above and below on any compact interval.

So we have

0 < p

₀

≤ p(x) ≤ p

₁

< ∞ for all x ∈ [0, 1].

In the sequel we need to fix p

₀

. We get immediately that M =

Z

+∞

−∞

(s(x)σ

²

(x))

⁻¹

dx ≤ 2M

₀

σ

²₀

e

2M0b0 σ2

0

2γ 2γ − σ

²₁

=: M

+

. This yields the following lower bound for all x ∈ [0, 1],

p(x) ≥ 1 M

+

1

σ

₁²

e

^−2b0/σ20

:= p

₀

. (2.2) In conclusion of this subsection, let us give an example of a diffusion process which fulfills Assumptions 2.2. Consider the solution of

dX

t

= − γX

_t

1 + X

_t²

dt + dW

t

, X

0

= x, γ > 31

2 .

It is positive recurrent with stationary distribution µ(dx) ∼ dx

(1 + x

²

)

^γ

and satisfies all the assumptions of 2.2. Remark that there is no evidence whether this diffusion is exponentially β-mixing.

2.2 Construction of the estimator.

In this section we introduce a nonparametric estimator of the unknown drift function b on an interval K. We use the penalized least-squares based ap- proach, where an estimator is constructed as a “projection” on some finite dimensional approximation space. We firstly address the non-adaptive case, where the statistician chooses himself the dimension of the approximation space. This choice can be done in an optimal way for example if the smooth- ness of the unknown function b is known. Secondly we address the adaptive estimation procedure. In this case the dimension of the approximation space is chosen automatically using some penalization procedure, based on the data.

Consider a collection {S

_m

; m ∈ M

_t

} of approximation spaces. Each of these spaces is a linear finite dimensional subspace of L

²

(K, dx). Here M

_t

is a set of indices. We suppose that there exists a space denoted by S

t

, belonging to the collection, such that S

_m

⊆ S

_t

for all m ∈ M

_t

. Denote by D

_m

the dimension of S

_m

and by D

_t

the dimension of S

_t

.

Put

khk

²_t

= 1 t

Z

t 0

h

²

(X

_s

) ds and denote the corresponding quadratic form by

T

_X

(h, f) = 1 t

Z

t 0

h(X

_s

)f (X

_s

) ds for all f, h ∈ S

_t

.

We firstly construct the non-adaptive estimator. To this end fix a linear subspace S

_m

⊂ S

_t

. We shall write shortly b

_K

(x) := b(x)1

_K

(x) for the re- striction of the function b to the interval K . The estimator ˆ b

_m

of b

_K

will be defined as trajectorial minimizer on S

_m

of the following contrast function :

γ

_t

(h) = khk

²_t

− 2 t

Z

t 0

h(X

_s

) dX

_s

.

To insure the existence of ˆ b

_m

we impose some condition under which T

_X

is a.s. positive-definite on S

_t

and hence on each S

_m

, m ∈ M

_t

. Denote by khk the L

²

(K, dx)-norm, and let

ρ

_t

(X) = inf

h∈S_t;khk=1

T

_X

(h, h).

Put

A

_t

=

ρ

_t

(X) ≥ t

^−1/2

. (2.3)

Note that, since S

t

is finite-dimensional, γ

t

is almost surely defined for all h ∈ S

_t

(see remark 2.3 below). We finally put

ˆ b

_m

= arg min

h∈S_m

γ

_t

(h) on A

_t

and ˆ b

_m

= 0 on A

^c_t

.

Clearly, for all ω ∈ A

_t

, T

_X

is a strictly positive-definite quadratic form on S

m

, m ∈ M

t

, and γ

t

is a difference between this strictly positive quadratic form and a linear form. Hence the minimizer of γ

_t

exists and is unique on S

_m

, m ∈ M

_t

. As it was explained, in the non-adaptive case the statistician chooses himself the approximation space.

In the adaptive case the dimension is chosen automatically using a model selection procedure. In order to describe this procedure, we have to define properly γ

t

(ˆ b

m

). Fix some basis {ϕ

1

, . . . , ϕ

Dm

} of S

m

. From the definition of γ

_t

it follows that on A

_t

,

ˆ b

m

=

Dm

X

i=1

ˆ α

i

ϕ

i

,

with random ˆ α = ( ˆ α

₁

, . . . , α ˆ

_Dm

)

^∗

(we denote by ∗ the usual matrix transpo- sition) satisfying

T

^ϕ

α ˆ = 1 t

Z

t 0

ϕ(X

_s

)dX

_s

, (2.4)

where T

^ϕ

is the D

_m

× D

_m

random matrix with elements T

_ij^ϕ

= 1

t Z

t

0

ϕ

_i

(X

_s

)ϕ

_j

(X

_s

) ds and where

Z

t 0

ϕ(X

_s

) dX

_s

=





 R

t

0

ϕ

1

(X

s

)dX

s

.. . R

t

0

ϕ

_Dm

(X

_s

)dX

_s





 .

Define on A

_t

γ

_t

(ˆ b

_m

) := k ˆ b

_m

k

²_t

− 2 t

Dm

X

i=1

ˆ α

_i

Z

t 0

ϕ

_i

(X

_s

) dX

_s

. (2.5) Now we are able to introduce the adaptive estimator. Define

ˆ

m := arg min

m∈M_t

h

γ

t

(ˆ b

m

) + pen(m) i

,

where the penalization term pen(m) will be given later, see (2.7). Then the estimator that we propose is the following adaptive estimator

ˆ b

_mˆ

:=

P

n

1

{m=n}ˆ

ˆ b

_n

on A

_t

0 on A

^c_t

.

Remark 2.3 The above considerations and in particular the definition of γ

_t

(ˆ b

_m

) of (2.5) do not depend on the special choice of bases.

Indeed, let {ϕ

1

, . . . , ϕ

n

} and {ψ

1

, . . . , ψ

n

} be two bases of S

t

(or S

m

), with n = D

_t

(resp. D

_m

), and let A = (a

_ij

) be the n × n matrix such that ϕ

_i

= P

j

a

_ij

ψ

_j

, for any 1 ≤ i ≤ n. We then have for a function h h =

n

X

i=1

α

_i

ϕ

_i

=

n

X

i=1

β

_i

ψ

_i

,

where β = A

^∗

α.

1. Hence, given a version of the stochastic integrals R

ϕ

_i

(X

_s

)dX

_s

, 1 ≤ i ≤ D

_t

, the equalities

Z

t 0

h(X

_s

)dX

_s

=

Dt

X

i=1

α

_i

Z

t

0

ϕ

_i

(X

_s

)dX

_s

= α

^∗

Z

t

0

ϕ(X

_s

)dX

_s

= α

^∗

A Z

t

0

ψ(X

_s

)dX

_s

=

Dt

X

i=1

β

_i

Z

t

0

ψ

_i

(X

_s

)dX

_s

determine automatically a version of any stochastic integral R

h(X

s

)dX

s

on S

_t

, that does not depend on the choice of the basis.

2. From the definition (2.5) of γ

_t

(ˆ b

_m

), we have γ

_t

(ˆ b

_m

) = k ˆ b

_m

k

²_t

− 2

t

ˆ α

^∗

Z

t 0

ϕ(X

_s

)dX

_s

= k ˆ b

_m

k

²_t

− 2 t

ˆ α

^∗

A

Z

t 0

ψ(X

_s

)dX

_s

= k ˆ b

_m

k

²_t

− 2 t

β ˆ

^∗

Z

t 0

ψ(X

_s

)dX

_s

where β ˆ = A

^∗

α. The equality ˆ (2.4) yields T

^ψ

β ˆ = A

⁻¹

T

^ϕ

(A

⁻¹

)

^∗

A

^∗

α ˆ = 1

t Z

t

0

ψ(X

_s

) dX

_s

,

hence β ˆ satisfies (2.4), when replacing all ϕ

_i

by ψ

_i

. This implies that the definition of ˆ b

_m

and of γ

_t

(ˆ b

_m

) does not depend on the choice of a basis in S

_m

.

2.3 Assumptions on linear subspaces of L ² (K, dx).

We assume that the approximation spaces satisfy the following conditions : Assumption 2.4 1. We suppose that there exists Φ

₀

> 0 such that

for all m ∈ M

t

, for all h ∈ S

m

,

||h||

∞

≤ Φ

₀

D

^1/2_m

||h||.

Recall that khk

²

= R

K

h

²

(x)dx is the usual L

²

(K, dx)−norm.

2. We suppose that

X

m∈M_t

e

^−Dm

≤ C, where the constant C does not depend on t.

3. Dimension condition.

D

t

≤ t.

4. We suppose that there exists an orthonormal basis {ϕ

1

, . . . , ϕ

Dt

} of S

_t

⊂ L

²

(K, dx) and a positive constant Φ

₁

such that for all i,

card {j : ||ϕ

_i

ϕ

_j

||

∞

6= 0} ≤ Φ

₁

.

5. We suppose that the cardinality of M

_t

satisfies card M

_t

≤ D

_t

.

2.4 Example for approximation spaces.

We present a collection of models that can be used for estimation. We con- sider the space of piecewise polynomials, as introduced for example in Baraud et al. [2], [3] and Comte et al. [5].

Take K = [0, 1] and fix an integer r ≥ 0. For p ∈ N , consider the dyadic subintervals I

_j,p

= [(j − 1)2

^−p

, j2

^−p

], for any 1 ≤ j ≤ 2

^p

. On each subinterval I

_j,p

, we consider polynomials of degree less or equal to r, so we have polyno- mials ϕ

_j,l

, 0 ≤ l ≤ r of degree l, such that ϕ

_j,l

is zero outside I

_j,p

. Then the space S

_m

, for m = (r, p), is defined as the space of all functions that can be written as

t(x) =

2^p

X

j=1 r

X

l=0

t

_j,l

ϕ

_j,l

(x).

Hence, D

_m

= (r + 1)2

^p

. Then the collection of spaces {S

_m

, m ∈ M

_t

} is such that

M

_t

= {m = (r, p), p ≥ 0, r ∈ {0, . . . , r

_max

}, 2

^p

(r

_max

+ 1) ≤ D

_t

}.

One possible choice of S

t

and D

t

is as follows : Take

p

_max

:= max{p : 2

^p

(r

_max

+ 1) ≤ t}, D

_t

= 2

^pmax

(r

_max

+ 1),

and let S

_t

be the space of piecewise polynomials associated to m

_max

:=

(r

_max

, p

_max

). Then it is evident that any of the spaces S

_m

, m ∈ M

_t

, is con- tained in S

_t

. Furthermore, card M

_t

= (p

_max

+ 1)(r

_max

+ 1) ≤ D

_t

≤ t.

It is well known, see for instance Comte et al. [5], that for this model the assumption of norm connection 2.4.1 is satisfied. Note moreover that for a fixed ϕ

_j,l

∈ S

_t

,

card {(j

⁰

, l

⁰

) : ϕ

_j⁰_,l⁰

ϕ

_j,l

6= 0} = card {(j, l

⁰

) : ϕ

_j,l⁰

ϕ

_j,l

6= 0} ≤ r

_max

+ 1,

which does not depend on t. Hence assumption 2.4.4 is satisfied. Finally, it

is easy to check that also assumption 2.4.2 holds : X

m∈Mt

e

^−Dm

=

rmax

X

r=0

X

p:2^p(rmax+1)≤Dt

e

^−(r+1)2p

≤

rmax

X

r=0

X

p:2^p(rmax+1)≤Dt

e

^−2p

≤ (r

_max

+ 1) X

k≥0

e

^−k

< +∞, where the last quantity does not depend on t.

Spaces generated by compactly supported wavelets, similar to those con- sidered by Hoffmann [10] and Baraud et al. [2] or [3] are also covered by Assumption 2.4. On the other hand, spaces spanned by the trigonometric basis do not fulfill Assumption 2.4.4 and therefore do not fit to our set-up.

2.5 Main results.

We have the following first result concerning the non-adaptive estimator.

Recall that b

_K

(x) = b(x)1

_K

(x) is the restriction of the function b to the interval K. We define the risk of the estimator ˆ b

m

as

E

x

k ˆ b

_m

− b

_K

k

²_t

= E

x

1 t

Z

t 0

(ˆ b

_m

− b

_K

)

²

(X

_s

)ds

.

Let b

_m

be the L

²

(K, dx)-projection of b

_K

onto S

_m

. Then the following holds.

Theorem 2.5 Suppose that t ≥ t

₀

:= 4/p

²₀

. Suppose that X satisfies Assumptions 2.1 and 2.2. Suppose that the collection of the approximation spaces satisfies Assumptions 2.4.1, 2.4.3–5. Then

E

x

k ˆ b

_m

− b

_K

k

²_t

≤ 3κkb

_m

− b

_K

k

²

+ 16σ

₁²

κ p

₀

D

m

t + Ct

⁻¹

. (2.6) Here, κ = κ(t) =

_σ²2

0

(

^2diam(K)_t

+

^2σ√1

t

+ 2b

₀

+

^σ₂¹²

) (see Proposition 3.1), and C is a positive constant depending on b

₀

, σ

₁

and Φ

₀

.

Let us give some comments on (2.6). It is natural to choose the dimension

D

_m

that balances the bias term ||b

_m

− b

_K

||

²

and the variance term which is

of order D

_m

/t. Assume that b

_K

belongs to some Besov space B

_2,∞^α

([0, 1]) and consider the space of piecewise polynomials S

_m

such that r > α − 1. Then it can be shown that ||b

m

− b

K

||

²

≤ CD

^−2α_m

, see for example Barron et al. [4], Lemma 12. Thus the best choice of D

_m

is to take

D

_m

= t

^2α+11

and then we obtain

E

x

(|| ˆ b

_m

− b

_K

k

²_t

) ≤ Ct

^−2α+12α

+ C

₁

t

⁻¹

,

and this yields exactly the classical nonparametric rate t

^−2α+12α

(compare for example to Hoffmann [10]). This choice however supposes the knowledge of the regularity α of the unknown drift function, and that is why an adaptive estimation scheme has to be used, in order to choose automatically the best dimension D

_m

in the case when the regularity α is not known.

Concerning the adaptive drift estimator, we have the following theorem.

Theorem 2.6 Suppose that X satisfies Assumptions 2.1 and 2.2. Sup- pose that the collection of the approximation spaces satisfies Assumption 2.4.

Suppose that t ≥ t

₀

, where t

₀

:= 4/p

²₀

. Let pen(m) = χσ

²₁

D

_m

t , (2.7)

where χ is a universal constant that will be given explicitly in (4.11). Then we have

E

x

|| ˆ b

_mˆ

− b

_K

||

²_t

≤ 3κ inf

m∈Mt

||b

_m

− b

_K

||

²

+ pen(m) + C

t , where κ = κ(t) =

_σ²2

0

(

^2diam(K)_t

+

^2σ√1

t

+ 2b

₀

+

^σ₂¹²

) (compare to Proposition 3.1) and where C is a positive constant not depending on t.

3 Probabilistic tools and auxiliary results.

In this section, we collect some probabilistic results and auxiliary lemmas

that are needed for the proofs of the main results.

3.1 Probabilistic tools.

In what follows we often need to compare empirical and theoretical norms.

One way of doing this is given by the next proposition.

Proposition 3.1 For any positive function f having support on a com- pact interval K, we have

1 t E

x

Z

t 0

f(X

_s

)ds ≤ κ(t) Z

K

f(x)dx, where κ(t) =

_σ²2

0

(

^2diam(K)_t

+

^2σ√1

t

+ 2b

₀

+

^σ₂¹²

).

Proof By the occupation time formula and since f has support in K, E

^x

Z

t 0

f (X

_s

)ds = Z

K

f (y) 2

σ

²

(y) E

^x

L

^y_t

dy.

We will derive a bound on E

x

L

^y_t

for y ∈ K. Let y

₀

be the leftmost point of K . We have

E

^x

L

^y_t⁰

− E

^x

|L

^y_t

− L

^y_t⁰

| ≤ E

^x

L

^y_t

≤ E

^x

L

^y_t⁰

+ E

^x

|L

^y_t

− L

^y_t⁰

| and

|L

^y_t

− L

^y_t⁰

| ≤ |y − y

₀

| + | Z

t

0

1

_{y0<Xs<y}

σ(X

_s

)dW

_s

| + Z

t

0

1

_{Xs∈K}

|b(X

_s

)|ds.

Taking expectation we obtain E

x

Z

t 0

1

{X_s∈K}

|b(X

_s

)|ds ≤ t b

₀

, and by norm inclusion and isometry,

E

x

| Z

t

0

1

_{y0<Xs<y}

σ(X

_s

)dW

_s

| ≤

E

x

( Z

t

0

1

_{y0<Xs<y}

σ(X

_s

)dW

_s

)

²

^1/2

≤

E

^x

( Z

t

0

1

{Xs∈K}

σ

²

(X

s

)ds)

^1/2

≤ σ

1

√ t.

In conclusion,

E

x

L

^y_t

≤ E

x

L

^y_t⁰

+ diam(K ) + σ

₁

√

t + tb

₀

= C

₀

+ L,

where L := diam(K) + σ

₁

√

t + tb

₀

and C

₀

= E

x

L

^y_t⁰

. We also have C

₀

− L ≤ E

x

L

^y_t

, so

t ≥ E

^x

Z

t

0

1

_K

(X

_s

)ds = Z

K

2 E

x

L

^y_t

σ

²

(y) dy ≥ 2(C

₀

− L) σ

₁²

, whence

C

0

≤ L + σ

₁²

t/2, and thus finally,

E

^x

L

^y_t

≤ 2L + σ

₁²

t/2 = 2(diam(K) + σ

1

√

t + tb

0

) + σ

₁²

t/2.

This concludes the proof. •

Now we give a useful deviation inequality for the one-dimensional ergodic diffusion process X, which is an immediate consequence of deviation inequal- ity obtained by Loukianova et al. [16]. For f : R → R denote as usually µ(f) = R

R

f dµ.

Theorem 3.2 (Deviation inequality.)

Let f be a measurable bounded function with compact support such that µ(f) 6= 0. Suppose that X satisfies Assumptions 2.1 and 2.2.1, 2.2.2. Then for all n ∈ N such that

n < γ σ

₁²

+ 1

2

and any 0 < ε ≤ 1, we have the following polynomial bound P

x

1 t

Z

t 0

f(X

_s

)ds − µ(f)

>≥

≤ K(n)t

^−n/2

ε

⁻ⁿ

µ(|f|)

ⁿ

,

where K(n) is positive and finite, depending on the coefficients of the diffusion and on n but not depending on f, t, ε.

This theorem follows directly from theorem 4.3 and theorem 5.5 of [16].

Corollary 3.3 Under Assumption 2.2.3 the previous theorem is satisfied

for all n ≤ 16.

3.2 Auxiliary results.

In what follows we also need to compare empirical and theoretical norms through the set

Ω

_t

=

∀h ∈ S

_t

, 1

2 µ(h

²

) ≤ khk

²_t

≤ 3 2 µ(h

²

)

, (3.1)

where any h ∈ S

_t

is defined as 0 outside of K. Recall that A

_t

is given by (2.3) and p

0

by (2.2).

Proposition 3.4 For all t ≥ 4/p

²₀

it holds that Ω

_t

⊆ A

_t

.

Proof Note that by the definition of A

_t

and Ω

_t

, under the assumption t ≥ 4/p

²₀

, the inequality µ(h

²

)/2 ≤ khk

²_t

implies khk

²_t

≥ p

0

khk

²

/2 ≥ t

^−1/2

, so

Ω

_t

⊆ A

_t

. •

Proposition 3.5 Suppose that X satisfies Assumptions 2.1, 2.2.1 and 2.2.2. Suppose that the collection of approximation spaces {S

m

, m ∈ M

t

} satisfies Assumptions 2.4.3, 2.4.4. Then for all

n < γ σ

₁²

+ 1

2 and for all x ∈ R we have that

P

x

(Ω

^c_t

) ≤ Ct

^−12(n−2)

,

where C depends on n, the constant Φ

1

given in Assumption 2.4.4 and on the coefficients of X, but does not depend on t.

Proof Recall that kfk denotes the usual L

²

(K, dx)−norm. For any function f, write

Z

_t

(f) := 1 t

Z

t 0

f (X

_s

)ds − µ(f ).

Since for f supported by K, kfk

²_µ

= 1 implies that ||f||

²

≤ p

⁻¹₀

, we have that P

x

(Ω

^c_t

) ≤ P

x

( sup

f∈St,kfk≤1

|Z

_t

(f

²

)| > 0, 5p

₀

).

Let {ϕ

₁

, . . . , ϕ

_Dt

} be an orthonormal basis of S

_t

⊂ L

²

(K, dx), satisfying Assumption 2.4.4, and note that any function f with ||f || ≤ 1 can be written as

f =

Dt

X

i=1

a

_i

ϕ

_i

with X

a

²_i

≤ 1.

Therefore,

P

x

(Ω

^c_t

) ≤ P

x

( sup

||f||≤1

|Z

_t

(f

²

)| > 0, 5p

₀

)

≤ P

x

sup

Pa²_i≤1

X

i,j

a

_i

a

_j

|Z

_t

(ϕ

_i

ϕ

_j

)| > 0, 5p

₀

! .

Write

C

_ij

:= µ(|ϕ

_i

ϕ

_j

)|

and fix some positive number ε. On the set

{|Z

_t

(ϕ

_i

ϕ

_j

)| ≤ C

_ij

ε, ∀i, j} , we have that

sup

Pa²_i≤1

X a

_i

a

_j

|Z

_t

(ϕ

_i

ϕ

_j

)| ≤ ε%(C),

where %(C) is the biggest eigenvalue of the matrix C. Then choosing ε :=

p

0

/(4%(C)), we conclude that

P

x

(Ω

^c_t

) ≤ P

x

(∃i, j : |Z

_t

(ϕ

_i

ϕ

_j

)| > C

_ij

ε) . By theorem 3.2, we have the upper bound

P

^x

(|Z

_t

(ϕ

_i

ϕ

_j

)| > C

_ij

ε) ≤ K(n)%(C)

ⁿ

t

^−n/2

.

Note that due to assumption 2.4.4 and since µ(|ϕ

_i

ϕ

_j

|) ≤ p

₁

, we have that

%(C) ≤ Φ

1

p

1

where the upper bound does not depend on t. Indeed, using that 2u

_i

u

_j

≤ u

²_i

+ u

²_j

, we have that

%(C) = sup

u∈R^Dt,||u||≤1

< Cu, u >= sup

u∈R^Dt,||u||≤1

X

i,j

C

_ij

u

_i

u

_j

≤ sup

u∈R^Dt,||u||≤1

X

i,j

C

ij

u

²_i

= sup

u∈R^Dt,||u||≤1

X

i

u

²_i

X

j:ϕiϕj6=0

µ(|ϕ

_i

ϕ

_j

|)

≤ sup

u∈R^Dt,||u||≤1

X

i

u

²_i

Φ

1

p

1

≤ Φ

1

p

1

. Using once more that

X

i

X

j

1

{ϕ_iϕj6=0}

≤ D

_t

· Φ

₁

, due to assumption 2.4.3 we conclude that

P

x

(Ω

^c_t

) ≤ C D

_t

t

^−n/2

≤ Ct

^−(n/2−1)

,

where C = K (n)Φ

ⁿ⁺¹₁

p

ⁿ₁

depends on n and coefficients of X, but does not

depend on t. •

4 Proofs of the main results.

4.1 Proof of Theorem 2.5.

The proof follows the lines of Comte et al. [5]. Recall that from the definition of γ

_t

it follows that on A

_t

,

ˆ b

_m

=

Dm

X

i=1

ˆ α

_i

ϕ

_i

, with random ˆ α = ( ˆ α

₁

, . . . , α ˆ

_Dm

)

^∗

satisfying

T α ˆ = 1 t

Z

t 0

ϕ(X

_s

)dX

_s

,

where T is the D

_m

×D

_m

random matrix and R

t

0

ϕ(X

_s

)dX

_s

is the D

_m

−dimensional random vector with elements

T

_ij

= 1 t

Z

t 0

ϕ

_i

(X

_s

)ϕ

_j

(X

_s

) ds,

Z

t 0

ϕ(X

_s

) dX

_s

=





 R

t

0

ϕ

₁

(X

_s

)dX

_s

.. . R

t

0

ϕ

_Dm

(X

_s

)dX

_s





 .

Observe that ˆ b

_m

is a F

_t

-measurable random variable with values in S

_m

. If for such a random variable

h(ω, x) =

Dm

X

i=1

α

_i

(ω)ϕ

_i

(x) we put

γ

_t

(h) = khk

²_t

− 2 t

Dm

X

i=1

α

_i

Z

t

0

ϕ

_i

(X

_s

) dX

_s

.

Then γ

_t

(h) − γ

_t

(ˆ b

_m

) ≥ 0 on A

_t

. This inequality is evidently valid for any basis of S

m

.

Finally, we define the risk of the estimator ˆ b

_m

as E

x

k ˆ b

_m

− b

_K

k

²_t

= E

x

1 t

Z

t 0

(ˆ b

_m

− b

_K

)

²

(X

_s

)ds

.

Let Ω

_t

be given by (3.1) and A

_t

given by (2.3). Recall that Ω

_t

⊆ A

_t

(Propo- sition 3.4.)

Now write

E

x

k ˆ b

_m

− b

_K

k

²_t

= E

x

k ˆ b

_m

− b

_K

k

²_t

1

_Ωt

+ E

x

k ˆ b

_m

− b

_K

k

²_t

1

_Ω^c

t

. We will treat separately the two terms on the right-hand side.

We start with the first one, recalling that Ω

_t

= Ω

_t

∩ A

_t

. In what follows it will be useful to use an orthonormal basis {ψ

₁

, . . . , ψ

_Dm

} of S

_m

viewed as a subspace of L

²

(K, dµ). Hence, our estimator can be rewritten as

ˆ b

_m

=

Dm

X

i=1

β ˆ

_i

ψ

_i

, and b

_m

=

Dm

X

i=1

β

_i

ψ

_i

.

Observe that a.s. on A

_t

0 ≤ γ

_t

(ˆ b

_m

) − γ

_t

(b

_m

) =

k ˆ b

_m

k

²_t

− kb

_m

k

²_t

− 2 t

Dm

X

i=1

( ˆ β

_i

− β

_i

) Z

t

0

ψ

_i

(X

_s

) (b(X

_s

) ds + σ(X

_s

) dW

_s

) =

T

_X

(ˆ b

_m

−b

_m

, ˆ b

_m

+b

_m

)−2T

_X

(ˆ b

_m

−b

_m

, b

_K

)− 2 t

Dm

X

i=1

( ˆ β

_i

−β

_i

) Z

t

0

ψ

_i

(X

_s

)σ(X

_s

) dW

_s

= k ˆ b

_m

− b

_K

k

²_t

− kb

_m

− b

_K

k

²_t

− 2

t

Dm

X

i=1

( ˆ β

_i

− β

_i

) Z

t

0

ψ

_i

(X

_s

)σ(X

_s

) dW

_s

, whence a.s. on A

_t

k ˆ b

_m

− b

_K

k

²_t

≤ kb

_m

− b

_K

k

²_t

+ 2

Dm

X

i=1

( ˆ β

_i

− β

_i

) 1

t Z

t

0

ψ

_i

(X

_s

)σ(X

_s

)dW

_s

. (4.1) Remark that P

Dm

i=1

( ˆ β

_i

− β

_i

)

²

= k ˆ b

_m

− b

_m

k

²_µ

. Using Cauchy-Schwartz in- equality we have

k ˆ b

_m

− b

_K

k

²_t

1

_Ωt

≤ kb

_m

− b

_K

k

²_t

1

_Ωt

+ 2

Dm

X

i=1

( ˆ β

_i

− β

_i

) 1

t Z

t

0

ψ

_i

(X

_s

)σ(X

_s

) dW

_s

1

_Ωt

≤ kb

_m

− b

_K

k

²_t

+ 1

8 k ˆ b

_m

− b

_m

k

²_µ

1

_Ωt

+8

Dm

X

i=1

1 t

Z

t 0

ψ

_i

(X

_s

)σ(X

_s

) dW

_s

2

. (4.2) Then on Ω

_t

1

8 k ˆ b

_m

− b

_m

k

²_µ

1

_Ωt

≤ 1

2 (k ˆ b

_m

− b

_K

k

²_t

+ kb

_m

− b

_K

k

²_t

)1

_Ωt

. Plugging this into (4.2) gives

k ˆ b

m

− b

K

k

²_t

1

Ωt

≤ 3kb

m

− b

K

k

²_t

+ 16

Dm

X

i=1

1 t

Z

t 0

ψ

i

(X

s

)σ(X

s

) dW

s

²

.

We have

E

x

k ˆ b

_m

− b

_K

k

²_t

1

_Ωt

≤ 3 t E

x

Z

t 0

(b

_m

− b

_K

)

²

(X

_s

)ds + 16σ

₁²

t

²

Dm

X

i=1

E

x

Z

t 0

ψ

_i²

(X

_s

) ds.

Using Proposition 3.1, we can write for any positive function f having support on K,

E

x

Z

t 0

f (X

_s

)ds ≤ κt Z

K

f dx,

where the constant κ is explicitly given in Proposition 3.1 and does only depend on the model constants b

₀

, σ

₀

, σ

₁

. Using this estimation, we obtain the following bound for the integrated risk restricted on Ω

_t

:

E

^x

k ˆ b

m

− b

K

k

²_t

1

Ωt

≤ 3κkb

m

− b

K

k

²

+ 16σ

₁²

κ p

₀

D

_m

t .

We now consider the risk restricted on Ω

^c_t

. Recall that A

^c_t

⊆ Ω

^c_t

and that ˆ b

_m

= 0 on A

^c_t

, and write

kb

_K

− ˆ b

_m

k

²_t

1

_Ω^c

t

= kb

_K

− ˆ b

_m

k

²_t

1

_Ω^c

t∩A_t

+ kb

_K

k

²_t

1

_A^c

t

(4.3)

Let ˜ b

_m

be the almost surely defined on Ω

^c_t

∩ A

_t

orthogonal projection of b

_K

onto S

m

w.r.t. k · k

t

. We have

kb

K

− ˆ b

m

k

²_t

1

Ω^c_t∩At

= kb

K

− ˜ b

m

k

²_t

1

Ω^c_t∩At

+ k ˜ b

m

− ˆ b

m

k

²_t

1

Ω^c_t∩At

≤ kb

_K

k

²_t

1

_Ω^c_t∩At

+ k ˜ b

_m

− ˆ b

_m

k

²_t

1

_Ω^c_t∩At

, which, combined with (4.3), implies

kb

_K

− ˆ b

_m

k

²_t

1

_Ω^c_t

≤ kb

_K

k

²_t

1

_Ω^c_t

+ k ˜ b

_m

− ˆ b

_m

k

²_t

1

_Ω^c_t∩At

. (4.4) Our Assumption 2.1.1 on b(x) yields

E

x

kb

_K

k

²_t

1

_Ω^c_t

≤ b

²₀

P

x

(Ω

^c_t

). (4.5) From the definition of ˜ b

m

it follows that ˜ b

m

= P

Dm

i=1

α ˜

i

ϕ

i

, with ˜ α satisfying T α ˜ = 1

t Z

t

0

ϕ(X

_s

)b(X

_s

)ds.

Recall that on A

_t

, ˆ b

_m

= P

Dm

i=1

α ˆ

_i

ϕ

_i

, with ˆ α given by (2.4). Hence on A

_t

, we can write ˆ α − α ˜ = T

⁻¹

M

_t

, where

M

_t

= 1 t

Z

t 0

ϕ(X

_s

)σ(X

_s

) dW

_s

=







1 t

R

t

0

ϕ

₁

(X

_s

)σ(X

_s

)dW

_s

.. .

1 t

R

t

0

ϕ

Dm

(X

s

)σ(X

s

)dW

s





 .

So on A

_t

we have ˆ b

_m

− ˜ b

_m

= ϕ

^∗

( ˆ α − α) = ˜ ϕ

^∗

T

⁻¹

M

_t

, where ϕ

^∗

= (ϕ

₁

, . . . ϕ

_Dm

), and (we denote by ∗ the matrix-transposition operation),

(ˆ b

_m

− ˜ b

_m

)

²

(X

_s

) = M

_t^∗

(T

^∗

)

⁻¹

ϕϕ

^∗

(X

_s

)T

⁻¹

M

_t

. So,

k ˜ b

_m

− ˆ b

_m

k

²_t

= 1 t

Z

t 0

(˜ b

_m

− ˆ b

_m

)

²

(X

_s

)ds =

M

_t^∗

(T

^∗

)

⁻¹

T T

⁻¹

M

_t

= M

_t^∗

(T

^∗

)

⁻¹

M

_t

=< T

⁻¹

M

_t

, M

_t

>, which gives, by the definition of A

_t

,

k ˜ b

_m

− ˆ b

_m

k

²_t

1

_Ω^c_t∩At

≤ 1

t

^−1/2

kM

_t

k

²

1

_Ω^c_t

= t

^1/2

Dm

X

i=1

1 t

Z

t 0

ϕ

i

(X

s

)σ(X

s

) dW

s

²

1

Ω^c_t

. (4.6) Using Burkholder-Davis-Gundy inequalities and the hypothesis kϕ

²_i

k

∞

≤ Φ

²₀

D

_m

, it follows from (4.6),

E

x

k ˜ b

_m

− ˆ b

_m

k

²_t

1

_Ω^c_t∩At

≤ t

^1/2

t

²

Dm

X

i=1

E

x

(

Z

t 0

ϕ

_i

(X

_s

)σ(X

_s

) dW

_s

)

²

1

_Ω^c_t

≤ t

^−3/2

Dm

X

i=1

s E

x

Z

t 0

ϕ

_i

(X

_s

)σ(X

_s

) dW s

4

P

x

(Ω

^c_t

)

≤ t

^−3/2

Dm

X

i=1

s

C(4) E

x

Z

t 0

ϕ

²_i

(X

_s

)σ

²

(X

_s

) ds

²

P

x

(Ω

^c_t

).

Here, C(4) is a Burkholder-Davis-Gundy constant. But Z

t

0

ϕ

²_i

(X

_s

)σ

²

(X

_s

) ds ≤ Φ

²₀

D

_m

σ

²₁

t,

hence

E

x

k ˜ b

_m

− ˆ b

_m

k

²_t

1

_Ω^c_t∩At

≤ p

C(4) t

^−3/2

Dm

X

i=1

q

Φ

⁴₀

D

²_m

σ

⁴₁

t

²

P

x

(Ω

^c_t

)

≤ p

C(4)σ

²₁

Φ

²₀

t

^−1/2

D

²_m

p

P

x

(Ω

^c_t

).

From (4.4) and (4.5) the integrated risk on Ω

^c_t

satisfies E

^x

kb

_K

− ˆ b

_m

k

²_t

1

_Ω^c

t

≤ b

²₀

+ Cσ

₁²

Φ

²₀

t

^−1/2

D

²_m

p

P

^x

(Ω

^c_t

)

≤ b

²₀

+ Cσ

₁²

Φ

²₀

t

^−1/2

D

_m²

p

P

x

(Ω

^c_t

). (4.7) As a consequence, since D

²_m

≤ t

²

, the full integrated risk satisfies

E

x

k ˆ b

_m

− b

_K

k

²_t

≤ 3κkb

_m

− b

_K

k

²

+ 16σ

₁²

κ p

₀

D

m

t + b

²₀

+ Cσ

₁²

Φ

²₀

t

^3/2

p

P

x

(Ω

^c_t

).

Finally, Proposition 3.5, applied with n = 12, yields P

^x

(Ω

^c_t

) ≤ C

t

⁵

for t ≥ t

₀

. This finishes the proof. •

Remark 4.1 In the case when X is in the stationary regime, i.e. starting from the invariant measure µ, (2.6) can be improved to

E

µ

k ˆ b

_m

− b

_K

k

²_t

≤ 3p

₁

kb

_m

− b

_K

k

²

+ 16σ

₁²

D

_m

t + Ct

⁻¹

.

4.2 Proof of Theorem 2.6.

Put

ν

_t

(f ) := 1 t

Z

t 0

f(X

_s

)σ(X

_s

)dW

_s

.

The same argument that yields (4.1) in the non-adaptive case gives for any m ∈ M

_t

,

|| ˆ b

_mˆ

− b

_K

||

²_t

1

_At

≤ ||b

_m

− b

_K

||

²_t

1

_At

+ 2ν

_t

(ˆ b

_mˆ

− b

_m

)1

_At

+ (pen(m) − pen( ˆ m)) 1

_At

.

(4.8)

Here, a special attention has to be paid to the term ν

_t

(ˆ b

_mˆ

− b

_m

), since it is not a priori clear that this stochastic integral is well-defined. On ˆ m = n, ˆ b

mˆ

− b

m

is an element of S

n

+ S

m

viewed as linear subspace of L

²

(K, µ).

Put k = dim(S

_n

+ S

_m

) and let {ψ

₁

, . . . , ψ

_k

} be an orthonormal basis of this subspace. Then 1

{m=n}ˆ

(ˆ b

_mˆ

− b

_m

) = 1

{m=n}ˆ

P

k

i=1

β ˆ

_i

ψ

_i

, and we define on ˆ

m = n,

ν

_t

(ˆ b

_mˆ

− b

_m

) :=

k

X

i=1

β ˆ

_i

ν

_t

(ψ

_i

).

Hence, ν

_t

(ˆ b

_mˆ

− b

_m

) is well-defined and linear. Thus we may write ν

_t

(ˆ b

_mˆ

−b

_m

) ≤ || ˆ b

_mˆ

−b

_m

||

_µ

·ν

_t

ˆ b

_mˆ

− b

_m

|| ˆ b

_mˆ

− b

_m

||

_µ

!

≤ || ˆ b

_mˆ

−b

_m

||

_µ

· sup

h∈Sm+S_mˆ,||h||µ=1

|ν

_t

(h)|.

Write for short

G

m

(m

⁰

) := sup

h∈S_m+S_m0,||h||_µ=1

|ν

_t

(h)|.

We now investigate (4.8). First, on A

_t

∩ Ω

_t

, using that 2ab ≤

¹₈

a

²

+ 8b

²

,

|| ˆ b

_mˆ

− b

_K

||

²_t

≤ ||b

_m

− b

_K

||

²_t

+ 2|| ˆ b

_mˆ

− b

_m

||

_µ

G

_m

( ˆ m) + [pen(m) − pen( ˆ m)]

≤ ||b

_m

− b

_K

||

²_t

+ 1

8 || ˆ b

_mˆ

− b

_m

||

²_µ

+ 8 G

²_m

( ˆ m) + [pen(m) − pen( ˆ m)]

≤ ||b

_m

− b

_K

||

²_t

+ 1 2

|| ˆ b

_mˆ

− b

_K

||

²_t

+ ||b

_K

− b

_m

||

²_t

+8 G

²_m

( ˆ m) + [pen(m) − pen( ˆ m)]

≤ 3

2 ||b

_m

− b

_K

||

²_t

+ 1

2 || ˆ b

_mˆ

− b

_K

||

²_t

+ 8 G

²_m

( ˆ m) + [pen(m) − pen( ˆ m)] . This yields finally, on A

t

∩ Ω

t

= Ω

t

,

|| ˆ b

_mˆ

− b

_K

||

²_t

≤ 3||b

_m

− b

_K

||

²_t

+ 16 G

²_m

( ˆ m) + 2 [pen(m) − pen( ˆ m)] . (4.9) Now, as in Comte et al. [5], put p(m, m

⁰

) := p(m) + p(m

⁰

), where

p(m) := χσ

₁²

D

m

t and where χ is a universal constant. Then

G

²_m

( ˆ m)1

Ωt

≤

(G

²_m

( ˆ m) − p(m, m))1 ˆ

Ωt

+

+ p(m, m) ˆ

≤ X

n∈Mt

(G

²_m

(n) − p(m, n))1

Ωt

+

+ p(m, m). ˆ