Asymptotic Behavior of The Maximum Likelihood Estimator For ...

Asymptotic Behavior of The Maximum Likelihood Estimator For Ergodic and Nonergodic Sqaure-Root Diffusions By Mohamed Ben Alaya and Ahmed Kebaier∗

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LAGA, CNRS (UMR 7539), Institut Galil´ee, Universit´e Paris 13, 99, av. J.B. Cl´ement 93430 Villetaneuse, France [email protected] [email protected] November 10, 2011

Abstract This paper deals with the problem of global parameter estimation in the Cox-IngersollRoss (CIR) model (Xt )t≥0 . This model is frequently used in finance for example to model the evolution of short-term interest rates or as a dynamic of the volatility in the Heston model. In continuity with a recent work by Ben Alaya and Kebaier [1], we establish new asymptotic results on the maximum likelihood estimator (MLE) associated to the global estimation of the drift parameters of (Xt )t≥0 . RTo do so,Rwe need to study first the asympt ds t totic behavior of the quadruplet (log Xt , Xt , 0 Xs ds, 0 X ). This allows us to obtain s various and original limit theorems on our MLE, with different rates and different types of limit distributions. Our results are obtained for both cases : ergodic and nonergodic diffusion. AMS 2000 Mathematics Subject Classification. 44A10, 60F05, 62F12. Key Words and Phrases. Cox-Ingersoll-Ross processes, nonergodic diffusion, Laplace transform, limit theorems, parameter inference.

1

Introduction

The Cox-Ingersoll-Ross (CIR) process is widely used in mathematical finance to model the evolution of short-term interest rates. It is also used in the valuation of interest rate derivatives. ∗

This research benefited from the support of the chair “Risques Financiers”, Fondation du Risque.

1

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It was introduced by Cox, Ingersoll and Ross [2] as solution to the stochastic differential equation (SDE) p dXt = (a − bXt )dt + 2σ|Xt |dWt , (1) where X0 = x > 0, a > 0, b ∈ R, σ > 0 and (Wt )t≥0 is a standard Brownian motion. Under the above assumption on the parameters that we will suppose valid through all the paper, this SDE has a unique non-negative strong solution (Xt )t≥0 (see Ikeda and Watanabe [8], p. 221). In the particular case b = 0 and σ = 2, we recover the square of a a-dimensional Bessel process starting at x. For extensive studies on Bessel processes we refer to Revuz and Yor [16] and Pitman and Yor [14] and [15]. The behavior of the CIR process X mainly depends on the sign of b. Indeed, in the case b > 0 there exists a unique stationary distribution, say π,R of X and 1 t 1 the stationary CIR processes R enjoy the ergodic property that is: for all h ∈ L (π), t 0 h(Xs )ds converges almost surely to R h(x)π(dx). In the case a ≥ σ, the CIR process X stays strictly positive; for 0 < a < σ, it hits 0 with probability p ∈]0, 1[ if b < 0 and almost surely if b ≥ 0, the state 0 is instantaneously reflecting (see e.g. G¨oing-Jaeschke and Yor [6] for more details). During the past decades, inference for diffusion models has become one of the core areas in statistical sciences. The basic results concerning the problem of estimating the drift parameters when a diffusion process was observed continuously are well-summarized in the books by Lipster and Shiryayev [13] and Kutoyants [11]. This approach is rather theoretical, since the real data are discrete time observations. Nevertheless, if the error due to discretization is negligible then the statistical results obtained for the continuous time model are valid for discrete time observations too. Most of these results concern the case of ergodic diffusions with coefficients satisfying the Lipschitz and linearity growth conditions. In the literature, only few results can be found for nonergodic diffusions or diffusions with nonregular coefficients such as the CIR. To our knowledge, one of the first papers having studied the problem of global parameter estimation in the CIR model is that of Fourni´e and Talay [4]. They have obtained a nice explicit formula of the maximum likelihood estimator (MLE) of the drift parameters θ := (a, b) and have established its asymptotic normality only in the case b > 0 and a > σ by using the classical martingale central limit theorem. Otherwise, for cases b ≤ 0 or a ≤ σ this argument is no more valid. Note that, in practice, the parameter σ is usually assumed to be known and one can estimate it separately using the quadratic variation of the process X. In a recent work Ben Alaya and Kebaier [1] use a new approach, based on Laplace transform technics, to study the asymptotic behavior of the MLE associated to one of the drift parameters in the CIR model, given that the other one is known, for a range of values (a, b, σ) covering ergodic and nonergodic situations. More precisely, they prove that • First case: when a is supposed to be known the MLE of θ = b is given by ˆbT = aTR+ x − XT . T Xs ds 0 √ If b > 0, the asymptotic theorem on the error ˆbT − b is obtained with a rate equal to T and the limit distribution is Gaussian. However, for b < 0 and b = 0, when the diffusion (Xt )t≥0 is not ergodic, the rate of convergence are respectively equal to e−bT /2 and T and the obtained limit distributions are not guassians (for more details see Theorem 1 of [1]) 2

• Second case: when b is supposed to be known, the MLE of θ = a, given by RT log XT − log x + bT + σ 0 Xdss a ˆT = , R T ds 0 Xs

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is well defined only for a ≥ σ. When on the error a ˆT − a √ a > σ, the asymptotic theorem √ is obtained with a rate equal to T if b > 0 and with a rate equal to log T if b = 0 and the limit distribution is Gaussian in both cases. When a = σ, the asymptotic theorem on the error a ˆT − a is obtained with a rate equal to T if b > 0 and with a rate equal to log T if b = 0 and the corresponding limit distributions are not guassians. However, for the case b < 0 the MLE estimator a ˆT is not consistent (for more details see Theorem 2 of [1]). Note that for the case b ≤ 0, when the diffusion (Xt )t≥0 is not ergodic, and even R 1 t 1 in the case b > 0 and a = σ, when the diffision is ergodic, the quantity t 0 Xs ds goes to infinity as t tends to infinity and consequently the classical technics based on central limt theorem for martingales fail. In order to pove the first case results, Ben Alaya and Kebaier [1] use the explicit Laplace RT transform of the couple (XT , 0 Xs ds) which is well known (see e.g. Lamberton and Lapeyre [12], p. 127), in view to study its asymptotic behavior (see Theorem 1). However, for the RT second case, they only use the explicit Laplace transform of 0 Xdss based on a recent work of Craddock and Lennox [3] to deduce its asymptotic behavior too (see Theorem 3). The aim of this paper is to study the global MLE estimator of parameter θ = (a, b) for a range of values (a, b, σ) covering ergodic and nonergodic situations. Indeed, it turns out that the MLE of θ = (a, b) is defined only when a ≥ σ and the associated estimation error is given by   R T ds  R T   Xs ds − T (XT − x − aT ) log X − log x + (σ − a) T  0 0 Xs   a ˆ − a = R R  T T T ds   Xs ds − T 2  0 Xs 0 θˆT − θ =  R  R T ds   RT  T ds  T log X − log x + bT + σ − X − x + b X ds  T T s X Xs  0 0 0 s  ˆbT − b =  . R R  T T ds  2 X ds − T s 0 Xs 0 Hence, the results obtained by [1] are no longer sufficient for our proposed study, since we need RT RT a precise description of the asymptotic behavior of the quadruplet (log XT , XT , 0 Xs ds, 0 Xdss ) appearing in the above error estimation. For this purpose, we use once again the recent work of [3]. The obtained results are stated in the second section. Then in the third section, we take advantage of this study to establish new asymptotic theorems on the error estimation θˆT − θ with different rates of convergence and different types of limit distributions that vary according to assumptions made on the parameters a, b and ˆ σ. More precisely, we prove that √ when a > σ, the asymptotic theorem√on the error θT − θ is obtained with a rate equal to T if b > 0 and with a rate equal to diag( log T , T ) if b = 0 and the limit distribution is Gaussian only for the first case. When a = σ, the asymptotic theorem 3

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√ on the error θˆT − θ is obtained with a rate equal to diag(T, T ) if b > 0 and with a rate equal to diag(log T, T ) if b = 0 and the corresponding limit distributions are not guassians. For b < 0 the MLE estimator θˆT is not consistent. Our results cover both cases : ergodic and nonergodic situations and are summarized in Theorems 5, 6 and 7. In the last section we study the problem of parameter estimation from discrete observations. These observations consist of a discrete sample (Xtk )0≤k≤n of the CIR diffusion at deterministic and equidistant instants (tk = k∆n )0≤k≤n . Our aim is to study a new estimator θˆt∆nn := (ˆ a∆tnn , ˆb∆tnn ), for θ = (a, b) based on discrete observations, under the conditions of high frequency, ∆n → 0, and infinite horizon, n∆n → ∞. Several authors have studied this estimation problem by basing the inference on a discretization of the continuous likelihood ratio, see Genon-Catalot [5] and Yoshida [17]. In our approach, we proceed in a slightly different way, we discretize the continuous time MLE instead of considering the maximum argument of the discretized likelihood. We give sufficient conditions on the stepsize ∆n under which the error θˆt∆nn − θˆtn correctly normalized tends to zero, so that the limit theorems obtained in the continuous time observations for θˆtn can be easily carried out for the discrete one θt∆nn . It turnsout that, for  a > 2σ, these limit theorems are satisfied if n2 ∆n → 0, when b > 0, or 3 2

n∆n max n2 ∆n , log(n∆ n)

→ 0, when b = 0 (see Theorems 8 and 9). The case 0 ≤ a ≤ 2σ, seems

to be much more harder to treat and requires a specific and more detailed analysis see the last Remark of section 4. For b > 0, the condition n2 ∆n → 0 is consistent with those of papers in the litterature dealing with the same problem for ergodic diffusions with regular coefficients  (see  3 2

n∆n Yoshida [17] and its references). However, for b = 0, the condition max n2 ∆n , log(n∆ n)

→0

seems to be quite original since it concerns a nonergodic case.

2

Asymptotic behavior of the triplet (Xt,

Rt

0 Xs ds,

Rt

ds 0 Xs )

Let us recall that (Xt )t≥0 denotes a CIR process solution to (1). It is relevant to consider separately the cases b = 0 and b 6= 0, since the process (Xt )t≥0 behaves differently.

2.1

Case b = 0

In this section, we consider the Cox-Ingersoll-Ross CIR process (Xt )t≥0 with b = 0. In this particular case, (Xt )t≥0 satisfies the SDE p dXt = adt + 2σXt dWt . (2) Note that for σ = 2, we recover the square of a a-dimensional Bessel process starting at x and denoted by BESQax . This process has been attracting considerably the attention of several studies (see Revuz and Ben Alaya and Kebaier study the seperate R t Yor [16]). Recently, Rt asymptotic behavior of (Xt , 0 Xs ds) and 0 Xdss (see Propositions 1 and 2 of [1]) and prove the following result Theorem 1 Let (Xt )t≥0 be a CIR process solution to (2), we have 4

Rt law 1. ( Xtt , t12 0 Xs ds) −→ (R1 , I1 ) as t tends to infinity, where (Rt )t≥0 is the CIR process startRt ing from 0, solution to (2) and It = 0 Rs ds. Z t  ds 2. Px < ∞ = 1 if and only if a ≥ σ. 0 Xs Z t 1 ds P 1 3. If a > σ then as t tends to infinity. −→ log t 0 Xs a−σ Z t ds law 1 4. If a = σ then −→ τ1 as t tends to infinity, where τ1 is the hitting time 2 (log t) 0 Xs associated with Brownian motion τ1 := inf{t > 0 : Wt = √12σ }.

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In the following, we study the asymptotic behavior of the triplet (Xt , makes sense only and olny if a ≥ σ and we prove the following result.

Rt 0

Xs ds,

Rt

ds ) 0 Xs

which

Theorem 2 Let (Xt )t≥0 be a CIR process solution to (2), we have 1. If a > σ then Xt 1 ( , 2 t t

t

Z

1 Xs ds, log t

0

Z

t

0

1 ds law ) −→ (R1 , I1 , ) as t tends to infinity. Xs a−σ

2. If a = σ then Xt 1 ( , 2 t t

Z

t

0

1 Xs ds, (log t)2

Z 0

t

ds law ) −→ (R1 , I1 , τ1 ) as t tends to infinity. Xs

Rt Here, (Rt )t≥0 denotes the CIR process starting from 0, solution to (2), It = 0 Rs ds, and τ1 is the hitting time associated with Brownian motion τ1 := inf{t > 0 : Wt = √12σ }. Moreover, The couple (R1 , I1 ) and the random time τ1 are independent. In order to prove we choose to compute the Laplace transform of the quadruplet R t thisRproposition, t (log Xt , Xt , 0 Xs ds, 0 Xdss ). Here is the obtained result. Proposition 1 For ρ ≥ 0, λ ≥ 0, µ > 0 and η ∈] − k −  Ex

Xtη e−ρXt −λ

Rt 0

Xs ds−µ

√

×

Rt

σλx √ σ sinh( σλt)

ds 0 Xs



− 12 , +∞[, we have

! √ √ Γ(η + k + ν2 + 12 ) η σλx = x exp − coth( σλt) Γ(ν + 1) σ

! ν2 + 12 −k−η

√ − ν2 − 12 −k−η √ √ √ ( σρ/ λ) sinh( σλt) + cosh( σλt) √ σλx

 × 1 F 1 η + k +

ν 2



ν 1 √   (3) + , ν + 1, √ √ √ √ 2 2 σ sinh( σλt) ( σρ/ λ) sinh( σλt) + cosh( σλt) 5

1p a ,ν= (a − σ)2 + 4µσ and 1 F1 is the confluent hypergeometric function defined 2σ σ Qn−1 P∞ un zn by 1 F1 (u, v, z) = k=0 (u + k) and n=0 vn n! , with u0 = v0 = 1, and for n ≥ 1, un = Qn−1 vn = k=0 (v + k). Rt Rt Consequently, the Laplace transform of the triplet (Xt , 0 Xs ds, 0 Xdss ) is obtained by taking η = 0 in relation (3) and we have the following result. where k =

Corollary 1 For ρ ≥ 0, λ ≥ 0 and µ > 0, we have 

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Ex e−ρXt −λ

Rt 0

Xs ds−µ

Rt

ds 0 Xs



! √ √ Γ(k + ν2 + 12 ) σλx = exp − coth( σλt) Γ(ν + 1) σ ! ν2 + 12 −k √ − ν2 − 12 −k √ √ √ ( σρ/ λ) sinh( σλt) + cosh( σλt)

√ σλx √ × σ sinh( σλt)   √ σλx ν 1 √   (4) × 1 F1 k + + , ν + 1, √ √ √ √ 2 2 σ sinh( σλt) ( σρ/ λ) sinh( σλt) + cosh( σλt) where k =

a 1p ,ν= (a − σ)2 + 4µσ. 2σ σ

Proof of Proposition 1: We apply Theorem 5.10 in [3] to our process. We have just to √ be careful with the misprint in formula (5.24) of [3]. More precisely, we have topreplace Axy, in the numerator of the first term in the right hand side of this formula, by Ax/y. Hence, for a > 0 and σ > 0, we obtain the so called fundamental solution of the PDE ut = σxuxx + aux − ( µx + λx)u, λ > 0, µ > 0 : ! ! √ √ √ √  y k−1/2 2 σλ xy σλ σλ(x + y) √ √ √ exp − p(t, x, y) = Iν , (5) σ sinh( σλt) x σ tanh( σλt) σ sinh( σλt) where Iν is the modified Bessel R t function R t ds of the first kind. This yields the Laplace transform of the quadruplet (log Xt , Xt , 0 Xs ds, 0 Xs ), since  Ex

Xtη e−ρXt −λ

Rt 0

Xs ds−µ

Rt

ds 0 Xs



Z = 0

6

∞

y η e−ρy p(t, x, y)dy.

Evaluation of this integral is routine, see formula 2 of section 6.643 in [7]. Therefore, we get 

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Ex

! √ 1 ν √ + ) Γ(η + k + σλx 2 2 Xtη e−ρXt −λ 0 Xs ds−µ = x−k exp − coth( σλt) Γ(ν + 1) σ η+k  √ σ sinh( σλt)  × √  √ √ √ √ σλ ( σρ/ λ) sinh( σλt) + cosh( σλt)   √ σλx √  × exp  √ √ √ √ 2σ sinh( σλt) ( σρ/ λ) sinh( σλt) + cosh( σλt)   √ σλx √   , (6) × M−η−k, ν2  √ √ √ √ σ sinh( σλt) ( σρ/ λ) sinh( σλt) + cosh( σλt) Rt

Rt

ds 0 Xs



where Ms,r (z) is the Whittaker function of the first kind given by 1 z 1 Ms,r (z) = z r+ 2 e− 2 1 F1 (r − s + , 2r + 1, z). 2

(7)

See [7] for more details about those special functions. We complete the proof by inserting relation (7) in (6).  Proof of Theorem 2 : The first assertion is straightforward from Theorem 1. For the last assertion and under notations ofthe above corollary, it is easy to check that  √  √  µ σλx √ √ R R   ρ exp log t ds σ log t tσ sinh( σλ) − X − λ t X ds− µ √ √ √ = lim √ lim Ex e t t t2 0 s (logt)2 0 Xs t→∞ ( σρ/ λ) sinh( σλ) + cosh( σλ) t→∞ √  µ exp √σ √ √ √ = √ ( σρ/ λ) sinh( σλ) + cosh( σλ)

= Ex e−ρR1 −λI1 Ex e−µτ1 . Which completes the proof. We now turn to the case b 6= 0.

2.2



Case b 6= 0

Let us resume the general model of the CIR given by relation (1) with b 6= 0, namely p dXt = (a − bXt )dt + 2σXt dWt ,

(8)

where X0 = x > 0, a > 0, b ∈ R∗ , σ > 0. Note that this process may be represented in terms
σ of a square Bessel process through the relation Xt = e−bt Y 2b (ebt − 1) , where Y denotes 7

2a

a BESQxσ . This relation results from simple properties of square Bessel processes (see e.g. G¨oing-Jaeschke and Yor [6] and Revuz and Yor [16]). In the same manner R t as for the case R t dsb = 0, Ben Alaya and Kebaier study the seperate asymptotic behavior of (Xt , 0 Xs ds) and 0 Xs (see propositions 3 and 4 of [1]) and prove the following result Theorem 3 Let (Xt )t≥0 be a CIR process solution to (8), we have Z 1 t P a 1. If b > 0 then Xs ds −→ as t tends to infinity. t 0 b   Z t law bt bt 2. If b < 0 then e Xt , e Xs ds −→ (Rt0 , t0 Rt0 ), as t tends to infinity, where t0 = −1/b

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0

and (Rt )t≥0 is the CIR process, starting from x, solution to (2).  Z t ds < ∞ = 1 if and only if a ≥ σ. 3. Px 0 Xs Z b 1 t ds P −→ as t tends to infinity. 4. If b > 0 and a > σ then t 0 Xs a−σ Z 1 t ds law 5. If b > 0 and a = σ then 2 −→ τ2 as t tends to infinity, where τ2 is the hitting t 0 Xs time associated with Brownian motion τ2 := inf{t > 0 : Wt = σ√b 2 }. Z t0 ds law −→ It0 := 6. If b < 0 and a ≥ σ then Rs ds as t tends to infinity, where t0 = −1/b 0 Xs 0 and (Rt )t≥0 is the CIR process, starting from x, solution to (2). Z

t

According to the third assertion of the above theorem our next result makes sense if and only if a ≥ σ and we have. Theorem 4 Let (Xt )t≥0 be a CIR process solution to (8), we have     Z t Z 1 t ds 1 a b P 1. If b > 0 and a > σ then −→ , as t tends to infinity. Xs ds, t 0 t 0 Xs b a−σ  Z t  Z 1 1 t ds law  a  Xs ds, 2 2. If b > 0 and a = σ then −→ , τ2 as t tends to infinity, where t 0 t 0 Xs b τ2 is the hitting time associated with Brownian motion τ2 := inf{t > 0 : Wt = √b2σ }. 

t

t

 ds law 3. If b < 0 and a ≥ σ then e Xt , e Xs ds, −→ (Rt0 , t0 Rt0 , It0 ), as t tends to 0 0 Xs infinity, where t = −1/b, (R ) is the CIR process, starting from x solution to (2), and 0 t t≥0 R t0 It0 := 0 Rs ds. bt

bt

Z

Z

8

Proof of Theorem The two first assertions are straightforward R t ds consequence from Theorem 3. For the case b < 0 and a ≥ σ, we have only to note that ( 0 Xs )t≥0 is an increasing process converging to a random variable with the same law as It0 . The result follows by assertion 2 of Proposition 3 of [1].  Nevertheless, the above theorem is not sufficient to prove all the asymptotic results we manage to establish in our next section. Therefore, we need to prove in extra the following result. Proposition 2 For ρ ≥ 0, λ ≥ 0 and µ > 0, we have 

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Ex

  Γ(k + ν2 + 12 ) b Ax = exp (at + x) − coth(At/2) e Γ(ν + 1) 2σ 2σ   ν2 + 21 −k  − ν2 − 12 −k 2σρ + b Ax × sinh(At/2) + cosh(At/2) 2σ sinh(At/2) A   ν 1 A2 x × 1 F1 k + + , ν + 1, (9) 2 2 2σ sinh(At/2) ((2σρ + b) sinh(At/2) + A cosh(At/2)) −ρXt −λ

where k =

Rt 0

Xs ds−µ

Rt

ds 0 Xs



√ a 1p , A = b2 + 4σλ and ν = (a − σ)2 + 4µσ. 2σ σ

Proof We apply Theorem 5.10 of [3], for b 6= 0. We obtain the fundamental solution of the PDE ut = σxuxx + (a − bx)ux − (λx + µx )u, µ > 0 and λ > 0 :  y a/(2σ)−1/2 A 2σ sinh(At/2) x     √ (10) A xy A(x + y) b [at + (x − y)] − Iν . × exp 2σ 2σ tanh(At/2) σ sinh(At/2)  Rt R t ds  R∞ Since Ex e−ρXt −λ 0 Xs ds−µ 0 Xs = 0 e−ρy p(t, x, y)dy, we deduce the Laplace transform of the Rt Rt triplet (Xt , 0 Xs ds, 0 Xdss ). In the same manner as in the proof of Proposition 3, formula 2 of section 6.643 in [7] gives us p(t, x, y) =



−ρXt −λ

Ex e

Rt 0

  Γ(k + ν2 + 12 ) −k b Ax = x exp (at + x) − coth(At/2) Γ(ν + 1) 2σ 2σ  k 2σ sinh(At/2) × (2σρ + b) sinh(At/2) + A cosh(At/2)   A2 x × exp 4σ sinh(At/2) ((2σρ + b) sinh(At/2) + A cosh(At/2))   A2 x × M−k, ν2 . (11) 2σ sinh(At/2) ((2σρ + b) sinh(At/2) + A cosh(At/2))

Xs ds−µ

Rt

ds 0 Xs



Finally, by inserting relation (7) in (11) we obtain the announced result.  9

3

Estimation of the CIR diffusion from continuous observations

Let us first recall some basic notions on the construction of the maximum likelihood estimator (MLE). Suppose that the one dimensional diffusion process (Xt )t≥0 satisfies

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dXt = b(θ, Xt )dt + σ(Xt )dWt ,

X0 = x0 ,

where the parameter θ ∈ Θ ⊂ Rp , p ≥ 1, is to be estimated. The coefficients b and σ are two functions satisfying conditions that guarantee the existence and uniqueness of the SDE for each θ ∈ Θ. We denote by Pθ the probability measure induced by the solution of the equation on the canonical space C(R+ , R) with the natural filtration Ft := σ(Ws , s ≤ t), and let Pθ,t := Pθ |Ft be the restriction of Pθ to Ft . If the integrals in the next formula below make sense then the measures Pθ,t and Pθ0 ,t , for any θ, θ0 ∈ Θ, t > 0, are equivalent (see Jacod [9] and Lipster and Shirayev [13]) and we are able to introduce the so called likelihood ratio Z t  Z dPθ,t b(θ, Xs ) − b(θ0 , Xs ) 1 t b2 (θ, Xs ) − b2 (θ0 , Xs ) θ,θ0 Lt := = exp dXs − ds . (12) dPθ0 ,t σ 2 (Xs ) 2 0 σ 2 (Xs ) 0   θ,θ0 The process Lt is an Ft −martingale. t≥0

In the present section, we observe the process X T = (Xt )0≤t≤T as a parametric model solution to equation (1), namely p dXt = (a − bXt )dt + 2σXt dWt , where X0 = x > 0, a > 0, b ∈ R, σ > 0. The unknown parameter, say θ, is involved only in the drift part of the diffusion and we consider the case θ = (a, b).

3.1

Parameter estimation θ = (a, b)

According to relation (12), the appropriate likelihood ratio, evaluated at time T with θ0 = (0, 0), RT makes sense when Pθ ( 0 Xdss < ∞) = 1 and is given by LT (θ) = LT (a, b) :=

0 Lθ,θ T

 = exp

1 2σ

T

Z 0

a − bXs 1 dXs − Xs 4σ

Z 0

T

 (a − bXs )2 ds . Xs

Hence, for a ≥ σ, the MLE θˆT = (ˆ aT , ˆbT ) of θ = (a, b) that maximizes LT (a, b) is well defined and we have  RT R T dXs X ds − T (XT − x)  s   a ˆT = 0 R T ds0 R TXs    Xs ds − T 2  0 Xs

     ˆ   bT

=

T

RT

0

dXs − (XT − 0 Xs R T ds R T Xs ds 0 Xs 0

10

x)

RT

ds 0 Xs

− T2

Hence, the error is given by  RT R T dWs RT √  √ X ds − T Xs dWs √  s 0 0 0 X  s  a ˆ − a = 2σ R R  T T T ds   Xs ds − T 2  0 Xs 0      ˆb − b    T

√ T = 2σ

R T ds R T √ dWs √ − Xs dWs 0 0 Xs 0 Xs R T ds R T Xs ds − T 2 0 Xs 0

RT

The above error is obviously of the form 

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θˆT − θ =

√

RT

dWs √ 0 Xs



  2σhM i−1  T MT , with MT =  RT √ − 0 Xs dWs

(13)

and (hM it )t≥0 is the quadratic variation of the Brownian martingale (Mt )t≥0 . If this quadratic variation, correctly normalized, converges in probability then the classical martingale central limit theorem can be applied. Here and in the following =⇒ means the convergence in distribution under Pθ . Theorem 5 For the case b > 0 and a > σ  b  o n√
−1 −1 a−σ ˆ T (θT − θ) =⇒ NR2 0, 2σC , with C = . Lθ −1 ab Proof : Since for a > σ and b > 0 the CIR process is ergodic and the stationary distribution is a Gamma law with shape a/σ and scale σ/b, the ergodic theorem yields Z Z a b 1 T ds hM iT 1 T Xs ds = , lim = and lim = C Pθ a.s.. lim T →∞ T →∞ T 0 b T →∞ T 0 Xs a−σ T o n Therefore, by the martingale central limit theorem we get Lθ √1T MT =⇒ NR2 (0, C). We complete the proof using identity (13).  In the following, it is relevant to rewrite, using Itˆo’s formula, the MLE error as follows   R T ds  R T   log XT − log x + (σ − a) 0 Xs 0 Xs ds − T (XT − x − aT )    a ˆ − a = R T ds R T  T   Xs ds − T 2  0 Xs 0 (14)  R  R T ds   RT  T  T log XT − log x + bT + σ 0 Xs − XT − x + b 0 Xs ds 0 Xdss    ˆ  . R T ds R T  bT − b =  2 X ds − T s 0 Xs 0 RT RT Consequently, the study of the vector (XT , 0 Xs , 0 Xdss ), done in the previous section, will be very helpful to investigate the limit law of the error. The asymptotic behavior of θˆT − θ can be summarized as follows. 11

Theorem 6 The MLE of θ = (a, b) is well defined for a ≥ σ and satisfies   n o 1 a − R1 ˆ 1. Case b = 0 and a = σ : Lθ diag(log T, T )(θT − θ) =⇒ , , where (Rt ) is τ I 1 1 Rt the CIR process, starting from 0, solution to (2), It = 0 Rs ds, and τ1 is the hitting time associated with Brownian motion τ1 = inf{t > 0 : Wt = √12σ }. The couple (R1 , I1 ) and the random time τ1 are independent.   n o p p a − R 1 2. Case b = 0 and a > σ : Lθ diag( log T , T )(θˆT − θ) =⇒ 2σ(a − σ)G, , I1 where (R1 , I1 ) is defined in the previous case, G is a standard normal random variable indepedent of (R1 , I1 ).

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Proof :

For the case b = 0 and a = σ    R T log XT −log x 1  × T 2 0 Xs ds − XTT−x−aT   log T log T      R   R  T T 1 1 ds 1    (log T )2 0 Xs × T 2 0 Xs ds − (log T )2 diag(log T, T )(θˆT − θ) =   R T ds RT  log XT −log x XT −x a 1  + − ×  2 2 2 (log T ) (log T ) X T (log T )  0 0 s      R   R T ds T  1  × 12 Xs ds − 1 2 2 (log T )

0 Xs

T

ds Xs



(log T )

0

By a scaling argument the process (X2t/a ) has the same distribution as a bidimensional square Bessel process, starting from x, BESQ2x . It follows that √ law law X2T /a = kBT + xk2 = T kB1 + x/ T k2 , where (Bt )t≥0 denotes a standard bidimensional Brownian motion. Hence, log XT / log T converges in law to one and consequently in probability. We complete the proof of the first case using the second assertion of Theorem 2. For the case b = 0 and a > σ, we have R ds   R  log XT −log x+(σ−a) 0T X T 1 s  √ × X ds − XTT√−x−aT  s  T2 0 log T log T      R   R  T ds T 1  × T12 0 Xs ds − log1 T  log T 0 Xs  p diag( log T , T )(θˆT − θ) = R ds   R T ds   log XT −log x+σ 0T X  XT −x 1 s  − ×  log T T log T 0 Xs      R  .  R  T ds T  1 1 1 × X ds − s log T 0 Xs T2 0 log T Note that for any u ∈ R, v > 0 and w > 0 we have " #! RT Z T log XT + (σ − a) 0 Xdss v w √ Ex exp u − 2 Xs ds − XT = T 0 T log T   Z η Ex Xt exp −ρXT − λ 0

12

T

Z Xs ds − µ 0

T

ds Xs



u √ with η = √log , λ = Tv2 and µ = u(a−σ) ,ρ= w . The moment generating-Laplace transform in T T log T the right hand side of the above equality is given by relation (3) of Proposition 1 . Consequently, using standard evaluations, it is easy to prove that the limit of the last quantity, when T tends to infinity, is equal to



 σa √ σv √ √ × √ σw sinh( σv) + σµ cosh( σv) s " !# 1 4uσ(a − σ) σ − a u lim exp − log T (a − σ)2 + √ + −√ T →+∞ 2σ 2σ log T log T  σa    √ σv σ 2 √ √ = exp u . √ a−σ σw sinh( σv) + σµ cosh( σv)

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It follows that log XT + (σ − a) √ log T

RT

ds 0 Xs

1 , 2 T

Z 0

T

XT Xs ds, T

! law

−→

r

2σ G, I1 , R1 a−σ

! as T tends to infinity.

Z T 1 1 ds P −→ (see the first We complete the proof of the second assertion using that log T 0 Xs a−σ log XT P assertion of Theorem 2) and that −→ 1 which is a starightforward consequence of the log T XT law −→ R1 .  above result, namely: T Theorem 7 The MLE of θ = (a, b) is well defined for a ≥ σ and satisfies   n o √ b √ ˆ 1. Case b > 0 and a = σ : Lθ diag(T, T )(θT − θ) =⇒ , 2bG , where G is a stanτ2 dard normal random variable indepedent of τ2 the hitting time associated with Brownian motion τ2 = inf{t > 0 : Wt = √b2σ }. 2. Case b < 0 and a ≥ σ : the MLE estimator θˆT is not consistent. Proof :

For the first case we have   R  T log XT −log x 1  × X ds − XT −x−aT  s T T 0 T    R   R    T ds T  1  × T1 0 Xs ds − T1  T 2 0 Xs  √ diag(T, T )(θˆT − θ) =  R R   XT −x−aT +b 0T Xs ds T log XT −log x+bT  √ √ − × T12 0   T T T    R   R    T ds T 1 1  × X ds − T1 s 2 T T 0 0 Xs

13

ds Xs



At first, note that the term XTT appearing in the first component of the above relation vanishes as T tends to infinity. This is straightforward by combining that √ Z T Z p XT 2a x+a b T = − Xs ds + Xs dWs , T T T 0 T 0

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with the classical large law number property for continuous martingales which applies here, RT P since T1 0 Xs ds −→ ab (see assertion 2 of Theorem 4). Now, note that for any u ∈ R and v > 0, we have " #! RT Z T XT − aT + b 0 Xs ds ds v √ Ex exp u − 2 T 0 Xs T    Z T Z T √ ds −au T =e Xs ds − µ Ex exp −ρXT − λ 0 Xs 0 with ρ = − √uT , λ = − √ubT and µ = Tv2 . The moment generating-Laplace transform in the right hand side of the above equality is given by relation (9) of Proposition 2 with a = σ. Using standard evaluations, it is easy to prove that the limit of the last quantity, when T tends to infinity, is equal to √

!!− T2 √vσ −1

s

√ bT T 4aub − au T sinh b2 − √ 2 2 T s !!   √ √ 2 v T 4aub bT − au T exp (− √ − 1) b2 − √ = lim exp T →∞ 2 2 T σ T  2 2 √  σ u b u − √ = exp b σ  q   R XT −aT +b 0T Xs ds 1 R T ds 2 √ converges in law to σG, τ when T tends Hence the couple , 2 T 2 0 Xs b T to infinity. On the other hand, using that the CIR process (Xt ) can be represented in terms of a BESQ2x as follows a law XT = e−bT BESQ2x ( (ebT − 1)) 2b 2 and that log BESQx (T )/ log T converges in probability to one, we obtain that (log XT − log x + log XT P bT )/T converges in probability to b and consequently we deduce −→ 0. We complete T Z T 1 P a Xs ds −→ (see assertion 2 of Theorem 4). the proof of the first assertion using that T 0 b For the second case, b < 0 and a ≥ σ, we have 1 exp T →∞ 2





lim

14

R T dWs RT √ √ X ds − T Xs dWs s 0 0 0 Xs a ˆT − a = 2σ R T ds R T Xs ds − T 2 0 Xs 0 RT √ R T dWs T ebT 0 Xs dWs √ − RT 0 Xs √ ebT 0 Xs ds 2σ = . R T ds T 2 ebT − RT 0 Xs ebT 0 Xs ds RT

√



R

t ds 0 Xs

is an increasing process converging to a finite random variable, we easily R  t dWs deduce the almost sure convergence of the Brownian martingale 0 √ . Thanks to the Xs t≥0 RT convergence in law of the term ebT 0 Xs ds, we finish the proof using that

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Since

2 2bT

T e

t≥0

Z E 0

2

T

p

Xs dWs

2 2bT

Z

=T e

T 2 2bT

Z

E(Xs )ds = T e 0

0

T

a −bs  + (x − )e ds −→ 0. T →∞ b b

a



4

Estimation of the CIR diffusion from discrete observations

In this section, we consider rather a discrete sample (Xtk )0≤k≤n of the CIR diffusion at deterministic and equidistant instants (tk = k∆n )0≤k≤n . Our aim is to study a new estimator for θ = (a, b) based on discrete observations, under the conditions of high frequency, ∆n → 0, and infinite horizon, n∆n → ∞. A common way to do that is to consider a discretization of the logarithm likelihood (see [10] and references there). In our case this method yields the contrast n−1 n−1 1 X (a − bXtk )2 1 X a − bXtk ∆n (Xtk+1 − Xtk ) − , 2σ k=0 Xtk 4σ k=0 Xtk

Our approach is slightly different since we discretize the continuous time MLE, obtained in the previous section, instead of considering the maximum argument of the above contrast. Doing so, we take advantage from limit theorems obtained in the continuous time observations. Hence, thanks to relation (14), the discretized version of the MLE is given by   Pn−1 ∆n  Pn−1  log X − log x + σ  tn k=0 Xtk k=0 ∆n Xtk − tn (Xtn − x)  ∆n   a ˆ = Pn−1 ∆n Pn−1  tn  2  k=0 Xtk k=0 ∆n Xtk − tn  (15)    P P  n−1 n−1  tn log Xtn − log x + σ k=0 X∆tn − (Xtn − x) k=0 X∆tn   k ∆n  ˆ  btn = . Pn−1 ∆n Pn−1 k   2 ∆ X − t n tk n k=0 Xt k=0 k

15

In order to prove limit theorems on the discrete estimator, θˆt∆nn := (ˆ a∆tnn , ˆb∆tnn ), we need to control R tn R tn ds Pn−1 Pn−1 ∆n the errors 0 Xs ds − k=0 ∆n Xtk and 0 Xs − k=0 Xt and somme moments behavior on k the increments of the CIR process are needed.

4.1

Moment properties of the CIR process

Let us first, yield some essential properties on moments of the CIR process. Proposition 3 For all η ∈] − σa , +∞[ we have 1. For b = 0, we have

a η η Γ( σ + η) η t , Ex (Xt )∼σ Γ( σa )

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and we have, sup0≤t≤1 Ex (Xtη ) < ∞ and supt≥1 2. For b > 0, we have Ex (Xtη )∼



σ2 2b

η

as t → +∞,

Ex (Xtη ) < ∞. tη

Γ( σa + η) , as t → +∞, Γ( σa )

and we have, supt≥0 Ex (Xtη ) < ∞. Proof : For the first assertion we take ρ = λ = 0 and let µ tend to 0 in relation (3) of Proposition 1, it follows that Ex (Xtη ) = (σt)η

 x a Γ( σa + η) a x exp − + η, , F 1 1 Γ( σa ) σt σ σ σt

and we get the result using that lim 1 F1

t→∞

a x + η, , = 1. σ σ σt

a

The second assertion is immediate using the previous assertion and that, for b > 0, we have the 2a
σ relation Xt = e−bt Y 2b (ebt − 1) , where Y denotes a BESQxσ . As the function t 7→ Ex (Xtη ) is continuous we obtain the last result.  In the following, C denotes a positive constant that may change from line to line. Proposition 4 In the case b > 0, let 0 ≤ s < t such that 0 < t − s < 1 we have 1. For all q ≥ 1,

q

Ex |Xt − Xs |q ≤ C(t − s) 2 . 2. For all a > 2σ, 1 1 ≤ C(t − s) 12 . Ex − Xt Xs 16

Proof :

First, note that by linearization technics we obtain p d(ebt Xt ) = aebt dt + ebt 2σXt dWt .

(16)

Hence, Xt − Xs =

a b

− Xs



1−e

−b(t−s)




t

Z

e−b(t−u)

+

p 2σXu dWu .

s

Let q ≥ 1, q

q−1

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Ex |Xt − Xs | ≤ 2

1−e


−b(t−s) q

Z t q a q p q−1 −b(t−u) Ex − Xs + 2 Ex e 2σXu dWu . b s

The first term in the right hand side is bounded by C(t − s)q since 1 − e−x ≤ x, for x ≥ 0, and supt≥0 Ex (Xtq ) < ∞ (see proposition 3). For the second term, the Burkholder-Davis-Gundy inequality yields Z t q  2q Z t p Ex e−b(t−u) 2σXu dWu ≤ Ex 2σe−2b(t−u) Xu du s

s

For q ≥ 2, we apply the Holder inequality on the integral to obtain Z t q Z t p q q −b(t−u) −1 Ex e e−bq(t−u) Ex (Xu2 ) du 2σXu dWu ≤ C(t − s) 2 s

s q 2

≤ C(t − s) , q

since the integrand is bounded using supu≥0 Ex (Xu2 ) < ∞. For 1 ≤ q < 2, we apply the Holder inequality on the expectation and we obtain  2q Z t q Z t p Ex e−b(t−u) 2σXu dWu 2σe−2b(t−u) Ex (Xu )du ≤ s

s q

≤ C(t − s) 2 , since the integrand is also bounded using once again supu≥0 Ex (Xu ) < ∞. Combining these three upper bounds with 0 < t − s < 1, we deduce the first assertion. Concerning the second assertion, we use the Holder inequality with 1q + p2 = 1 and 2 < p < σa which ensures the boundedness of all the terms. We obtain 1



1 ≤ kXt − Xs k Xt−1 Xs−1 ≤ C(t − s) 21 , Ex − q p p Xt Xs q

since by Proposition 3 we have supt≥0 Ex (Xt−p ) < ∞, for p < σa , and Ex |Xt − Xs |q ≤ C(t − s) 2 , for q ≥ 2.  Proposition 5 In the case b = 0, let 0 ≤ s < t such that 0 < t − s < 1 we have 1. For all q ≥ 2,

q

q

Ex |Xt − Xs |q ≤ C(t − s) 2 sup Ex (Xu2 ). s≤u≤t

17

2. For all 1 ≤ q < 2,

q

q

Ex |Xt − Xs |q ≤ C(at + x) 2 (t − s) 2 . 3. For all a > 2σ, there exists q ≥ 2 and 2 < p < σa , such that  q 1 1



1 1 q 2

Xt−1 Xs−1 . 2 Ex E (X − sup ≤ C(t − s) ) u x p p Xt Xs s≤u≤t Proof :

First, we have Z tp 2σXu dWu . Xt − Xs = a(t − s) + s

Let q ≥ 1, q

q−1

Ex |Xt − Xs | ≤ 2

q

(a(t − s)) + 2

q−1

Z t q p Ex 2σXu dWu .

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s

For the second term, the Burkholder-Davis-Gundy inequality yields q  2q Z t p Z t 2σXu dWu ≤ Ex Ex 2σXu du s

s

For q ≥ 2, we apply the Holder inequality on the integral to obtain Z t p q Z t q q −1 Ex (Xu2 ) du Ex 2σXu dWu ≤ C(t − s) 2 s

s

≤ C(t − s)

q 2

q

sup Ex (Xu2 ),

s≤u≤t

which completes the second assertion. For 1 ≤ q < 2, we apply the Holder inequality on the expectation and we obtain  2q q Z t Z t p 2σXu dWu 2σEx (Xu )du Ex ≤ s

s q

q

≤ C(at + x) 2 (t − s) 2 , using Ex (Xu ) = au + x. This completes the second assertion. Concerning the last one, we use the Holder inequality with 1q + p2 = 1 and 2 < p < σa which ensures the boundedness of all the terms. We obtain

1

1 ≤ kXt − Xs k Xt−1 Xs−1 − Ex q p p X t Xs 1  q 



1 q ≤ C(t − s) 2 sup Ex (Xu2 ) Xt−1 p Xs−1 p , s≤u≤t

q

q

since for q ≥ 2, Ex |Xt − Xs |q is bounded by C(t − s) 2 sups≤u≤t Ex (Xu2 ). 18



4.2

Parameter estimation of θ = (a, b)

The task now is to give sufficient conditions on the frequency ∆n in order to get the same asymptotic results of Theorem 5 and Theorem 6, when we replace the continuous MLE estimator θˆtn := (ˆ atn , ˆbtn ) by the discrete one θˆt∆nn := (ˆ a∆tnn , ˆb∆tnn ).

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Theorem 8 Under the above notations, for b > 0 and a > 2σ, if n∆2n → 0 then we have ! ! Z Z n−1 n−1 √ √ 1 X 1 X ∆n 1 tn 1 tn 1 Xs ds − ds − −→ 0, tn ∆n Xtk −→ 0 and tn tn 0 tn k=0 tn 0 Xs tn k=0 Xtk (17) in L1 and in probability. Consequently  b  o n√
−1 −1 ∆ n a−σ , with Γ = Lθ tn (θˆtn − θ) =⇒ NR2 0, 2σΓ . −1 ab Proof : For the first convergence in relation (17), thanks to the first assertion of Proposition 4, we consider the L1 norm and we write Z n−1 n−1 Z tn X 1 X tk+1 1 √ Ex ∆n Xtk ≤ √ Ex |Xs − Xtk | ds Xs ds − tn 0 t n t k k=0 k=0 Z n−1 C X tk+1 √ ≤ √ s − tk ds tn k=0 tk n−1 √ C X 32 √ ≤ ∆n = C n∆n −→ 0, tn k=0

since n∆2n −→ 0. In the same manner, thanks to the second assertion of Propostion 4, we have Z n−1 n−1 Z tn ds X 1 ∆ 1 1 1 X tk+1 n ds √ Ex − Ex − ≤ √ Xs Xtk tn 0 Xs k=0 Xtk tn k=0 tk √ ≤ C n∆n −→ 0. √ √ √ Finally, as tn (θˆt∆nn − θ) = tn (θˆt∆nn − θˆtn ) + tn (θˆtn − θ) and thanks to Theorem 5, it is √ sufficient to prove tn (θˆt∆nn − θˆtn ) −→ 0 in probability. This last convergence is easily obtained by standard arguments using the first part of the theorem. In fact, for instance to prove the convegence of the first compenent we write ˆtn √tn (Aˆ∆t n − Aˆtn ) − Aˆtn √tn (B ˆt∆n − B ˆtn ) √ B ∆n n n tn (ˆ atn − a ˆtn ) = . ˆt2 + B ˆtn (B ˆt∆n − B ˆtn ) B n n where Aˆtn = ˆtn = B





R tn ds R tn log Xtn −log x σ 1 +  R tn   tnR 0 Xs  tn 0 tn ds tn 1 1 Xs ds − 1, tn 0 Xs tn 0 19



Xs ds −

Xtn −x tn

ˆt∆n ) is simply the discretization version of the couple (Aˆtn , B ˆtn ). From the first part and (Aˆ∆tnn , B n √ ˆ∆n ˆt∆n − B ˆtn ) towards zero and from we have the convergence in probability of tn (Atn − Aˆtn , B n ˆtn ). Which completes Theorem 4 we have the convergence in probabilty of the couple (Aˆtn , B the proof.  Theorem 9 Under the above notations, for b = 0 and a > 2σ, 1. if n∆2n → 0 then we have tn

1 t2n

tn

Z 0

n−1 1 X Xs ds − 2 ∆n Xtk tn k=0

! −→ 0, in L1 and in probability.

3

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n∆n2 2. if → 0 then we have log(n∆n ) tn

1 log tn

Z 0

tn

n−1 1 1 X ∆n ds − Xs log tn k=0 Xtk

! −→ 0, in L1 and in probability.

3

n∆n2 → 0 and 3. Consequently, if → 0 then we have log(n∆n )   n o p p a − R1 ∆n ˆ Lθ diag( log tn , tn )(θtn − θ) =⇒ 2σ(a − σ)G, , I1 Rt where (Rt ) is the CIR process, starting from 0, solution to (2), It = 0 Rs ds, and G is a standard normal random variable indepedent of (R1 , I1 ). n∆2n

Remarks : Proof : For the first convergence, thanks to the second assertion of Proposition 5, we consider the L1 norm and we write Z n−1 Z n−1 tn X 1 1 X tk+1 Ex |Xs − Xtk | ds Ex Xs ds − ∆n Xtk ≤ tn 0 t n t k k=0 k=0 n−1 Z tk+1 X √ √ C ≤ as + x s − tk ds tn k=0 tk 1 Z ∆n2 tn √ as + xds ≤ C tn 0 √ ≤ C n∆n −→ 0,

20

since n∆2n −→ 0. In the same manner, thanks to the second assertion of Propostion 4, we have Z n−1 n−1 Z tn ds X 1 tn ∆n tn X tk+1 1 Ex − Ex − ds ≤ log tn 0 Xs k=0 Xtk log tn k=0 tk Xs Xtk √ n−1 Z  q 1

−1 tn ∆n X tk+1 q

Xs ds, sup Ex (Xu2 ) Xt−1 ≤ C k p p log tn k=0 tk tk ≤u≤s using the third assertion of Proposition 5. Thanks to the first assertion of Proposition 3, in one hand, for s < 1 the integrand is bounded, and in the other hand, for s ≥ 1,  q 1



1 q

≤ Ct−1 ≤ 2Cs−1 when ∆n ≤ 1 . Ex (Xu2 ) ≤ Cs 2 , Xs−1 p ≤ Cs−1 , and Xt−1 k k p 2 tk ≤u≤s sup

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 1q

kXs−1 k ≤ Cs− 23 . Consequently, for n Ex (Xu ) Xt−1 p k p q 2

It follows that for s ≥ 1, suptk ≤u≤s large enough, we get Z √ n−1 tn ds X tn ∆n 1 ∆n tn Ex − (1 + √ ) ≤ C log tn 0 Xs k=0 Xtk log tn tn √ tn ∆n ≤ C −→ 0, log tn 3

√ √ n∆n2 since → 0. Finally, as diag( log tn , tn )(θˆt∆nn − θ) = diag( log tn , tn )(θˆt∆nn − θˆtn ) + log(n∆n ) √ diag( log tn√ , tn )(θˆtn − θ) and thanks to the second assertion of Theorem 6, it is sufficient to prove diag( log tn , tn )(θˆt∆nn − θˆtn ) −→ 0 in probability. In order to show this convegence, a similar analysis to that in the previous proof can be done. Indeed, we rewrite the both compenents as follows log tn (ˆ a∆tnn − a ˆtn ) = tn (ˆb∆tnn − ˆbtn ) where Aˆtn = ˆtn = B ˆ tn = D

=

ˆt ) ˆ ∆n −D ˆt log tn (D ˆt )−A ˆ∆n −A ˆ t log tn (A D n n n n tn tn ∆n 2 ˆ ˆ ˆ ˆ Dtn +Dtn (Dtn −Dtn ) ˆt tn (D ˆ ∆n −D ˆt ) ˆ t tn (B ˆ ∆n −B ˆt )−B D n

n

tn

n

n

tn

ˆt ) ˆ 2 +D ˆ t (D ˆ ∆n −D D n n tn tn





,



Rt R tn log Xtn −log x 1 n −x + tσ2 0 n Xdss Xs ds − tXn tlog t2n log tn 0 tn n R tn ds   R (log Xtn −log x+σ 0 Xs ) Xtn −x t n 1 ds − t2 tn log tn log tn 0 Xs n    R tn R tn ds 1 1 Xs ds − log1tn , log tn 0 Xs t2n 0

ˆt∆n , D ˆ t∆n ) is simply the discretization version of the triplet (Aˆtn , B ˆtn , D ˆ tn ). From the and (Aˆ∆tnn , B n n ∆ ∆ ˆt n − B ˆtn , D ˆ t∆n − D ˆ tn ) above assertions the rate of convergence in probability of (Aˆtnn − Aˆtn , B n n towards zero is tn , using Theorem 2 we have the convergence in distribution of the couple √ ˆ ˆ ˆ (Atn , Btn , Dtn ), and now it is easy to check that diag( log tn , tn )(θˆt∆nn − θˆtn ) vanishes. Which completes the proof.  21

5

Conclusion

We have investigate the asymptotic behavior of the MLE for square-root diffusions in ergodic and nonergodic cases. However these estimators does not make sense when a < σ and are not consistent when b < 0. One can wonder: how to overcome this problem by constructing, in these particular cases, new consistent estimators with explicit asymptotic behavior.

References [1] Mohamed Ben Alaya and Ahmed Kebaier. Parameter Estimation for the Square-root Diffusions : Ergodic and Nonergodic Cases. hal-00579644 (submitted), 2011.

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