Local Asymptotic Normality Complexity Arising in a Parametric Statistical Lévy Model

Abstract

We consider statistical experiments associated with a Lévy process observed along a deterministic scheme . We assume that under a probability , the r.v. , has a probability density function , which is regular enough relative to a parameter . We prove that the sequence of the associated statistical models has the LAN property at each , and we investigate the case when is the product of an unknown parameter by another Lévy process with known characteristics. We illustrate the last results by the case where is attracted by a stable process.

1. Introduction

This work is a part of an ambitious program consisting in the estimation of the parameter intervening in the stochastic differential equation driven by a known Lévy process :

These kinds of models are motivated by mathematical finance problems ([1]). In this context, the property of local asymptotic normality property (LAN) has become an important issue [2]. The LAN property is described as follows: a sequence of families of probabilities indexed by an open set is said to have the LAN property at each point with speed , if the sequence of probabilities localized around ,converges, in the sense of weak convergence of the associated likelihood processes, to a Gaussian shift ; see Section 2 for a precise definition. The LAN property allows to recover the so-called asymptotic Fisher information quantity . This quantity is crucial in any estimation procedure, since provides the lower bound of the variance of any estimator of .

The LAN property was investigated by Akritas [3] in models associated with Lévy processes observed continuously in time over the interval . He obtained the property under the assumption of differentiability, according to the parameter , of the characteristics of . With the same asymptotic, Luschgy [4] obtained the local asymptotic mixed normality (LAMN) property on models associated with semimartingales. As a notion, LAMN property is more general than the LAN because it allows the Fisher information quantity to be random. With the asymptotic , the estimation methods do not seem to be feasible in practice, for this reason, several recent works focused on discretized schemes, i.e., observations of the process along the discrete scheme

In practice, the most interesting case of the discretization path turns out to be relatively difficult. The classical case of a Brownian motion in (1) has been widely treated [5]. Clément and Gloter [6] studied the LAN for the model in (1), in the case where is a Lévy process attracted by a symmetric stable process with index . Aït-Sahalia and Jacod [7], Masuda [8], and Kawai and Masuda [9, 10] studied LAN property for the model in (1) in case of constant coefficients, i.e.,

Our investigation goes to same direction of Aït-Sahalia and Jacod [11], who studied the LAN property and the problem of estimation of the parameter involved in the model of a log-asset price , solution of (3) with being a standard symmetric stable process with index . Section 4 completes their situation in case where is a general stable process, eventually mixed. The last direction was initiated Rammeh [12] with observations according to random schemes for the scale model:where is a real unknown real parameter, and is a symmetrical standard -stable process. Rammeh showed that the LAN property always occurs, and his main arguments strongly rely to the linearity in to the fact that stable processes have the temporal scaling property and to the asymptotic behavior of the stable densities. Theorem 2 generalizes Rammeh’s results in the context of deterministic discrete scheme .

Because of the intricacy of the case (1), we first focus on the following model, which contains (5) and intercepts (1): we assume that for all , under , is a Lévy process, null at , such that its Lévy exponent is given by the so-called Lévy-Khintchine formula:where , and is a positive measure on which integrates .

For sake of clarity, we take is the open interval . As in the precited literature, we will assume the following.(i)The existence of densities , such that is regular enough,(ii)The convergence, as , of some integrals depending on .

Theorem 1 and Corollary 1 provide conditions ensuring the LAN property for the model (6), when the process is observed along the discrete scheme (3). Denoting , the logarithmic derivative of relative to , the asymptotic Fisher information quantity at each should satisfy

It is difficult to find Lévy processes fulfilling (7), and the reasons are numerous, for instance, the existence of the densities , the fact that they are not explicit in general, and their degeneracy as . For these reasons, Corollary 1 focuses on the linear dependance (4) of the characteristics relative to . In this case, we may assume, without loss of generality, that contains a reference value, 1 for example, and the value 0 is excluded in order to avoid trivialities. In this case, we only need to assume some regularities of the function and conditions of the kind (7) for . Let and be the logarithmic derivatives of . The asymptotic Fisher information quantity should then satisfy

The case of the discretization with constant path is quite obvious since the scale model (5) becomes a regular i.i.d. one, that is, to say is finite and nonnull. If , the situation is more intricate because degenerates when . It turns out that even the linear model (5) is falsely simple to handle. Intuitively, one looks at special Lévy processes attracted by stable processes on the sense of (10). The price to pay is to exhibit refined controls on the probability density function of . In a second step, we restrict our attention to the scale model (5). For simplicity’s sake, it is easier in this case to express the probabilities in the form (5) rather than considering them as solutions of martingale problems associated with the family of characteristics because of the intricacy inherent in the truncation functions [13]. Generic examples of Lévy processes are stable processes. They characterized Lévy exponent as follows. Let and

A stable process, with parameters , is a Lévy process , such that the corresponding Lévy exponent is given by

The parameter is the stability coefficient, is the skewness coefficient, is the scale coefficient, and is the drift parameter. The corresponding triplet of characteristics is given by

See [14]. In model (6), a candidate for the unknown parameter could be any the parameters , or . Since stable processes enjoy the scaling property, with ,then a candidate for the unknown parameter in the model (5) is clearly the parameter . For more account on Lévy processes, the reader is referred to [13] or [15] and for stable distributions, we suggest [16] and also [17]. Section 4 provides nontrivial examples of LAN models associated with Lévy processes attracted by stable ones. That means that there exist measurable functions and a nondegenerate distribution , such that

In [18], Far focused on the LAMN property for the model (5) discretized along the scheme , when the process is of the form , the sum of a standard Brownian motion and an independent compound Poisson process. She obtained LAMN property under the condition that the Lévy measure of has no diffuse singular part and that if is absolutely continuous, then the model has the LAN property. Our development in Section 5 constitutes a complement to Corollary 1 for the scale model (5) and also to Far’s work [18] and illustrates how to build a LAN scale model from another LAN scale model.

2. Definition of the LAN Property

In Section 3, we provide some theoretical results on models associated with observations, at times , of the process , and to illustrate by some examples. To this end, we consider the sequence of i.i.d. random variables and the family of -fields:

Denoting and , we introduce the sequence of filtered statistical models:

For any fixed , we denoteand we introduce the statistical experiments localized around :where the last statistical experiment is a Gaussian Shift. By a Gaussian shift, we mean, that for all is the unique probability on equivalent to on each and that its associated likelihood process is the geometric Brownian motion defined bywhere is a Wiener process, and then, under , the process is again a Wiener process. The quantity is called the asymptotic Fisher information quantity; it is a positive constant related to the sequence of statistical experiments in (17) and has to be determined. The asymptotic Fisher information quantity is crucial in any estimation procedure. Indeed, under the LAN property, is the lower bound of the variance of any estimator of . More precisely, HAJEK’s asymptotic convolution theorem says that if satisfiesthen the distribution is the convolution product , where and is a probability measure on . See [19] for more.

Local asymptotic normality of the sequence of models in (17), in a value , is actually equivalent to the weak functional convergence in time of the sequence of statistical experiments to the Gaussian shift in (17). This fact is explained as follows: let and be the likelihood processes defined, for all and at each time , bywith the convention , if . According to [5], the likelihood process of the statistical experiment is represented by

The notion of weak functional convergence in time was introduced by Lecam [2] and developed by Strasser [19] and Jacod [20]. It is expressed as follows: for every finite subset of , and every , we havein the sense of the weak convergence for the Skorohod topology.

3. When Does LAN Property Hold for Lévy Models?

Our aim is to give sufficient conditions on the p.d.f. of under , ensuring the LAN property for the sequence of filtered statistical models .

3.1. LAN Property for the Model (6)

In this section, we will consider the model (6). If , integrates , and (respectively, ), then the support of the distribution of is

In all other cases, the distribution of has a support equal to . There are many situations in which for all has a probability p.d.f. which is infinitely differentiable in . For instance, the latter holds iffor any and for some and some , see [21]. Later on, we may assume the following:

We denote and we define, on the interior of , the following functions:

When the number appears, it is always understood that is big enough so that and are in . For all , and , we denote

For statisticians, is a familiar quantity and corresponds to the Fisher information quantity at stage . The quantity is less intuitive; it is a localized quantity around the true value and corresponds to the rest of Taylor approximations at the order 1 of Hellinger integrals of the model.

We are now able to state our first result, that is, the LAN property for the model (6).

Theorem 1. Assume (H0) and the following conditions:

Then, the sequence of sequence of filtered statistical models (15), corresponding to (6), has the LAN property at with the speed and the asymptotic Fisher information quantity .

Remark 1. (i)Cauchy–Schwarz inequality gives , and both conditions (H1) and (H2) are implied by(ii)Under different conditions and a different proof, Masuda obtained ([8], Theorem 2.12) the same conclusion as in Theorem 1.

Genon-Catalot and Jacod [5] exhibited discretized models according to random sampling schemes associated with a diffusion process driven by Brownian motions (with coefficients dependent on and by an homogeneous way on ) and proved the LAMN property under conditions similar to (H0), that is, differentiability to the third order relative to and integrability of the densities of the processes. Their proofs have a general vocation in the sense that they only use the Markovian property of the processes and are based on a method of approximation of the log-likelihood. Because of the intricate form 16 of the likelihood processes, we show the weak functional convergence of to via the convergence of the Hellinger processes, and with a tool, one can find in [20].

Proof of Theorem 1. Fix . The Hellinger process of order between and , relative to , is deterministic and has the formAccording to ([20], Theorem 5.3), it is enough to show that the Hellinger processes between and , relative to , satisfy the following: there exists , such that for every , , and , and the convergence in lawholds. We will use this method because in our framework, the processes are also deterministic and have the following quite simple form one can find in [5]: with being the integer part of , we have(1) For , , take , and observe thatAccording to (32) and (33), it is enough to show that , and , and we have(2) Assume (H0), (H1), and (H2) for a fixed . Applying Taylor expansion at the first order of for and big enough, we get for , the representation of on :where for all , the functionsare defined on the interior of . Also, observe the following relations:Because of (38) and (39), one hasUsing (36), finally write(3) For all and large enough, so that , writeThen, use (39) and obtainAccording to (41), we havewhere(4a) We will now prove the following convergence that will imply (35): for all ,(4b). To prove (46), we use both (26) and (37), and for all and , we have the representation and the control.Let , and let . Using (26), (37), and Taylor expansion at the first order of , we obtain the galloping control, valid for all :Taking in (48), applying Cauchy–Schwarz inequality and assuming (H1) and (H2), we obtainWe then can writeand clearly (50) implies (46).
(4c). To prove (47), we use same the same arguments as in (4b), the Taylor expansion at the first order of , and the representationBy (48) and Cauchy–Schwarz inequality, we have the following control, valid for all :The latter impliesUsing (48), we also haveand reproducing the method, we used to get (54), and we obtainAccording to (38) and (54), we also haveFurthermore, (38), (48), and Cauchy–Schwarz inequality implyFinally, according to (45), (54), (56)–(58), we obtain the controland we conclude with the fact that assumptions (H1) and (H2) imply (47).

3.2. LAN Property for the Scale Model (5)

As a consequence of Theorem 1, we obtain a result for the scale model (5). The main argument is that the r.v. has a p.d.f under if and only if has the p.d.f. under . From now on, the functions , and denote, respectively, , the p.d.f. of under , the first and the second derivatives of . For and , we have the expression

Therefore, if we want (H0) to be satisfied, we need to impose . Then, for all , we havewhere the functions and are given by

Making a change of variables, we see that the quantities and , defined in (27), satisfy

Using Cauchy–Schwarz inequality and again a change of variables, we obtain the following: for all , the quantities , defined in (28), satisfy

Corollary 1. For the scale model (5), the sequence of statistical models has the LAN property with speed at each if the following conditions are satisfied:In this case, the asymptotic Fisher information quantity is .

Proof. It is immediate that (C0) implies (H0). Representation (63) shows that (C1) implies (H1) and (64) shows that (C2) implies (H2).

Remark 2. Since is the p.d.f. of , then we should be aware that the discretization path intervene in the assumptions (C1) and (C2). In Theorem 2, we will see that in some favorable cases, the effect of the path has a quite negligible effect. In general, the LAN property depends strongly on the asymptotic of .

4. Examples of LAN Property for Lévy Processes Attracted by Stable Processes

In this section, we provide some examples of Lévy processes satisfying the conditions of Corollary 1. This corollary provides the LAN property for the scale model (5), , under regularity and integrability conditions on the p.d.f. of . Unfortunately, these p.d.f., if they exist, are not explicit in general; for this reason, we focus on processes which belong to the domain of attraction of stable processes.

We recall that a process is stable, if it is a Lévy process with Lévy exponent given in (10). In the case and , the process is a subordinator, i.e., a positive increasing the Lévy process, and the distribution of , has a support equal to . In all cases, , has a p.d.f. which is infinitely differentiable and is explicit only for the particular values . These values correspond, respectively, to the first passage times of the Brownian motion, the Cauchy process, and the Brownian Motion. Otherwise, is be expressed only as the inverse Fourier transform of . As announced, the introduction, we focus in this section on the case where the Lévy processes , in the scale model (5), is attracted by a stable process, i.e., satisfying (13). Bertoin and Doney [22] showed that if (10) holds, then the process is attracted by a stable one, and then, the following holds:(i)There exists , , and a slowly varying function (i.e., a function satisfying ), such that and (ii), where is the p.d.f. of some stable r.v.