Marie Du

Roy De Chaumaray

WEIGHTED LEAST SQUARES ESTIMATION FOR THE SUBCRITICAL HESTON PROCESS

Keywords: 2000 Mathematics Subject Classification. 62M05, 60G20, 60G44 Squared radial Ornstein-Uhlenbeck process, Heston process, parameters estimation, weighted least squares estimates, consistency, asymptotic normality

We simultaneously estimate the four parameters of a subcritical Heston process. We do not restrict ourself to the case where the stochastic volatility process never reaches zero. In order to avoid the use of unmanageable stopping times and natural but intractable estimator, we propose to make use of a weighted least squares estimator. We establish strong consistency and asymptotic normality for this estimator. Numerical simulations are also provided, illustrating the good performances of our estimation procedure.

Introduction

Introduced in 1973, as an hedging tool, the Black-Scholes model uses a geometric Brownian motion to represent asset prices. The implied volatility is supposed to be constant over time, which turned out to be inaccurate to fit real market data, especially during the crash in 1987, see [START_REF] Stein | Overreactions in the options market[END_REF]. Several alternative models have been constructed to take into account the so-called smile effect associated to deep in-the-money or out-of-money options. A particular attention has been drawn to the study of stochastic volatility processes in which the volatility is also given as a solution of some stochastic differential equation, see [START_REF] Stein | Stock price distributions with stochastic volatility: an analytic approach[END_REF], [START_REF] Lewis | Option valuation under stochastic volatility[END_REF] and [START_REF] Gatheral | The Volatility Surface: A Practitioner's Guide[END_REF] for financial accuracy. Among them, Heston process [START_REF] Heston | A closed-form solution for options with stochastic volatility with applications to bond and currency options[END_REF] is one of the most popular, due to its computational tractability. For example, [START_REF] Lee | Option pricing by transform methods: Extensions, unification, and error control[END_REF] easily computes call option prices using Fourier inversion techniques. Numerous results about the asymptotic volatility smile can be found in the very recent literature: see e.g. [START_REF] Forde | The large-maturity smile for the Heston model[END_REF], [START_REF] Forde | The small-time smile and term structure of implied volatility under the Heston model[END_REF], [START_REF] Jacquier | Large-maturity regimes of the Heston forward smile[END_REF].

We denote by Y t the logarithm of the price of a given asset and by X t its instantaneous variance, and we consider the following Heston process (1.1) # dX t " pa `bX t q dt `2? X t dB t dY t " pα `βX t q dt `2? X t ´ρ dB t `a1 ´ρ2 dW t with a ą 0, pb, α, βq P R 3 and ρ Ps ´1, 1r, where pB t , W t q is a 2-dimensional standard Wiener process and the initial state px 0 , y 0 q P R `ˆR. In this process, the volatility X t is driven by a generalized squared radial Ornstein-Uhlenbeck process, also known as the CIR process, firstly studied by Feller [START_REF] Feller | Two singular diffusion problems[END_REF] and introduced in a financial context by Cox, Ingersoll and Ross [START_REF] Cox | A theory of the term structure of interest rates[END_REF] to compute short-term interest rates. The asymptotic behavior of this process has been widely investigated and depends on the values of both coefficients a and b.

Once a model has been chosen for its realistic features, it needs to be calibrated before being used for pricing. Our goal in this paper is to estimate parameters pa, b, α, βq at the same time using a trajectory of pX t q and pY t q over the time interval r0, T s. Azencott and Gadhyan [START_REF] Azencott | Accurate parameter estimation for coupled stochastic dynamics[END_REF] developed an algorithm to estimate some parameters of the Heston process based on discrete time observations, by making use of Euler and Milstein discretization schemes for the maximum likelihood. However, in the special case of an Heston process, the exact likelihood can be computed. It allows us to construct the maximum likelihood estimator (MLE) without using sophisticated approximation methods, which is necessary for many stochastic volatility models, see [START_REF] Aït-Sahalia | Maximum likelihood estimation of stochastic volatility models[END_REF]. The MLE of pa, b, α, βq has been recently investigated in [START_REF] Barczy | Asymptotic properties of maximum-likelihood estimators for Heston models based on continuous time observations[END_REF], together with its asymptotic behavior in the special case where a ě 2. Denote by τ 0 the stopping-time given by

(1.2) τ 0 " inf " T ą 0 ˇˇˇż T 0 X ´1 t dt " 8 * .
For any a ą 0, the MLE r θ T " ´r a T , r b T , r α T , r β T ¯is given, for T ă τ 0 , by:

(1.3) r θ T " ˆG´1 T 0 0 G ´1 T ˙ˆr U T r V T ẇhere r U T " ´şT 0 X ´1 t dX t , ş T 0 dX t ¯⊺, r V T " ´şT 0 X ´1 t dY t , ş T 0 dY t ¯⊺ and G T " ˜şT 0 X ´1 t dt T T ş T 0 X t dt ¸.
One can observe that pr a T , r b T q coincides with the MLE of the parameters pa, bq of the CIR process based on the observation of pX T q over the time interval r0, T s. The asymptotic behavior of this latter estimator is well-known, see for example [START_REF] Fournié | Application de la statistique des diffusions à un modèle de taux d'intérêt[END_REF], [START_REF] Overbeck | Estimation for continuous branching processes[END_REF] and [START_REF] Ben Alaya | Asymptotic Behavior of The Maximum Likelihood Estimator For Ergodic and Nonergodic Square-Root Diffusions[END_REF]. In the supercritical case b ą 0, Overbeck [START_REF] Overbeck | Estimation for continuous branching processes[END_REF] has shown that r b T converges a.s. to b whereas there exists no consistent estimator for a. Hence, we will focus our attention on the geometrically ergodic case b ă 0. Furthermore, the value of a governs the behavior at zero of pX T q: for a ě 2, the process almost surely never reaches zero , whereas for 0 ă a ă 2, zero is quite frequently visited and

(1.4) P pτ 0 ă 8q " 1, see for instance [START_REF] Lamberton | Introduction au calcul stochastique appliqué à la finance[END_REF] or [START_REF] Overbeck | Estimation for continuous branching processes[END_REF]. For a ą 2, the MLE converges a.s. to θ " pa, b, α, βq and satisfies the following Central Limit Theorem (CLT)

? T ´r θ T ´θ¯L Ý Ñ N p0, 4R b Σ ´1q
where the matrix R and Σ are respectively given by

R " ˆ1 ρ ρ 1 ˙and Σ " ˆ´b a´2 1 1 ´a b ˙,
and b stands for the Kronecker product.

A large deviation principle for the couple pr a T , r b T q was recently established in [START_REF] Du Roy De Chaumaray | Large deviations for the squared radial Ornstein-Uhlenbeck process[END_REF]. In the particular case where one parameter is known and the other one is estimated, large deviations can be found in [START_REF] Zani | Large deviations for squared radial OrnsteinUhlenbeck processes[END_REF], while moderate deviations are given in [START_REF] Gao | Moderate deviations for squared Ornstein-Uhlenbeck process[END_REF].

By contrast, in the case where 0 ă a ă 2, (1.4) implies the non-integrability of X ´1 T for large values of T so that the MLE does not converge for T going to infinity. Consequently, this case has been less investigated even though it is often of interest in finance, to compute long dated interest rates for instance, as explained in [START_REF] Anderson | Simple and efficient simulation of the heston stochastic volatility model[END_REF], or in FX-markets, see [START_REF] Janek | FX smile in the Heston model[END_REF]. In the case of the CIR process, Overbeck [START_REF] Overbeck | Estimation for continuous branching processes[END_REF] used accurate stopping times to build a strongly consistent estimator based on the MLE:

(1.5) 1l T ăτ 0 ˆr a T r b T ˙`1l τ 0 ďT ¨lim tÒτ 0 S t Σ ´1 t ´şT 0 X s ds ¯´1 ˆXT ´T lim tÒτ 0 S t Σ ´1 t ˙‹ '
where S t " ş t 0 X ´1 s dX s , Σ t " ş t 0 X ´1 s ds and τ 0 is given by (1.2). The aim of this paper is to investigate a new strongly consistent weighted least squares estimator (WLSE) for the quadruplet of parameters θ (and for pa, bq as a consequence). The weighting allows us to circumvent the explosion for X T reaching zero and consequently avoid us to make use of stopping times, which are not easy to handle in practice. It generalizes to continuous time the original work of Wei and Winnicki [START_REF] Wei | Estimation of the means in the branching process with immigration[END_REF] for branching processes with immigration, inspired by an analogy with first order autoregressive processes. Our results answer, by the way, the question of Ben Alaya and Kebaier in the conclusion of [START_REF] Ben Alaya | Asymptotic Behavior of The Maximum Likelihood Estimator For Ergodic and Nonergodic Square-Root Diffusions[END_REF] regarding the CIR process.

Following the seminal work of [START_REF] Wei | Estimation of the means in the branching process with immigration[END_REF], denote C T " X T `c where c is some positive constant. Our new couple of weighted least squares estimator is given by

(1.6) p θ T " ˆΓ´1 T 0 0 Γ ´1 T ˙ˆU T V T ẇhere U T " ´şT 0 1 Ct dX t , ş T 0 Xt Ct dX t ¯⊺, V T " ´şT 0 1 Ct dY t , ş T 0 Xt Ct dY t ¯⊺ and Γ T " ˜şT 0 1 Ct dt ş T 0 Xt Ct dt ş T 0 Xt Ct dt ş T 0 X 2 t Ct dt ¸.
We do not restrict ourselves to the case where c " 1 as it may lower sometimes the variance of the estimators. In the particular case where c " 0, one can observe that the new estimator coincides with the MLE.

The paper is organized as follows. The second section contains our main results: the strong consistency of this new couple of estimators as well as its asymptotic normality. The third section deals with a comparison with the MLE, while the remaining of the paper is devoted to the proofs of our main results, as well as their illustration by some numerical simulations.

Main results

Our main results are as follows.

Theorem 2.1. Assume that a ą 0 and b ă 0. Then, the four-dimensional WLSE p θ T is strongly consistent: for T going to infinity,

(2.1) p θ T a.s. Ý Ý Ñ θ.
For T going to infinity, X T converges in distribution to a random variable X with Gamma Γpa{2, ´b{2q distribution, see Lemma 3 of [START_REF] Overbeck | Estimation for continuous branching processes[END_REF] for instance. Additionally, we denote by C the limiting distribution of X T `c, as T goes to infinity. Theorem 2.2. Assume that a ą 0 and b ă 0. Then, for T going to infinity, the estimator p θ T satisfies the following CLT

(2.2) ? T ´p θ T ´θ¯L Ý Ñ N p0, 4Λq ,
where the asymptotic variance Λ is defined as a block matrix by

(2.3) Λ " ˆALA ρALA ρALA ALA ˙,
with the matrix A and L respectively given by

A " pE rCs E r1{Cs ´1q ´1 ˆE rX 2 {Cs ´E rX{Cs ´E rX{Cs E r1{Cs ˙ and L " ˆE rX{C 2 s E rX 2 {C 2 s E rX 2 {C 2 s E rX 3 {C 2 s ˙.
We deduce from the previous theorems the following result for the MLE of the two parameters of the CIR process pX T q.

Corollary 2.1. Assuming that a ą 0 and b ă 0, the WLSE pp a T , p b T q of parameters pa, bq is strongly consistent for T going to infinity,

ˆp a T p b T ˙a.s. Ý Ý Ñ ˆa b ˙.
and satisfies the following CLT

? T ˆp a T ´a p b T ´b ˙L Ý Ñ N p0, 4ALAq . Remark 2.1.
In the remaining of this paper, we denote

(2.4) ψ c " ˆ´bc 2 ˙a{2 e ´bc{2 Γ p1 ´a{2, ´bc{2q ,
where Γ is the upper incomplete gamma function defined for all y P R and α P R `by Γpα, yq " ż `8 y e ´tt α´1 dt, and extended, for y ‰ 0, to any real α by holomorphy. To simplify following expressions, we also define

(2.5) ϕ c " ψ c ´1 ´a bc ¯´1.
In the proof of Theorem 2.2, we evaluate the two matrices A and L involving c and we obtain that

(2.6) A " ϕ ´1 c ˆc pψ c ´1q ´a b ψ c ´1 ψ c ´1 ψc c ȧnd (2.7) L " 1 2
¨a c ψ c `b p1 ´ψc q pa `2 ´bcq p1 ´ψc q ´a pa `2 ´bcq p1 ´ψc q ´a c pψ c ´1q pa `4 ´bcq `ac ´2a b '.

By a straightforward computation, we deduce that ALA " pϕ c q ´2 ˆσ11 σ 12 σ 12 σ 22 ˙where the variances σ 11 and σ 22 are respectively given by

σ 11 " a b pψ c ´1q 2 ´a2 2b ϕ c , σ 22 " ϕ c ˆψc c ´b 2 ˙`ψ c c pψ c ´1q ,
and the covariance σ 12 is given by σ 12 " pψ c ´1q 2 ´a 2 ϕ c .

Remark 2.2. For c going to zero (for which we need a to be greater than 2) , we obtain the same covariance matrix than for the MLE. Indeed, using well-known asymptotic results about the incomplete Gamma function Γ, which could be found in [START_REF] Luke | The Special Functions and their Approximations[END_REF], we have that, as soon as a ą 2,

(2.8) Γ p1 ´a{2, ´bc{2q ˆ´bc 2 ˙a{2´1 ÝÑ cÑ0 ´1 1 ´a{2 " 2 a ´2.
Thus ψ c goes to zero for c tending to zero, ψc c converges to ´b a´2 and ϕ c tends to 2 a´2 . Hence, we easily obtain that, for c going to zero,

σ

Asymptotic variance

Even though we considered the weighted least squares estimators in order to investigate the case 0 ă a ă 2 for which the MLE is not consistent, it is interesting to compare the asymptotic variances in the CLT of this new estimators and of the MLE, in the case where a ą 2. This comparison requires a lot of technical calculation as the asymptotic variances depends on the value of a, b and c. However, it is quite easy to compare variances for the MLE of the parameters of the CIR process in the case where we suppose one of the parameter to be known and we estimate the other one, as it simplifies substantially the expression of the estimators. On the one hand, if a is known, the MLE for b is given by

(3.1) q b T " X T ´x0 ´aT ş T 0 X t dt
Ý Ñ N p0, 4{E " X ´1‰ q
with E rX ´1s " ´b{pa ´2q. Whereas, the weighted least squares estimators are respectively given by

p b T " ş T 0 Xt Ct dX t ´a ş T 0 Xt Ct dt ş T 0 X 2 t Ct dt and p a T " ş T 0 1 Ct dX t ´b ş T 0 Xt Ct dt ş T 0 1 Ct dt . Proposition 3.1.
Assume that a ą 0 and b ă 0. For T going to infinity, p b T satisfies the following CLT:

(3.3) ? T ´p b T ´b¯L Ý Ñ N ´0, 4 E " X 3 {C 2 ‰ `E " X 2 {C ‰˘´2 ¯.
Proof. Replacing dX t by its expression (1.1), we easily get that

? T ´p b T ´b¯" 2 ˆ1 T ż T 0 X 2 t C t dt ˙´1 n T ? T where n T is a martingale given by n T " ż T 0 X t ? X t C t dB t and xny T " ż T 0 X 3 t C 2 t dt.
Using the ergodicity of the process, we obtain for T going to infinity

(3.4) xny T T a.s. Ý Ý Ñ E " X 3 {C 2 ‰ .
Thus, by the CLT for martingales, we obtain the following convergence in distribution

(3.5) n T ? T L Ý Ñ N `0, E " X 3 {C 2 ‰˘.
Consequently, (3.3) follows from (3.5), Slutsky's lemma and the fact that, by the ergodicity of the process, 1 T ş T 0 X 2 t {C t dt converges a.s. to E rX 2 {Cs for T going to infinity. Proposition 3.2. Assume that a ą 0 and b ă 0. For T going to infinity, p a T satisfies the following CLT:

(3.6) ? T pp a T ´aq L Ý Ñ N `0, 4 E " X{C 2 ‰ pE r1{Csq ´2˘.
Proof. It works as in the previous proof. One can observe that

? T pp a T ´aq " 2 ˆ1 T ż T 0 1 C t dt ˙´1 m T ?
T where m T is a martingale term given by

m T " ż T 0 ? X t C t dB t and xmy T " ż T 0 X t C 2 t dt.
Thus, for T going to infinity,

(3.7) xmy T T a.s. Ý Ý Ñ E " X{C 2 ‰
which implies the following convergence in distribution

(3.8) m T ? T L Ý Ñ N `0, E " X{C 2 ‰˘.
Finally, (3.8) leads to (3.6) thanks to the ergodicity of the process and Slutsky's lemma.

Proposition 3.3. Assume that a ą 2 is known and b ă 0. Then, the MLE of b satisfies a CLT with a smaller asymptotic variance than the weighted least squares estimator.

Proof. Using Cauchy-Schwarz Inequality, we notice that `E "

X 2 {C ‰˘2 " ´E " ? X ˆX3{2 {C ı¯2 ď E rXs E " X 3 {C 2 ‰
which immediately leads to the result.

Proposition 3.4. Assume that a ą 2 and b ă 0 is known. Then, the MLE of a satisfies a CLT with a smaller asymptotic variance than the weighted least squares estimator.

Proof. Using Cauchy-Schwarz Inequality, we notice that

pE r1{Csq 2 " `E " X ´1{2 ˆX1{2 {C ‰˘2 ď E r1{Xs E " X{C 2 ‰
which immediately leads to the result.

Remark 3.1. Thus, the weighted least squares estimator is less efficient than the MLE in the case where this later is easily manageable. This could seem to be contradictory to Remark 4.4 of [START_REF] Wei | Estimation of the means in the branching process with immigration[END_REF] which deals with the discrete-time counterpart of the process. In fact, they compare the weighted least squares with the conditional least squares estimator which does not coincide with the MLE.

Remark 3.2. One could wonder how to choose which estimator to use, as the parameter a is unknown. However, we suppose that we observe the whole trajectory of the process over the time interval r0, T s. Thus, if we are able to detect some local time at level zero, we know that a ă 2 and we should use the WLSE instead of the MLE.

Technical Lemmas

First of all, we rewrite (1.6) using (1.1):

(4.1) p θ T " θ `ˆΓ ´1 T 0 0 Γ ´1 T ˙ˆM T N T ˙,
where M T and N T are martingales respectively given by

M T " ¨ż T 0 2 ? X t C t dB t ż T 0 2 ? X t X t C t dB t ‹ ‹ ' and N T " ¨ż T 0 2 ? X t C t d r B t ż T 0 2 ? X t X t C t d r B t ‹ ‹ ' , with d r B t " ρ dB t `a1 ´ρ2 dW t . We denote by M T the martingale M T " pM T , N T q. As A dB t , d r B t E
" ρdt, we easily obtain that the increasing process of M T is given by

(4.2) xMy T " ˆxMy T ρ xMy T ρ xMy T xMy T ˙,
where the increasing process xMy T of M T is given by

xMy T " 4 ¨xmy T ż T 0 X 2 t C 2 t dt ż T 0 X 2 t C 2 t dt xny T ‹ ‹ ' ,
with xmy T and xny T respectively given by (3.7) and (3.4).

In order to prove Theorems 2.1 and 2.2, we need to investigate the almost sure convergence of all the integrals involved in the definition of the estimators. Overbeck recalls in Lemma 3(i) of [START_REF] Overbeck | Estimation for continuous branching processes[END_REF] that, for T going to infinity, X T converges in distribution to X with Gamma Γpa{2, ´b{2q distribution, whose probability density function is given by (4.3) f pxq " pΓpa{2qq ´1 p´b{2q a{2 x a{2´1 e xb{2 1 xą0 .

Thus, by Lemma 3(ii) of [START_REF] Overbeck | Estimation for continuous branching processes[END_REF], for T going to infinity,

1 T ż T 0 g pX t q dt a.s. Ý Ý Ñ E rgpXqs " ż `8 0 gpxqf pxq dx.
for any function g such that the right-hand side exists. We recall two properties of the incomplete gamma function that will be very useful in the following proof: We are now able to prove the following lemma. The first three points give us the almost sure limit of the matrix T Γ ´1 T as T goes to infinity, while the remaining deals with the increasing process of the four-dimensional martingale M T given by (4.2). Lemma 4.1. With ψ c given by (2.4), we have that

(i) E r1{Cs " ψc c . (ii) E rX{Cs " 1 ´ψc . (iii) E rX 2 {Cs " c pψ c ´1q ´a b . (iv) E rX{C 2 s " a 2c ψ c `b 2 p1 ´ψc q . (v) E rX 2 {C 2 s " 1 2 ppa `2 ´bcq p1 ´ψc q ´aq. (vi) E rX 3 {C 2 s " c 2 pa `4 ´bcq pψ c ´1q `ac 2 ´a b . Proof. (i) We have (4.6) E r1{Cs " ż `8 0 1 x
`c f pxq dx, where f is given by (4.3). Formula 3.383(10) of [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF] gives that ż `8

0 1 x
`cx a{2´1 e xb{2 dx " c a{2´1 e ´bc{2 Γpa{2qΓp1 ´a{2, ´bc{2q, which leads to

E r1{Cs " 1 c ˆ´bc 2 ˙a{2 e ´bc{2 Γ p1 ´a{2, ´bc{2q
and ensures the announced result.

(ii) As in the previous proof, we

`c f pxq dx. By formula 3.383 [START_REF] Feller | Two singular diffusion problems[END_REF] of [START_REF] Gradshteyn | Table of integrals, series, and products[END_REF], we know that ¸.

Combining (4.7), (4.8), (4.9) and the fact that Γpa{2 `1q " a{2 ˆΓpa{2q, we deduce the announced result.

(iii) We have

E " X 2 C  " E « pX `c ´cq 2 X `c ff " E rXs ´c `c2 E " 1 C  ,
and we conclude using (i) and the fact that E rXs " ´a{b.

(iv) By the very definition of f given by (4.3), we have

(4.10) E " X{C 2 ‰ " ż `8 0 x px `cq 2 f pxq dx " p´b{2q a{2 Γpa{2q ż `8 0 x a{2 px `cq 2 e xb{2 dx.
Integrating the right-hand side of (4.10) by part, we obtain that

E " X{C 2 ‰ " p´b{2q a{2 Γpa{2q " a 2
ż `8 0 x a{2´1 x `c e xb{2 dx `b 2 ż `8 0 x a{2 x `c e xb{2 dx  .
We have already computed both integrals in the proofs of respectively (i) and (ii), which leads to

(4.11) E " X{C 2 ‰ " a 2 E r1{Cs `b 2 E rX{Cs " a 2c ψ c `b 2 p1 ´ψc q .
(v) Integrating by parts and using (iii) and (iv),

E " X 2 {C 2 ‰ " p´b{2q a{2 Γpa{2q ż `8 0 x a{2`1 px `cq 2 e xb{2 dx " p´b{2q a{2 Γpa{2q " a `2 2 ż `8 0 x a{2 x `c e xb{2 dx `b 2 ż `8 0 x a{2`1 x `c e xb{2 dx  " `pa `2q E rX{Cs `b E " X 2 {C ‰˘{ 2
" ´pa `2q p1 ´ψc q `b ´c pψ c ´1q ´a b ¯¯{2.

(vi) Noticing that X 3 " X pX `cq 2 ´2cX 2 ´c2 X, we obtain that

E " X 3 {C 2 ‰ " E rXs ´2c E " X 2 {C 2 ‰ ´c2 E " X{C 2 ‰
and we conclude using (iv) and (v).

Proof of the strong Consistency

We are now in the position to prove Theorem 2.1.

Proof of Theorem 2.1. First of all, we have 1

T 2 det Γ T " 1 T ż T 0 1 C t dt ˆ1 T ż T 0 X 2 t C t dt ´ˆ1 T ż T 0 X t C t dt ˙2 .
Thus, as the process is ergodic, we obtain for T going to infinity, ( A straightforward application of Lemmas 4.1 (i) to (iii) gives that

A " 1 ψ c `1 ´a bc ˘´1 ˆc pψ c ´1q ´a b ψ c ´1 ψ c ´1 ψc c ˙.
Besides, using the strong law of large numbers for martingale, we obtain that the martingale M T satisfies for T going to infinity (5.4) M T T a.s.

Ý Ý Ñ 0.

As a matter of fact, by convergences (3.4) and (3.7), we know that a.s. xny T " O pT q and xmy T " O pT q. It ensures that for T going to infinity, n T T a.s.

Ý Ý Ñ 0 and m T T a.s.

Ý Ý Ñ 0.

As N T and M T share the same increasing process, this result remains true by replacing M T by N T . Finally, the almost sure convergence (2.1) follows from (4.1), (5.2) and (5.4).

Proof of the asymptotic normality

Proof of Theorem 2.2. First of all, we deduce from (4.1) that (6.1)

? T ´p θ T ´θ¯" ˆT Γ ´1 T 0 0 T Γ ´1 T ˙ˆM T { ? T N T { ? T ˙,
We already saw that T Γ ´1 T converges almost surely as T goes to infinity and its limit A is given by (5.3). We now have to establish the asymptotic normality of M T ? T . By the ergodicity of the process, we obtain that xMy T T a.s.

Ý Ý Ñ 4L where L " ˆE rX{C 2 s E rX 2 {C 2 s E rX 2 {C 2 s E rX 3 {C 2 s ˙.
As a straightforward consequence of Lemmas 4.1 (iv) to (vi), we obtain that

L " 1 2
¨a c ψ c `b p1 ´ψc q pa `2 ´bcq p1 ´ψc q ´a pa `2 ´bcq p1 ´ψc q ´a ψ c c pa `4 ´bq ´4c ´bc 2 ´2a b '.

We easily obtain the following a.s. convergence xMy T T a.s.

Ý Ý Ñ 4L

where L is a block matrix given by

L " ˆL ρL ρL L ˙.
and we deduce from the CLT for martingales that (6.2)

M T ? T L Ý Ñ N p0, 4Lq ,
Finally, the asymptotic normality (2.2) follows from (6.1) and (6.2) together with Slutsky's Lemma.

Numerical simulations

The efficient discretization of the CIR process is a challenging question, see for example [START_REF] Anderson | Simple and efficient simulation of the heston stochastic volatility model[END_REF] and [START_REF] Alfonsi | High order discretization schemes for the CIR process: application to affine term structure and Heston models[END_REF]. We choose to implement the QE-algorithm based on quadratic-exponential approximations proposed in [START_REF] Anderson | Simple and efficient simulation of the heston stochastic volatility model[END_REF]. Andersen introduced this algorithm to deal with the case a ă 2, for which common discretization schemes are not accurate. 7.2. Choice of the constant c. We have chosen to introduce a constant c in our weighting, instead of only considering the case c " 1 (as done in the discrete-time case in [START_REF] Wei | Estimation of the means in the branching process with immigration[END_REF]) with the aim of lowering the variance of the estimators. However, this raises the question of the optimal choice of the constant c, which depends on the values of parameters a and b. We set a " 1 and b " ´4 and simulate 500 trajectories of the process over the time interval r0, 50s. We compute the empirical variance of the estimators given by each trajectory for c varying between 10 ´10 and 1. It appears that one should choose a small value of c. The value should not be to small to avoid the growth illustrated by the figures on the right-hand side, which might however be a consequence of the discretized version of the CIR process we used. For p a T , there is a significant difference (factor 5) between the empirical variances obtained with c " 0.01 and c " 1. However, for p b T both empirical variances do not significantly differ. In what follows, we only consider the first equation as the results for the second one can be derived the same way. Suppose that we want to compute the usual LSE r µ n of the couple pℓ, bq based on discrete time observations pX 0 , X 1 , . . . , X n q, which would in fact be a weighted LSE due to the presence of C t in the above equations. It is the solution of the

(4. 4 )

 4 Γpα `1, xq " x α e ´x `αΓ pα, xq and (4.5) Γpα `2, xq " x α e ´x px `α `1q `α pα `1q Γ pα, xq .

  a{2 e xb{2 dx " c a{2 e ´bc{2 Γpa{2 `1qΓp´a{2, ´bc{2q. With formula (4.4), we easily obtain that (

7. 1 .

 1 Asymptotic behavior for c " 1. The two following figures illustrate our main results (strong consistency and asymptotic normality) in the case a " 1 and b " ´2, with the weighting parameter c " 1. The red curves in the second figure displays the standard normal distribution. We denote by v a (respestively v b ) the variance of p a T (resp. p b T ).

Figure 1 .Figure 2 .

 12 Figure1. Strong consistency: pp a T q in black and p p b T q in blue.

  [START_REF] Forde | The large-maturity smile for the Heston model[END_REF] ÝÑ where σ 11 , σ 22 and σ 12 are defined in Remark 2.1. This leads to

	cÑ0	´2a bpa ´2q	, σ 22 ÝÑ cÑ0	´2b pa ´2q 2 , and σ 12 ÝÑ cÑ0 ´2 a ´2 ,
		ALA ÝÑ cÑ0	Σ ´1	where	Σ "	ˆ´b 1 ´a b a´2 1	˙.

  Variance of p a 50 as a function of c.Variance of p b 50 as a function of c.Appendix : Motivation of the choosen estimatorWe rewrite Equation 1.1 by making use of the weighting C Using the fact that X t " C t ´c, we obtain the following equations:

	1.4e-02											9e-03
	1.3e-02
	1.2e-02											8e-03
	1.1e-02
												7e-03
	1e-02
	9e-03											6e-03
	8e-03
	7e-03											5e-03
	6e-03
												4e-03
	5e-03
	4e-03											3e-03
	3e-03
	2e-03											2e-03
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	0	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008	0.009	0.01
	5.8e-01											4.4e-01
	5.6e-01
												4.3e-01
	5.4e-01
	5.2e-01											4.2e-01
	5e-01
												4.1e-01
	4.8e-01
	4.6e-01											4e-01
	4.4e-01
												3.9e-01
	4.2e-01
	4e-01											3.8e-01
	3.8e-01
												3.7e-01
	3.6e-01
	3.4e-01											3.6e-01
	0	0.1	0.4	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1	0	0.001	0.002	0.003	0.004	0.005	0.006	0.007	0.008	0.009	0.01
																	´1{2 t	:
	(7.1)			$ & %	C C ´1{2 t ´1{2 dX t " pa `bX t q C t dY t " pα `βX t q C ´1{2 t ´1{2 dt `2 C t dt `2 C ´1{2 t ´1{2 a X t dB t t a X t d r B t .
	(7.2)			$ ' & ' %	C C ´1{2 t ´1{2 dX t " ´ℓ C t dY t " ´λ C ´1{2 t ´1{2 t	`b C `β C 1{2 t ¯dt `2 C	´1{2 t	a	X t dB t

1{2 t ¯dt `2 C ´1{2 t a X t d r B t ,

where ℓ " a ´cb and λ " α ´cβ.

following minimization problem:

We investigate the critical points and easily obtain that

and

.

Besides, as ℓ " a ´cb, we obtain an estimator r a n for a given by r a n " r ℓ n `c r b n . Motivated by those equations, we introduce a WLSE based on the continuous time observations pX t q tďT that is given by

, which easily rewrites as in Equation 1.6 pp a T , p t T q " Γ ´1 T U T , using the fact that X t " C t ´c.