Parameter maximum likelihood estimation problem for time periodic modulated drift Ornstein Uhlenbeck processes

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Parameter maximum likelihood estimation problem for time periodic modulated drift Ornstein Uhlenbeck

processes

Dominique Dehay

To cite this version:

Dominique Dehay. Parameter maximum likelihood estimation problem for time periodic modulated drift Ornstein Uhlenbeck processes. Statistical Inference for Stochastic Processes, Springer Verlag, 2015, 18 (1), pp.69-98. �10.1007/s11203-014-9104-7�. �hal-00794615v2�

Parameter maximum likelihood estimation problem for time periodic modulated drift

Ornstein Uhlenbeck processes

Dominique Dehay

Institut de Recherche Mathématique de Rennes, CNRS umr 6625 Université Rennes 2

email : [email protected] december 2014

Abstract In this paper we investigate the large-sample behaviour of the maximum likelihood estimate (MLE) of the unknown parameterθ for processes following the model

dξ_t=θf(t)ξ_tdt+dB_t,

where f :R→ R is a continuous function with period, say P >0. Here the periodic function f(·)is assumed known. We establish the consistency of the MLE and we point out its minimax optimality. These results comply with the well-established case of an Ornstein Uhlenbek process when the function f(·) is constant. However the case when RP

0 f(t)dt = 0 and f(·) is not identically null presents some special features. For instance in this case whatever is the value of θ, the rate of convergence of the MLE is T as in the case when θ= 0 and R_P

0 f(t)dt6= 0.

Keywords Langevin stochastic differential equation . Time-inhomogeneous diffusion process . Periodicity . Ornstein Uhlenbeck process . Maximum likelihood estimator . Local asymptotic minimax bound .

Mathematics Subject Classification (2000) 62F12

1 Introduction

The non-stationary and seasonal behaviour is quite common for many random phenomena observed in time. This can be due to the influence of external oscillation forces acting on the system under study, or to some internal forces that exist within the system (rotating machinery, waves, cyclical phenomenae, seasonality, time-periodic modulation, etc.). The periodicity may be hidden in the structure of the process, for instance in the covariance structure for a non- stationary periodically correlated process called also cyclostationary signal in signal theory (Gardner et al. 2006), or as in the present paper, in the coefficients of a time-inhomogeneous diffusion process (see also Höpfner and Kutoyants 2010). Many applications of such models can be found in mechanics, communication theory, climatology, econometrics, biology to name but a few (see e.g. Antoni 2009 ; Chaari et al. 2014 ; Collet and Martinez 2008 ; Gardner et al.

2006 ; Serpedin et al. 2005). A large amount of publications have been devoted to discrete time linear models with coefficients which are periodically time-dependent (see e.g. Hurd and Miamee 2007, and references therein). In the present work we consider the continuous-time counterpart of the periodically autoregressive time series model of order 1 (PAR(1) model), introducing a time-periodic modulation in the drift of the Ornstein Uhlenbeck model.

The main purpose of the paper is the maximum likelihood estimation problem for the unknown parameter θ ∈ R in the so called P-periodic Langevin SDE (stochastic differential equation)

dξ_t=θf(t)ξ_tdt+dB_t, ξ₀, t≥0, (1) from the observation of a continuous sample path of the P-OU-process {ξ_t, t ≥ 0}. Here {B_t, t ≥ 0} is a Brownian motion which is independent with respect to the initial random variableξ₀. The time-dependent modulationf :R→R is some known, no identically null,P- periodic and continuous function. Thus the periodP > 0 is known. For simplicity of exposure we have assumed that the diffusion coefficient is equal to 1. The non-parametric problem of estimation of the modulation functionf(·) is out of the scope of the paper and will be subject to another work. Parameter estimation problem for models of SDE with time-dependent drift have been considered by many authors (for instance see Mishra and Prakasa Rao 1985 ; Liptser and Shiryaev 2001 ; and recently, Barczy and Pap 2010 ; Höpfner and Kutoyants 2010 ; Dehling et al. 2010). None considers the case of SDE (1) which cannot be reduced to known models.

Such SDE (1) admits a unique solution for which we exhibit a completely explicit expres- sion (6). This permits us to develop the forthcoming analysis exploiting the periodic structure of the model. In this paper a solution of SDE (1) is called P-periodic Ornstein Uhlenbeck type process and for brevity notedP-OU-process.

In Section 2 we present the main features about this model (Dehay 2014). These proper- ties help us to understand the structure of the model under consideration, and are used in the following sections. The diversity of the statements corresponds to the different recurrence properties of the solution{ξ_t, t≥0}of SDE (1). More precisely the model possesses a periodic Markov structure which can be described with the help of the associated P-segments chain

{X_n := {ξ_nP+t, t ∈ [0, P]}, n ∈ N} as defined by Höpfner and Kutoyants (2010). According to the signum of F(P) := RP

0 f(t)dt, the model has a recurrent or transient behaviour. The case whereF(P) = 0 is of particular interest, it corresponds to a null recurrent Markov model.

We refer to (Meyn and Tweedie 2009 ; Revuz 1984) for the notions of recurrent, transient and Harris recurrent Markov chains. For recurrent or transient continuous-time processes we refer to (Has’minskiˇı 1980 ; Revuz and Yor 1994).

Next in Section 3 we takle with the problem of estimation of the parameterθin SDE (1) by the maximum likelihood method, from the observation of a sample path of the process along the finite interval [0, T], as T → ∞. Liptser and Shiryaev (2001, Theorem 17.2) have given conditions for evaluating the bias and the quadratic error of the maximum likelihood estimator (MLE) of the parameterθfor a large class of models including theP-periodic Langevin SDE (1).

Although this result can be applied successfully to model (1) withf(·)≡1(Liptser and Shiryaev 2001, Theorem 17.3), to carry out any checks when f(·) is not constant requires untractable computations. In a recent paper Barczy and Pap (2010) have investigated the MLE for a time- inhomogeneous diffusion process given by the SDE

dξ_t=θf(t)ξ_tdt+σ(t)dBt, ξ₀, t≥0,

where f(·) and σ(·) are known continuous functions. However due to the periodicity of the function f(·) and so the periodic structure of the process in the present work, their results do not apply to our context except to the Harris recurrent case where θF(P) < 0. In Section 3, thanks to the asymptotic behaviour of theP-OU-processes, we establish the strong consistency of the MLEθb_T whatever is the value ofθF(P) (Theorem 1). We also study the limit law of the scaled deviationδ_T(θ) θb_T −θ

. The normalizing factor δ_T(θ) as well as the limit law are quite different according to the signum of θF(P) (Section 3.3). When θF(P)<0 or θF(P)>0 the results are plain generalizations of the classical case wheref(·)≡1(Theorems 2 and 5). When F(P) = 0, the limit law of the scaled deviation T θb_T −θ

coincides with the mixed Gaussian law of

R1

0 BudB⁰_u R1

0 B²_udu

up to a factor c(θ)>0whatever is the value of θ∈R,{B_t, t∈[0,1]}and {B⁰_t, t∈[0,1]} being two independent Brownian motions (Theorem 3), while when θ= 0 andf(·)≡1, the limit law of Tθb_T coincides with the law of

R1

0 BudBu

R1

0 B²_udu = B²₁−1 2R1

0 B²_udu.

Furthermore we investigate the case whenθ= 0 andf(·) is not identically null, establishing a link between the results for the two casesF(P) = 0 versusθ= 0and f(·)≡1 (Theorem 4).

Finally Section 4 is devoted to the optimality of the MLE in the sense of local asymptotic minimax property (see Ibragimov and Has’minskiˇı, 1981 ; Jeganathan 1995 ; Le Cam and Yang 1990 ; see also Hájek 1972 ; Le Cam 1969 and 1986). WhenθF(P)<0, the recurrence property

of the P-OU-process entails that the model is locally asymptotically normal (LAN), the scale being √

T, and the MLE is locally asymptotically minimax for a large class of loss functions (Theorem 6). In the case θF(P) ≥ 0, we establish that the model is locally asymptotically mixed normal (LAMN), and the MLEθb_T has also some local asymptotic optimality (Theorems 7 and 9). Hereδ_T(θ) =T whenθF(P) = 0, and δ_T(θ) =e^nθF(P) when θF(P)>0, T =nP +t, n∈N,t∈[0, P). Moreover the size of the small neighbourhoodU_θ ofθin the local asymptotic minimax risk (14) is proportional toδ_T(θ)⁻¹ so tends to 0 as T → ∞in contrast to the LAN case θF(P) < 0 where in relation (16) the size of U_θ does not depend on T. For the specific quadratic loss function, when θF(P) = 0 we deal with van Trees inequality to establish a Bayesian version of the Cramér-Rao lower bound (Gill and Levit 1995) and to state that the MLE is locally asymptotically minimax with the size of the small neighbourhoodU_θ that does not depend onT as in the LAN case θF(P)<0 (Theorem 8).

For an easier reading and understanding of the statements of the paper, the proofs are collected in Section 5.

2 Background : Time periodic Ornstein Uhlenbeck process

Here we present some useful properties of the solution of the following P-periodic Langevin SDE

dξ_t=f(t)ξ_tdt+dB_t, ξ₀, t≥0.

As the modulation function f : R → R is periodic and continuous, the usual conditions for the existence and the unicity of the strong solution of SDE when the initial value is fixed, are satisfied (Liptser and Shiryaev 2001). Moreover the strong solution can be expressed easily

ξ_t=e^F(t)

ξ₀+ Z t

0

e^−F(u)dBu

,

whereF(t) := Rt

0 f(u)du. Remark that as the modulation function f(·) isP-periodic,F(nP + t) =nF(P) +F(t) for any t >0 and any integern.

The process {ξ_t, t ≥ 0} is inhomogeneous Markovian whose transition probability density ps,t(x,·)coincides with the density of the normal lawN

xe^F(t)−F(s),Rt

se^{2(F(t)−F(u))}du

. Notice that the transition probability density is periodic in time : p_s+P,t+P(x, y) = p_s,t(x, y), for all s, t, x and y. Furthermore for each t∈[0, P]the chain{ξ_nP+t}_n∈

N is homogeneous Markovian, and can be decomposed as

ξ_nP+t = e^F(P)ξ_(n−1)P+t+e^F(P)+F(t) Z _P

t

e^−F(u)dB^((n−1)P_u ⁾+e^F(t) Z _t

0

e^−F(u)dB^(nP)_u

where the processes B^(nP) := {B_nP+u −B_nP, u ∈ [0, P]}, n ∈ N, are independent Brownian motions on[0, P]. Following Höpfner and Kutoyants (2010, Section 2) we define theP-segments sequence{X_n}_n∈

N,Xn:={ξ_nP+t, t∈[0, P]}. Then the previous decomposition can be rewritten as

X (t) =e^F(P)X (t) +e^F(P)+F(t)(Z (P)−Z (t)) +e^F(t)Z (t),

where

Z_n(t) :=

Z t 0

e^−F(u)dB^(nP)_u . (2)

The sequence (Xn)n∈N is a homogeneous Markov chain with state space C[0, P], the space of real-valued continuous functions defined in[0, P]endowed with the uniform distance. Moreover L[Z_n(t)] =N(0, G(t)), whereG(t) :=Rt

0e^−2F(u)du for anyt.

ForF(P)≤0, that ise^F(P)≤1, theP-segments Markov chain(Xn)n∈Nis Harris recurrent, more precisely there is a positive measure λ on (C[0, P],C_P) such that the sets of positive λ- measure are visited infinitely often whatever is the initial state X0 ∈ C[0, P]. Here CP denotes the Borelσ-field of the separable metric space C[0, P].

When F(P) < 0, this positive measure λcan be chosen as the limit probability law given in limit (3) below and it is invariant for the Markov chain (Xn)n∈N. Thus this Harris recur- rent Markov chain is positive, that is for any A ∈ C_P such that λ[A] > 0, the expectation of the amount of steps between two visits in A is finite (Meyn and Tweedie 1993, Theorem 10.4.9). Furthermore the Markov chain(Xn)n∈Nfulfils some ergodicity properties (Höpfner and Kutoyants 2010, Theorem 2.1) and

n→∞lim L[Xn] = lim

n→∞L ξ_nP+·

=Lh

e^F(·)(ζ₁+Z) i

inC[0, P]. (3) whereζ₁ is a real-valued Gaussian variableN

0, ^G(P)

e^−2F(P)−1

,Z:={Z(t), t∈[0, P]}is a Gauss- sian process with representation Z(t) = Rt

0e^−F(u)dB⁰_u, {B⁰_u, u∈ [0, P]} is a Brownian motion on [0, P], independent with respect to the random variableζ₁.

When F(P) = 0, then the Harris recurrent chain (X_n)n∈N admits a σ-finite (unbounded) invariant measure, so it is null Harris recurrent and for any A∈C_P such that 0< λ[A]<∞, the expectation of the amount of steps between two visits in A is infinite. Notice that in this case for each t∈[0, P], the Markov chain{ξ_nP+t, n∈N} is a random walk onR. Furthermore

n→∞lim L X√n

n

= lim

n→∞L

ξ_nP+·

√n

=Lh e^F(·)ζ₂

i

inC[0, P] (4) where ζ₂ is a real-valued Gaussian variable N(0, G(P)). Then ξ_T/√

T converges in law to the Gaussian lawN(0, G(P)/P).

IfF(P)>0, theP-segments chainXand theP-OU process{ξ_t, t≥0}are transient (Meyn and Tweedie 1993, Theorem 8.0.1 ; Has’minskiˇı 1980, Chap.IV.2 pp.113). Moreover

n→∞lim e^−nF(P)X_n= lim

n→∞e^−nF(P)ξ_nP+· =e^F(·)(ξ₀+ζ₃) a.e. in C[0, P] (5) where ζ₃ is a real-valued Gaussian variable N

0, _1−e^G(P−2F(P)⁾

, ζ₃ and ξ₀ being independent.

Thuse^−F(T)ξ_T converges a.e. toξ₀+ζ₃ asT → ∞.

3 Parameter estimation

Henceforth we deal with the parameter estimation problem for model (1). Consider a dif- fusion process observed on the interval [0, T]following the P-periodic Langevin SDE (1) with

parameterθ∈Rand with initial variableξ₀independent with respect to{B_t, t≥0}and on the parameterθfor simplicity of exposure. Then we turn to the problem of estimating the unknown parameter θ ∈ R on the basis of the observation {ξ_t, t ∈ [0, T]}. More precisely we study the asymptotic behaviour of the maximum likelihood estimator (MLE) ofθas the observation time T goes to infinity.

Throughout the following we assume that the modulation function f(·) is any periodic continuous function with period P >0, except for specified cases. Then the strong solution of theP-periodic SDE (1) exists, is unique and admits an explicit expression

ξ_t:=ξ_t^(θ)=e^θF(t)ξ₀+ Z t

0

e^{θ(F(t)−F(u))}dBu, (6)

for t ∈ [0,∞). Notice that ξ_t⁽⁰⁾ = ξ₀+ B_t. For simplication we write ξ_t for ξ^(θ)_t when there is no possibility of confusion. Next in the notations P_θ,E_θ and L_θ, the index indicates that we consider that the true value of the parameter is θ. We need also the following functions

G_θ(t) :=

Z t 0

e^−2θF(u)du and H_θ(t) :=

Z t 0

f(u)²e^2θF(u)du

for t∈[0,∞) and θ∈R.

3.1 MLE for Ornstein Uhlenbeck processes

First we recall some well-established results on the maximum likelihood parameter estima- tion for Ornstein Uhlenbeck processes. Assume that the modulation functionf(·)is identically equal to 1, f(·) ≡1. The problem of estimation of the parameter θ for this model have been subject to a very large amount of contributions. In this classical case the MLE ofθ is equal to

θb_T = RT

0 ξ_udξ_u RT

0 ξ²_udu =θ+ RT

0 ξ_udB_u RT

0 ξ_u²du

and its behaviour as T → ∞ is well-known (see e.g. Basawa and Scott 1983 ; Bishwal 2008 ; Brown and Hewitt 1975 ; Feigin 1979 ; Kutoyants 2004 ; Liptser and Shiryaev 1977).

(i) For θ < 0, the process {ξ_t, t ≥ 0} is positive recurrent, ergodic with invariant measure N 0,1/(2|θ|)

, and

Tlim→∞L_θh√

T θb_T −θi

=N (0,2|θ|). (ii) Forθ= 0, the process {ξ_t, t≥0} is null recurrent

Tlim→∞L₀h Tθb_Ti

=L

" R1 0 B_tdB_t R1

0 B²_tdt

#

=L

"

B²₁−1 2R1

0 B²_tdt

#

. (7)

(iii) For θ >0, the process is transient :|ξ_t| → ∞ with probability 1

Tlim→∞L_θ e^θT

√

2θ θb_T −θ

=L ν

ξ₀+ζ^(θ)

on {ξ₀+ζ^(θ) 6= 0}, where L[ν] = N(0,1) and L ζ^(θ)

=N 0,1/(2θ)

. The random variables ξ ,ν andζ^(θ) are independent.

3.2 MLE for P-OU processes

Now we consider the general P-periodic model (1). As the function f(·) is continuous and almost every sample path of the process{ξ_t, t≥0}={ξ^(θ)

t , t≥0} is also continuous we have Z T

0

f(u)²ξ_u²du <∞ P_θ-a.e.

for any θ∈Rand anyT >0. Thus P_θ

Z T 0

f(u)²ξ_u²du <∞

= P0

Z T 0

f(u)²(ξ₀+ Bu)²du <∞

= 1.

Thanks to (Liptser and Shiryaev 2001, Theorem 7.7 and formula (17.24)) the law P^T_θ of {ξ_t= ξ_t^(θ), t∈[0, T]}is absolutely continuous with respect to the lawP^T₀ of{ξ⁽⁰⁾

t =ξ₀+ B_t, t∈[0, T]}, and the log-likelihood ratio ofP^T_θ to P^T₀ verifies

Λ^(θ)_T =θ Z T

0

f(u)ξ_udξ_u−θ² 2

Z T 0

f(u)²ξ_u²du=θ Z T

0

f(u)ξ_udB_u+θ² 2

Z T 0

f(u)²ξ_u²du. (8) Since

Z T 0

f(u)²ξ²

udu >0 P_θ-a.e.,

the maximum likelihood estimator bθ_T ofθ is well defined and equal to

θb_T = RT

0 f(u)ξ_udξ_u RT

0 f(u)²ξ²_udu =θ+ RT

0 f(u)ξ_udB_u RT

0 f(u)²ξ_u²du. (9)

3.3 Consistency of the MLE Assume θ∈Rbe fixed. Let

Ut:=

Z t 0

f(u)ξ_udBu (10)

and let{F_t, t≥0}be the filtration generated byξ₀ and the Brownian motion{B_t, t≥0}. Then {U_t, t ≥ 0} is a zero-mean continuous martingale with respect to {F_t, t ≥ 0}. Its quadratic variation is equal to

J_t:=

Z _t

0

f(u)²ξ²_udu and

E_θ U_t²

= E_θ[J_t] = Z t

0

f(u)²E_θh ξ²_ui

du <∞.

The strong law of large numbers for martingales (see Liptser and Shiryaev 2001, Lemma 17.4) entails that

Tlim→∞J_T⁻¹UT = 0 P_θ-a.e. on {J_∞=∞}

where J∞ := lim_T→∞J_T. Hence to prove the strong consistency of the MLE θb_T it suffices to show that Pθ[J∞<∞] = 0. This is done thanks to the asymptotic results in Section 2.

Theorem 1 For anyθ∈R, the MLE bθ_T is strongly consistent

Tlim→∞bθ_T =θ P-a.e.

Hence the MLEθb_T is a consistent estimator of θ

Tlim→∞P_θh θb_T −θ

> i

= 0

for any > 0, whatever is the value of F(P). Actually, in the following we will see that this consistency is uniform with respect to θ varying in any compact set contained in {θ⁰ ∈ R : θ⁰F(P) < 0} or in {θ⁰ ∈ R : θ⁰F(P) > 0} if F(P) 6= 0, and in any compact subset of R if F(P) = 0, the function f(·) being not identically null.

Next we deal with the asymptotic behaviour of the deviationθb_T−θaccording to the signum of θF(P).

3.3.1 Case θF(P)<0

When θF(P) <0, the ergodicity and recurrent properties of the P-OU process {ξ_t, t≥0}

imply that the maximum likelihood estimator θb_T has a standard asymptotic behaviour. Thus the result complies with the well-known case of the stationary Ornstein Uhlenbeck processes.

Theorem 2 Let K− be any compact subset of R contained in{θ:θF(P)<0}. Then

Tlim→∞L_θhp

J_T θb_T −θi

=N(0,1)

and

Tlim→∞L_θh√

T θb_T −θi

=N 0, I(θ)⁻¹

(11) uniformly with respect to θ varying in K−. The Fisher information I(θ) is equal to

I(θ) := G_θ(P)H_θ(P) P e^−2θF(P)−1 + 1

P Z P

0

f(u)²e^2θF(u)G_θ(u)du >0. (12) 3.3.2 Case θF(P) = 0

WhenθF(P) = 0, the MarkovP-OU process{ξ_t, t≥0}has a null recurrent behaviour. The maximum likelihood estimatorθb_T is consistent. In the following theorem, we see that the rate of convergence isT and the asymptotic law of the scaled deviationT θb_T −θ

is not standard.

However using the random scale√

J_T instead ofT, the limit law is parameter free : it does not depend on the model.

Theorem 3 Assume that F(P) = 0 and the functionf(·) is not identically null. Then

Tlim→∞L_θhp

J_T θb_T −θi

=N(0,1)

and

Tlim→∞L_θh

T θb_T −θi

=L





P pG_θ(P)H_θ(P)

ζ q

R1 0 B²_udu





where the real-valued random variableζ is Gaussian N(0,1)and is independent with respect to the Brownian motion {B_t, t∈[0,1]}. The limits being uniform with respect to θvarying in any compact subset of R.

It is worth to notice that in Theorem 3 when F(P) = 0 and f(·) not identically null we introduce a Gaussian variable ζ independent with respect to the Brownian motion in order to define the limit law, whatever is the value ofθ∈R. While forθ= 0and f(·)≡1, soF(P)6= 0, we have the well-established limit (7) expressed with only one Brownian motion. With the following theorem, we point out that this limit (7) forθ= 0can be seen as a particular case of the more general case where f(·) is any periodic continuous non identically null function. We recall that here the basic model is theP-periodic Langevin SDE (1) and the MLEθb_T is defined by relation (9).

Theorem 4 Assume that θ= 0 and the functionf(·) is not identically null. Then

Tlim→∞L₀h Tθb_T

i

=L





pP H₀(P)−F(P)² H0(P)

ζ q

R1 0 B²_udu

+ F(P) 2H0(P)

(B²₁−1) R1

0 B²_udu





where the random variableζ is Gaussian N(0,1), and is independent with respect to the Brow- nian motion {B_t, t∈[0,1]}. Recall thatF(P) =RP

0 f(u)du andH₀(P) =RP

0 f(u)²du.

Thus, whenF(P) = 0and the functionf(·)is not identically null, we obtain the limit law stated in Theorem 3 with θ = 0. When the function f(·) is constant non null, Theorem 4 reduces to the classical usual limit (7).

3.3.3 Transient case : θF(P)>0

The following result generalizes the well-known case when f(·)≡1and θ >0. The random local scale√

JT gives a parameter free limit law.

Theorem 5 Let K₊ be any compact subset of R contained in{θ:θF(P)>0}. Then

Tlim→∞L_θhp

JT θb_T −θi

=N(0,1)

and

n→∞lim L_θh

K_θ(·)e^nθF(P) θb_nP+·−θi

=L

"

ζ ξ₀+ζ₃^(θ)

#

in C[0, P] (13) uniformly with respect to θ varying in the compact set K₊. The real-valued random variable ζ is Gaussian N(0,1) and ζ₃^(θ) is Gaussian N

0,_1−e^G−2θF(P)^θ(P)

. Moreover ξ₀, ζ and ζ₃^(θ) are inde- pendent. Here

Kθ(t) :=

s

Hθ(P)

e^2θF(P)−1+Hθ(t).

Notice that the function (θ, t) 7→ Kθ(t) is positive, continuous and Kθ(P) = e^θF(P)Kθ(0), and recall that F(nP +t) = nF(P) +F(t). Thus, denoting T_P := T modulo P, Theorem 5 implies thatKθ T_P

e^θF(T) θb_T −θ

converges in law to ζ/ ξ₀+ζ₃^(θ)

uniformly with respect to θ varying in the compact setK+ asT → ∞.

Wheneverξ₀ = 0, limit law (13) is the Cauchy law with density functionx7→ ^cθ

π(c²θ+x²) where c_θ =

q1−e^−2θF(P)

Gθ(P) .

4 Optimality

Now we takle the problem of optimality for the maximum likelihood estimator θb_T. When the size of the sample path observation T is large, we would like to know how good or optimal is the maximum likelihood procedure for the estimation of the parameterθ. In the following we denote byP^T_θ, the law of the observed process{ξ_t, t∈[0, T]}when the value of the parameter is θ. Recall that for simplicity we assume that the law of the random variableξ₀ does not depend on the parameterθ.

We consider here the asymptotic optimality in the sense of local asymptotic minimax lower bound of the risk of {θ¯_T}:={θ¯_T, T >0}for the estimation ofθ, that is

R_θ({θ¯_T}) := lim

U_θ lim inf

T→∞ sup

θ⁰∈U_θ

E_θ⁰

L δ_T(θ⁰)⁻¹(¯θ_T −θ⁰)

(14) whereθ¯_T is any statistic function of the observation {ξ_t, t∈[0, T]}. HereU_θ is some neighbou- rhood ofθ which decreases to{θ} in some way depending on the model,δ_T(θ) is some positive coefficient calledlocal scale, andL(·) is a loss function in R,L(·)∈ L,L being the set of Borel functions L : R → [0,∞) symmetric, continuous at 0 with L(0) = 0, and non-decreasing on [0,∞)(see Hájek 1972 ; Ibragimov and Has’minskiˇı 1981, pp.18–19 ; Le Cam and Yang 1990 ; see also Jeganathan 1995, pp.838). Notice that the set of discontinuity of any bounded loss function L∈ Lis at most countable. Clearly all functionsL(x) =|x|^p,p >0, as well asL(x) =1(x > a), a >0, belong to L. (Here1(x > a) denotes the indicator function.)

Let θ ∈ R be fixed, and let δ_T(θ) → 0 as T → ∞. Considering local optimality, denote by Λ^(θ,u)_T the log-likelihood ratio function of the law P^T_θ+uδT(θ) of

n

ξ_t^{(θ+uδT(θ))}, t∈[0, T] o

with respect to the law P^T_θ of

n

ξ_t^(θ), t∈[0, T] o

, these laws being equivalent. Thanks to relation (8), we have

Λ^(θ,u)_T := ln

dP^T_θ+

uδT(θ)

dP^T_θ

!

ξ^(θ)

=uδ_T(θ)UT −1

2u²δ_T(θ)²JT. (15) In the light of the previous results, it is not surprising to see in the following that the log- likelihood ratioΛ^(θ,u)_T has a quite different behaviour according to the signum ofθF(P).

4.1 Case θF(P)<0

When θF(P) <0, thanks to Theorem 2 we take the local scale δ_T(θ) := T^−1/2. Then the log-likelihood ratioΛ^(θ,u)_T can be decomposed as

Λ^(θ,u)_T =u∆^(θ)_T −1

2u²I(θ) +r_T(θ, u) where∆^(θ)_T :=T^−1/2UT is asymptotically normal

Tlim→∞L_θh

∆^(θ)_T i

=I(θ)^1/2N(0,1),

the Fisher information I(θ) is defined by relation (12) and P_θ− lim

T→∞r_T(θ, u) = 0

uniformly with respect toθvarying in any compact subset of{θ:θF(P)<0}. Thus the family of laws {P^T_θ :θF(P) <0},T >0, is LAN (locally asymptotically normal) at each point θ∈R such thatθF(P)<0(see e.g. Hájek 1972 ; Ibragimov and Has’minskiˇı 1981 ; Le Cam and Yang 1990). Then the local asymptotic lower bound result (Hájek 1972, Theorem 4.1 ; Ibragimov and Has’minskiˇı (1981), Theorem II.12.1) entails that

lim→0lim inf

T→∞ sup

|θ−θ⁰|<

Eθ⁰

h L

√

T(¯θ_T −θ⁰) i

≥ 1

√ 2π

Z

R

L

y I(θ)^−1/2

e^−y2/2dy (16) for any loss function L(·) ∈ L, and for any family {θ¯_T, T > 0} of estimators of θ, that is θ¯_T is measurable with respect to the observation {ξ_t, t ∈ [0, T]} (see also Le Cam and Yang 1990, Section 5.6 Theorem 1). Notice that in this case the small neighbourhood U_θ of θ in the expression (14) of the local asymptotic minimax risk, is {θ⁰ :|θ⁰−θ| ≤ } and its size 2 >0 does not depend on T. Besides Theorem 2 gives

Tlim→∞L_θh√

T θb_T −θi

=I(θ)^−1/2N(0,1)

uniformly with respect to θ varying in any compact subset of {θ : θF(P) < 0}. So thanks to the convergence result from Ibragimov and Has’minskiˇı (1981, Appendix I Theorem 8) the next statement follows immediately.

Theorem 6 Assume thatF(P)6= 0. Then the MLEθb_T is locally asymptotically minimax with local scale T^−1/2 at any θ such that θF(P) <0 and for any bounded loss functionL(·)∈ L, in the sense that its local asymptotic minimax risk Rθ(θ)b is equal to the lower bound right hand side of inequality (16)

lim→0 lim

T→∞ sup

|θ−θ⁰|<

E_θ⁰h L√

T(bθ_T −θ⁰)i

= 1

√2π Z

R

L

y I(θ)^−1/2

e^−y2/2dy.

Then for a loss function L(·) ∈ L whose growth as |u| → ∞ is lower than any one of the functionse^|u|, >0, we readily deduce that

b→∞lim lim

→0 lim

T→∞ sup

|θ−θ⁰|<

Eθ⁰

h b∧L

√

T(θb_T −θ⁰) i

= 1

√ 2π

Z

R

L

y I(θ)^−1/2

e^−y2/2dy.

Thus for the quadratic risk with L(u) =u², we obtain

b→∞lim lim

→0 lim

T→∞ sup

|θ−θ⁰|<

E_θ⁰h

b∧ T(bθ_T −θ⁰)²i

=I(θ)⁻¹.

4.2 Case F(P) = 0, f(·) non identically null

When F(P) = 0, in Lemma 2 we state that

Tlim→∞L_θ J_T

T² , U_T T

=Lh

I(θ),I(θ)^1/2ζi

inR² uniformly with respect toθvarying in any compact subset of R. Here the random Fisher information is defined byI(θ) := ^Gθ(P_P^)H₂^θ(P)R1

0 B²_udu, and the real-valued randomζis Gaussian N(0,1)independent with respect to the Brownian motion{B_t:t∈[0,1]}, so independent with respect to I(θ). Then the log-likelihood ratioΛ^(θ,u)_T can be expressed as

Λ^(θ,u)_T =u∆^θ_T −1

2u²I(θ) +r_T(θ, u)

where ∆^θ_T := T⁻¹UT converges in law to the mixed Gaussian law of I(θ)^1/2ζ, and r_T(θ, u) converges in probability to 0 as T goes to infinity. We deduce that the family of laws {P^T_θ : θF(P) > 0}, T > 0, has a likelihood ratio which is LAMN (locally asymptotically mixing normal) (see e.g. Davies 1985 ; Jeganathan 1982, Definition 3). Following (Jeganathan 1995, Theorem 8 ; Le Cam and Yang 1990, Section 5.6), for each family {θ¯_T} of estimators of θ the local minimax riskR_θ(¯θ) can be bounded asymptotically as follows

Mlim→∞lim inf

T→∞ sup

|θ−θ⁰|<M T⁻¹

E_θ⁰

L T θ¯_T −θ⁰

≥E



L





P pGθ(P)Hθ(P)

ζ q

R1 0 B²_udu







 (17) for any L ∈ L. Thanks to Theorem 3 and to (Ibragimov and Has’minskiˇı 1981, Appendix I Theorem 8), we can assert that

Theorem 7 Assume that F(P) = 0, and f(·) is not identically null. Then the MLE θb_T is locally asymptotically minimax with local scale T⁻¹ at any θ and for any bounded loss function L(·)∈ L, in the sense that

Mlim→∞ lim

T→∞ sup

|θ−θ⁰|<M T⁻¹

E_θ⁰h L

T

θb_T −θ⁰i

= E



L





P pGθ(P)Hθ(P)

ζ q

R1 0 B²_udu







.

Notice that in relation (17) and in Theorem 7, the small neighbourhood U_θ of θ in the expression (14) of the local asymptotic minimax risk, is {θ⁰ : |θ⁰ −θ| ≤ M T⁻¹}. Its size is proportional to T⁻¹ and tends to 0 as T → ∞. However when the loss function is quadratic (L(x) = x²), thanks to van Trees inequality (Gill and Levit 1995) we state the following local asymptotic optimality of the MLE θb_T with small neighbourhood U_θ with size > 0 as in the LAN case.

Theorem 8 Assume that F(P) = 0, f(·) is not identically null. Then for eachθ∈R

lim→0lim inf

T→∞ sup

|θ⁰−θ|<

E_θ⁰h

T² θ¯_T −θ⁰2i

≥ 2P²

G_θ(P)H_θ(P) (18) whereθ¯_T is any family of estimators ofθ. Furthermore the MLEθb_T is asymptotically minimax for the quadratic loss function in the sense that

b→∞lim lim

→0 lim

T→∞ sup

|θ⁰−θ|<

E_θ⁰h

b∧T² θb_T −θ⁰2i

= 2P²

G_θ(P)H_θ(P).

4.3 Case θF(P)>0

From Lemma 3 and the proof of Theorem 5, we know that I^(θ)(t) := lim

n→∞e^−2nθF(P)J_nP+t=K_θ(t)²

ξ₀+ζ₃^(θ)2

inL¹(P_θ) and

n→∞lim L_θh

e^−2nθF(P)JnP+t, e^−nθF(P)UnP+t

i

=Lh

I^(θ)(t),I^(θ)(t)^1/2ζ i

in R² the limits being uniform with respect to t ∈[0, P] and to θ varying in any compact subset of {θ:θF(P)>0}. The random variable ζ is GaussianN(0,1)and independent with respect to {I^(θ)(t) :t ∈[0, P]}. Thus taking the local scale δ_nP+t(θ) := e^−nθF(P), from equality (15) we obtain that

Λ^(θ,u)_nP+t=u∆^(θ)_nP+t−1

2u²I^(θ)(t) +rnP+t(θ, u)

where for any t ∈ [0, P], the law of ∆^(θ)_nP+t := e^−nθF(P)U_nP+t converges to the mixed law of I^(θ)(t)^1/2ζ, andr_nP+t(θ, u) converges in probability to 0 as n goes to infinity. Notice that the local scale depends on θ.

We deduce that for each t ∈[0, P], the sequence of families of laws {P^nP_θ ^+t: θF(P)> 0}, n≥1, has a likelihood ratio which is LAMN. Then for each sequence {θ¯_nP+t} of estimators of θ, the local asymptotic minimax risk (14) can be bounded as follows.

M→∞lim lim inf

n→∞ sup

|θ−θ⁰|<M e^−nθF(P)

Eθ⁰

h L

e^nθ0F(P) θ¯nP+t−θ⁰i

≥E

"

L ζ

K_θ(t) ξ₀+ζ₃^(θ)

!#

(19) for any t ∈ [0, P] and any L ∈ L, each estimator θ¯nP+t being measurable with respect the observation {ξ_s:s∈[0, nP+t]}. Thanks to Theorem 5 we can assert the following optimality of the MLE.

Theorem 9 Assume thatF(P)6= 0. Then for eacht∈[0, P], the MLEθb_nP+t is locally asymp- totically minimax with local scale e^−nθF(P) at anyθ such θF(P)>0, and for any bounded loss function L(·)∈ L, in the sense that

Mlim→∞ lim

n→∞ sup

|θ−θ⁰|<M e^−nθF(P)

E_θ⁰h L

e^nθ0F(P)

θb_nP+t−θ⁰i

= E



L





ζ

K_θ(t) ξ₀+ζ₃^(θ)







.

The previous minimax result in the case θF(P) > 0 concerns the subsequences {bθ_nP+t}_n, t∈[0, P], of the family of MLE{bθ_T, T >0}ofθ. We can deduce the local asymptotic minimax optimality for the whole family{bθ_T, T >0} as follows.

Letδ_T(θ) := K_θ(t)e^nθF(P)−1

whereT =nP+twithn∈Nandt∈[0, P). From the lower bound inequality (19) as well as the fact that the function(θ, t)7→K_θ(t) is positive continuous on {θ:θF(P)>0} ×[0, P), we obtain that

M→∞lim lim inf

T→∞ sup

|θ−θ⁰|<M δT(θ)

E_θ⁰

L δ_T(θ⁰)⁻¹ θ¯_T −θ⁰

≥E

"

L ζ

ξ₀+ζ₃^(θ)

!#

for any θ ∈ R such θF(P) > 0, any bounded loss function L(·) ∈ L and any family {θ¯_T} of estimators ofθ. Then Theorem 9 implies that the MLEθb_T is optimal in the sense that

Mlim→∞ lim

T→∞ sup

|θ−θ⁰|<M δT(θ)

Eθ⁰

h L

δ_T(θ⁰)⁻¹

θb_T −θ⁰ i

= E

"

L ζ

ξ₀+ζ₃^(θ)

!#

.

Remark that the limit law in relation (13) is not square-integrable. Hence when θF(P)>0, we cannot get a finite local asymptotic minimax lower bound for the quadratic risk with small neighbourhoodU_θ of type{θ⁰ :|θ⁰−θ| ≤M e^−nθF(P)}.

5 Proofs

5.1 Proofs of results in Section 3 5.1.1 Proof of Theorem 1

From the remarks before the statement of Theorem 1 it is only sufficient to prove that J∞=∞ a.e.. Recall that the functionf(·) is not identically null and notice that the quadratic variationJT increases to J∞ a.e. asT → ∞.

(i) First assume thatθF(P)<0. From Section 2 we know that in this case theP-segments Markov sequence {X_k} is positive Harris recurrent. Then the ergodic theorem (Höpfner and Kutoyants 2010, Theorem 2.1) and limit (3) entail that

Tlim→∞

JT

T = lim

T→∞

1 T

Z T 0

f(u)²ξ²_udu=I(θ)>0 Pθ-a.e.

whereI(θ) is defined in relation (12). This implies thatJ∞=∞ a.e.

(ii) Assume now that θF(P) = 0. For any n >0 we have J_nP ≥

Z P 0

f(u)²ξ_nP² _+udu.

In another hand limit (4) implies that

n→∞lim P_θ 1

n Z _P

0

f(u)²ξ_nP² _+udu≥a

= P

H_θ(P) ζ₂^(θ)2

≥a

for anya >0, and whereL ζ₂^(θ)

=N(0, G_θ(P)). As the functionf(·)is not identically null, we have G_θ(P)H_θ(P) >0 so the random variable ζ₂^(θ) is non-degenerate Gaussian and we readily obtain that J_nP converges in P_θ-probability to infinity as n → ∞. As J_nP increases to J∞, Pθ-a.e. asT → ∞, we deduce that Pθ[J∞<∞] = 0.

(iii) Finally assume that θF(P)>0. According to limit (5)

n→∞lim e^−2nθF(P) Z ·

0

f(u)²ξ_nP² _+udu=H_θ(·)

ξ₀+ζ₃^(θ)2

P_θ-a.e. in C[0, P] where ζ₃^(θ) is the sum of the P-a.e. convergent series P

ke^−kθF(P)Zk(P). Thus L ζ₃^(θ)

= N

0,_1−e^G_−2θF(P^θ(P ₎

and the real-valued random variables ξ₀ and ζ₃^(θ) are independent. Besides JnP can be expressed as

JnP =

n−1

X

k=0

Z P 0

f(u)²ξ_kP² _+udu=

n−1

X

k=0

e^2kθF(P)e^−2kθF(P) Z P

0

f(u)²ξ_kP² _+udu

for any positive integer n. Applying Toeplitz lemma on series convergence we obtain

n→∞lim e^−2nθF(P)JnP = Hθ(P) e^2θF(P)−1

ξ₀+ζ₃^(θ) 2

P_θ-a.e. inR. Sinceξ₀andζ₃^(θ)are independent andζ₃^(θ)is non-degenerate Gaussian, we havePθ

h

ξ₀+ζ₃^(θ) = 0 i

= 0. Hence J∞=∞ P_θ-a.e. This completes the proof of the theorem.

5.1.2 Proof of Theorem 2

Recall that J_T(bθ_T −θ) = U_T where {J_t, t ≥ 0} is the quadratic variation process of the martingale{U_t, t≥0} defined by relation (10). The recurrence property of the process{ξ_t, t >

0}, implies that J_T converges almost surely to I(θ)asT → ∞. Then the central limit theorem for integrales with respect to Brownian motion (see e.g. Feigin 1976 ; see also Kutoyants 2004, Theorem 1.19 ; Barzcy and Pap 2010) entails the convergence in law

Tlim→∞L_θhp JT

θb_T −θ

i

=N(0,1)

for anyθsuch thatθF(P)<0. To establish the uniform convergence with respect toθ, requires a little more computations concerning the asymptotic behaviour of JT/T.

Let n∈N andt∈[0, P]. Then JnP+t=

Z nP+t 0

f(u)²ξ²_udu=

n−1

X

k=0

Z P 0

f(u)²ξ²_kP+udu+ Z t

0

f(u)²ξ_nP² _+udu

Owing to expression (6) we readily obtain by induction the following decomposition of ξ_nP+u ξ_kP+u=e^θF(u)

e^kθF(P) ξ₀+Sk−1

+Z_k(u)

(20) where

S−1 := 0 and Sk−1:=

k−1

X

j=0

e^−jθF(P)Zj = Z kP

0

e^−θF(v)dBv

for any k≥1, andZj :=Zj(P)for j∈N. Thus we can writeJnP+t as J_nP+t = H_θ(P)

n−1

X

k=0

e^2kθF(P) ξ₀+Sk−12

+ Z P

0

f(u)²e^2θF(u)

n−1

X

k=0

Z_k(u)²du

+ 2

n−1

X

k=0

e^kθF(P) ξ₀+Sk−1

Z P

0

f(u)²e^2θF(u)Zk(u)du+ Hθ(t)e^2nθF(P) ξ₀+Sn−12

+ Z t

0

f(u)²e^2θF(u)Z_n(u)²du+ 2e^nθF(P) ξ₀+Sn−1 Z t

0

f(u)²e^2θF(u)Z_n(u)du

:= A₁+A₂+A₃+A₄+A₅+A₆. (21)

Using the facts that the processesZ_ns are independent, and each of them is zero-mean Gaussian with independent increments, var_θ[Z_k(u)] = G_θ(u) for u ∈ [0, P], we will establish in the following Lemma 1 that (A1 +A2)/n converges in P_θ-quadratic mean to P I(θ), and A3/n, converges to 0. With the same argument we can prove thatA₄/n,A₅/nand A₆/n converge to 0. Each convergence being uniform with respect to t∈[0, P]and toθin any compact subset of {θ∈R:θF(P)<0}.

For each intergern, theP-segments processZ_n={Z_n(t), t∈[0, P]}is a continuous martin- gale with respect to the filtration generated by the Brownian motionB^(nP)={B_nP+t−B_nP, t∈ [0, P]}, and Doob maximal equality for matingales entails that

E_θ

t∈[0,P]max Zn(t)²

≤2 Z P

0

e^−2θF(u)du= 2G_θ(u). (22) Thus we obtain that

Tlim→∞

JT

T =I(θ)

inP_θ-quadratic mean uniformly with respect toθvarying in any compact subset of{θ:θF(P)<

0}. Thanks to the uniform central limit theorem for integrales with respect to Brownian motion (Kutoyants 2004, Theorem 1.20) we deduce the assertions of the theorem.

To complete the proof of Theorem 2 we state the convergence of A /n,A /n and A /n.