Mátyás Barczy
email: barczy@math.u-szeged.hu

Mohamed Ben Alaya
email: mohamed.ben-alaya@univ-rouen.fr

Ahmed Kebaier
email: kebaier@math.univ-paris13.fr

Gyula Pap

Asymptotic properties of maximum likelihood estimator for the growth rate of a stable CIR process based on continuous time observations

Keywords: 2010 Mathematics Subject Classifications: 60H10, 91G70, 60F05, 62F12 stable Cox-Ingersoll-Ross process, maximum likelihood estimator

We consider a stable Cox-Ingersoll-Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. In all cases we prove strong consistency of the MLE in question, in the subcritical case asymptotic normality, and in the supercritical case asymptotic mixed normality are shown as well. In the critical case the description of the asymptotic behavior of the MLE in question remains open.

Introduction

We consider a jump-type Cox-Ingersoll-Ross (CIR) process driven by a standard Wiener process and a spectrally positive strictly α-stable Lévy process given by the SDE dY t = (a -bY t ) dt + σ Y t dW t + δ α Y t-dL t , t ∈ [0, ∞), (1.1) with an almost surely non-negative initial value Y 0 , where a ∈ [0, ∞), b ∈ R, σ ∈ [0, ∞), δ ∈ (0, ∞), α ∈ (1, 2), (W t ) t∈[0,∞) is a 1-dimensional standard Wiener process, and (L t ) t∈[0,∞) is a spectrally positive α-stable Lévy process such that the characteristic function of L 1 takes the form

E(e iθL 1 ) = exp ∞ 0 (e iθz -1 -iθz)C α z -1-α dz , θ ∈ R, (1.2)
where C α := (αΓ(-α)) -1 and Γ denotes the Gamma function. In fact, (L t ) t∈[0,∞) is a strictly α-stable Lévy process, see, e.g., Sato [35, part (vi) of Theorem 14.7]. We suppose that Y 0 , (W t ) t∈[0,∞) and (L t ) t∈[0,∞) are independent. Under the given conditions together with E(Y 0 ) < ∞, there is a (pathwise) unique strong solution of the SDE (1.1) with P(Y t ∈ [0, ∞) for all t ∈ [0, ∞)) = 1. As a matter of fact, the SDE (1.1) is a special case of the SDE (1.8) in Fu and Li [START_REF] Fu | Stochastic equations of non-negative processes with jumps[END_REF] (with the special choice z 1 ≡ 0), for which the existence of a pathwise unique non-negative strong solution has been proved (see Fu and Li [START_REF] Fu | Stochastic equations of non-negative processes with jumps[END_REF]Corollary 6.3]). Eventually, the process (Y t ) t∈[0,∞) given by the SDE (1.1) is a continuous state and continuous time branching process with immigration (CBI process), see (ii) of Proposition 2.1. We call Y an α-stable CIR process (or Alpha-CIR process), which is a generalization of the usual CIR process (given by the SDE (1.1) formally with δ = 0).

Stable CIR processes become more and more popular in stochastic modelling, and it is an interesting class of CBI processes on its own right as well. Carr and Wu [9, equation (31)] considered a stochastic process admitting an infinitesimal generator which coincides with the corresponding one of an α-stable CIR process with σ = 0, see (iv) of Proposition 2.1.

Li and Ma [START_REF] Li | Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model[END_REF] proved exponential ergodicity for the process (Y t ) t∈[0,∞) provided that a ∈ (0, ∞) and b ∈ (0, ∞), for more details, see (ii) of Theorem 2.5. Li and Ma [START_REF] Li | Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model[END_REF] also described the asymptotic behavior of the conditional least squares estimator (LSE) and weighted conditional LSE of the drift parameters (a, b) of an α-stable CIR process given by the SDE (1.1) with σ = 0, based on (discretely observed) low frequency observations in the subcritical case (i.e., when b ∈ (0, ∞)). In the region α ∈ (1, 1+ √ 5

2 ), Li and Ma [START_REF] Li | Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model[END_REF] showed that the normalizing factor for the LSE of (a, b) is n (α-1)/α 2 , which is quite different from the √ n-normalization being quite usual for subcritical models. On the top of it all, Li and Ma [START_REF] Li | Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model[END_REF] also proved that the corresponding normalizing factor for the weighted LSE of (a, b) is n (α-1)/α (being different from the one for the (usual) LSE) in the whole region α ∈ (1, 2). Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF] investigated several properties of α-stable CIR processes such as integral representations, branching property in the pathwise sense, necessary and sufficient conditions for strictly positiveness and they made an analysis of the jumps of the process. Further, they used α-stable CIR processes for interest rate modelling and pricing by pointing out that these processes can describe some recent phenomena on sovereign bond market such as large fluctuations at a local extent together with the usual small oscillations, for more details, see the Introduction of Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF]. Very recently, Jiao et al. [START_REF] Jiao | A branching process approach to power markets[END_REF] have proposed concrete examples of applications and investigated a factor model for electricity prices, where α-stable CIR processes may appear as factors of the model in question.

Peng [START_REF] Peng | A stable Cox-Ingersoll-Ross model with restart[END_REF] introduced and studied a so-called α-stable CIR process with restart, by which one means that the process in question behaves as an α-stable CIR process given by the SDE (1.1) with σ = 0, it is killed at the boundary 0 of [0, ∞), and according to an exponential clock it jumps to a new point in [0, ∞) according to a given probability distribution on [0, ∞). As it was pointed out in Peng [START_REF] Peng | A stable Cox-Ingersoll-Ross model with restart[END_REF], restart phenomenon appears in internet congestion as well: whenever a web page takes too much time to appear, it is useful to press the reload button and then usually the web page appears immediately.

Yang [START_REF] Yang | Maximum likelihood type estimation for discretely observed CIR model with small α-stable noises[END_REF] studied α-stable CIR processes with small α-stable noises given by the SDE dY ε t = (a -bY ε t ) dt + δε q Y ε t-dL t , t ∈ [0, ∞), (1.3) with a non-negative deterministic initial value Y ε 0 = y 0 ∈ [0, ∞), where q ∈ 0, 1 1-1/α and ε ∈ (0, ∞). The asymptotic behavior of an approximate maximum likelihood estimator (MLE) of (a, b, δ) has been described based on discrete time observations at n regularly spaced time points k n , k ∈ {1, . . . , n}, on a fixed time interval [0,[START_REF] Amann | Ordinary Differential Equations. An Introduction to Nonlinear Analysis[END_REF]. Tending ε to 0 and n → ∞ at a given rate, for some restricted parameter set, Yang [START_REF] Yang | Maximum likelihood type estimation for discretely observed CIR model with small α-stable noises[END_REF]Theorem 2.4] proved asymptotic normality of the approximate MLE in question. In some sense it is surprising, since this restricted parameter set contains parameters belonging to critical (b = 0) and supercritical (b ∈ (-∞, 0)) models as well both with normal limit distributions, and for critical models, the limit distribution, in general, is not even mixed normal.

Ma and Yang [START_REF] Ma | Small noise fluctuations of the CIR model driven by α-stable noises[END_REF] investigated asymptotic behavior of the LSE of a for the model (1.3) (all the other parameters are supposed to be known) based on discrete time observations as in Yang [START_REF] Yang | Maximum likelihood type estimation for discretely observed CIR model with small α-stable noises[END_REF] described above. They described the asymptotic behavior of the LSE in question and derived large and moderate deviation inequalities for it as well, see Ma and Yang [START_REF] Ma | Small noise fluctuations of the CIR model driven by α-stable noises[END_REF]Theorems 2.1,.

In this paper, supposing that a ∈ [0, ∞), σ, δ ∈ (0, ∞) and α ∈ (1, 2) are known, we study the asymptotic properties of the MLE of b ∈ R based on continuous time observations (Y t ) t∈[0,T ] with T ∈ (0, ∞), starting the process Y from some known non-random initial value y 0 ∈ [0, ∞).

The paper is organized as follows. Section 2 is devoted to some preliminaries. First, we recall some useful properties of the stable CIR process (Y t ) t∈[0,∞) given by the SDE (1.1) such as the existence of a non-negative pathwise unique strong solution, the forms of the Laplace transform and the infinitesimal generator or conditions on the strictly positiveness of the process or the integrated process, see Proposition 2.1. We derive a so-called Grigelionis form of the semimartingale (Y t ) t∈[0,∞) , see Proposition 2.2. Based on the asymptotic behavior of the expectation of Y t as t → ∞, we distinguish subcritical, critical or supercritical cases according to b ∈ (0, ∞), b = 0 or b ∈ (-∞, 0), see Proposition 2.3 and Definition 2.4. In Proposition 2.3 it also turns out that the parameter b can be interpreted as a growth rate of the model. We recall a result about the existence of a unique stationary distribution for the process (Y t ) t∈[0,∞) in the subcritical and critical cases, and about its exponential ergodicity in the subcritical case, due to Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF], Li and Ma [START_REF] Li | Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model[END_REF] and Jin et al. [START_REF] Jin | Exponential ergodicity of an affine two-factor model based on the α-root process[END_REF], see Theorem 2.5. We call the attention that there exists a unique stationary distribution for (Y t ) t∈[0,∞) in the critical case as well. Remark 2.6 is devoted to give an alternative proof for the weak convergence of Y t as t → ∞ in Theorem 2.5 in case of σ ∈ (0, ∞), giving more insight as well. In Remark 2.7, we give a statistic for σ 2 using continuous time observations (Y t ) t∈[0,T ] with an arbitrary T ∈ (0, ∞), and due to this result we do not consider the estimation of the parameter σ, it is supposed to be known. In Section 3, we derive a formula for the joint Laplace transform of Y t and t 0 Y s ds, where t ∈ [0, ∞), using Theorem 4.10 in Keller-Ressel [START_REF] Keller-Ressel | Affine Processes -Theory and Applications in Finance[END_REF], see Theorem 3.1. We note that this form of the joint Laplace transform in question is a consequence of Theorem 5.3 in Filipović [START_REF] Filipović | A general characterization of one factor affine term structure models[END_REF], a special case of Proposition 3.3 in Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF] as well, and it is used for describing the asymptotic behavior of the MLE of b in question in the critical and supercritical cases. Section 4 is devoted to prove the existence and uniqueness of the MLE of b (provided that σ ∈ (0, ∞)) deriving an explicit formula for it as well, see Proposition 4.2. In Remark 4.3, under the additional assumption a ∈ σ 2 2 , ∞ , we prove that L t is a measurable function of (Y u ) u∈[0,T ] for all t ∈ [0, T ] with any T ∈ (0, ∞). In Section 5, provided that a ∈ (0, ∞), we prove strong consistency and asymptotic normality of the MLE of b in the subcritical case, see Theorem 5.1. The asymptotic normality in question holds with a usual square root normalization ( √ T ), but as usual, the asymptotic variance depends on the unknown parameter b, as well. To get around this problem, we also replace the normalization √ T by a random one

1 σ t 0 Y s ds 1/2
(depending only on the observation, but not on the parameter b) with the advantage that the MLE of b with this random scaling is asymptotically standard normal, so one can give asymptotic confidence intervals for the unknown parameter b, which is desirable for practical purposes. Section 6 is devoted to prove the strong consistency of the MLE of b in the critical case, provided that a ∈ (0, ∞), (see Theorem 6.2) using the limit behavior of the unique locally bounded solution of the differential equation (3.1) at infinity described in Proposition 6.1. We call the attention to the fact that for the α-stable CIR process (Y t ) t∈[0,∞) , the critical case (b = 0) is somewhat special (compared to the original CIR process with b = 0), since there still exists a unique stationary distribution for (Y t ) t∈[0,∞) , however its expectation is infinite unless a = 0 (see Theorem 2.5), and surprisingly, we can prove strong consistency of the MLE in question not only weak consistency usually proved for critical models. In the critical case the description of the asymptotic behavior of the MLE remains open. In Section 7, for the supercritical case, provided that a ∈ (0, ∞), we prove that the MLE of b is strongly consistent and asymptotically mixed normal with the deterministic scaling e -bT /2 , and it is asymptotically standard normal with the random scaling

1 σ t 0 Y s ds 1/2
, see Theorem 7.4. We point out that the limit mixed normal law in question is characterized in a somewhat complicated way, namely in its description a positive random variable V comes into play of which the Laplace transform contains a function related to the branching mechanism of the CBI process (Y t ) t∈[0,∞) , see Theorem 7.1. We give two proofs for the derivation of the Laplace transform of V , and the second one is heavily based on the general theory of CBI processes, for which we will refer to Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]. We close the paper with three Appendices, where we recall certain sufficient conditions for the absolute continuity of probability measures induced by semimartingales together with a representation of the Radon-Nikodym derivative (Appendix A), some limit theorems for continuous local martingales (Appendix B) and in case of a 3 2 -stable CIR process we present some explicit formulae for the Laplace transform of the unique stationary distribution in the subcritical and critical cases, of Y t , t ∈ [0, ∞), in all the cases of b ∈ R, and of V in the supercritical case, respectively (Appendix C).

Finally, we summarize the novelties of the paper. According to our knowledge, maximum likelihood estimation based on continuous time observations has never been studied before for the α-stable CIR process (Y t ) t∈[0,∞) , and since these processes become more and more popular in financial mathematics and market models for electricity prices, the problem of estimating its parameters is an important question as well. Further, in the critical case, somewhat surprisingly, we can prove strong consistency of the MLE of b, which can be considered as a new phenomenon, since for other critical financial models, such as for the usual CIR process or for the Heston process, only weak consistency is proved in the critical case, see Overbeck [32, Theorem 2, parts (iii) and (iv)] and Barczy and Pap [6,Remark 4.4], respectively.

Preliminaries

Let N, Z + , R, R + , R ++ , R -, R --and C denote the sets of positive integers, non-negative integers, real numbers, non-negative real numbers, positive real numbers, non-positive real numbers, negative real numbers and complex numbers, respectively. For x, y ∈ R, we will use the notations x ∧ y := min(x, y) and x ∨ y := max(x, y). The integer part of a real number x ∈ R is denoted by ⌊x⌋. By x and A , we denote the Euclidean norm of a vector x ∈ R d and the induced matrix norm of a matrix A ∈ R d×d , respectively. By B(R + ), we denote the Borel σ-algebra on R + . We will denote the convergence in probability, in distribution and almost surely, and almost sure equality by

P -→, D -→, a.s.
-→ and a.s. = , respectively. By C 2 c (R + , R) and C ∞ c (R + , R), we denote the set of twice continuously differentiable real-valued functions on R + with compact support and the set of infinitely differentiable real-valued functions on R + with compact support, respectively.

Let Ω, F, (F t ) t∈R + , P be a filtered probability space satisfying the usual conditions, i.e., (Ω, F, P) is complete, the filtration (F t ) t∈R + is right-continuous, F 0 contains all the P-null sets in F, and F = σ t∈R + F t . Let (W t ) t∈R + be a standard Wiener process with respect to the filtration (F t ) t∈R + , and (L t ) t∈R + be a spectrally positive strictly α-stable Lévy process with respect to the filtration (F t ) t∈R + such that the characteristic function of L 1 is given by (1.2). We assume that W and L are independent. Recall that the Lévy-Itô's representation of L takes the form

L t = (0,t] (0,∞) z µ L (ds, dz) = γt + (0,t] (0,1] z µ L (ds, dz) + (0,t] (1,∞) z µ L (ds, dz) (2.1) for t ∈ R + , where µ L (ds, dz) := u∈R + ½ {∆Lu =0} ε (u,∆Lu) (ds, dz) is the integer-valued Poisson random measure on R 2
++ associated with the jumps ∆L u := L u -L u-, u ∈ R ++ , ∆L 0 := 0, of the process L, and ε (u,x) denotes the Dirac measure at the point (u, x) ∈ R 2 + , µ L (ds, dz) := µ L (ds, dz) -ds m(dz), where m(dz) := C α z -1-α ½ (0,∞) (z) dz, and

γ := -(1,∞) z ds m(dz) = -C α ∞ 1 z -α dz = Cα 1-α .
The measure m is nothing else but the Lévy measure of L. We also note that (L t ) t∈R + is a martingale and consequently E(L t ) = 0, t ∈ R + .

The next proposition is about the existence and uniqueness of a strong solution of the SDE (1.1) stating also that Y is a CBI process with explicitly given branching and immigration mechanisms and we also collect some other useful properties of Y based on Dawson and Li [START_REF] Dawson | Skew convolution semigroups and affine Markov processes[END_REF], Fu and Li [START_REF] Fu | Stochastic equations of non-negative processes with jumps[END_REF], Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF] and Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF].

2.1 Proposition. Let η 0 be a random variable independent of (W t ) t∈R + and (L t ) t∈R + satisfying

P(η 0 ∈ R + ) = 1 and E(η 0 ) < ∞. Let a ∈ R + , b ∈ R, σ ∈ R + , δ ∈ R ++ ,
and α ∈ (1, 2). Then the following statements hold.

(i) There exists a pathwise unique strong solution

(Y t ) t∈R + of the SDE (1.1) such that P(Y 0 = η 0 ) = 1 and P(Y t ∈ R + for all t ∈ R + ) = 1.
(ii) The process (Y t ) t∈R + is a CBI process having branching mechanism

R(z) = σ 2 2 z 2 + δ α α z α + bz, z ∈ R + ,
and immigration mechanism

F (z) = az, z ∈ R + .
(iii) For all t ∈ R + and y 0 ∈ R + , the Laplace transform of Y t takes the form

E(e -λYt | Y 0 = y 0 ) = exp -y 0 v t (λ) - t 0 F (v s (λ)) ds (2.2)
for all λ ∈ R + , where R + ∋ t → v t (λ) ∈ R + is the unique locally bounded solution to

(2.3) ∂ ∂t v t (λ) = -R(v t (λ)), v 0 (λ) = λ. If t ∈ R + , y 0 ∈ R + and λ ∈ R ++ \ {θ 0 } with θ 0 := inf{z ∈ R ++ : R(z) ∈ R + } ∈ R + , then we have (2.4) E(e -λYt | Y 0 = y 0 ) = exp -y 0 v t (λ) + vt(λ) λ F (z) R(z) dz .
Especially, (2.4) holds for all λ ∈ R ++ whenever b ∈ R + .

(iv) The infinitesimal generator of Y takes the form

(Af )(y) = (a -by)f ′ (y) + σ 2 2 yf ′′ (y) + δ α y ∞ 0 f (y + z) -f (y) -zf ′ (y) C α z -1-α dz, (2.5) where y ∈ R + , f ∈ C 2 c (R + , R)
, and f ′ and f ′′ denote the first and second order partial derivatives of f .

(v) If, in addition, P(η 0 ∈ R ++ ) = 1 or a ∈ R ++ , then P t 0 Y s ds ∈ R ++ = 1 for all t ∈ R ++ . (vi) If, in addition, σ ∈ R ++ and a σ 2 2 , then P Y t ∈ R ++ for all t ∈ R ++ = 1.
(vii) If, in addition, P(η 0 ∈ R ++ ) = 1, a = 0 and b ∈ R + , then P(τ 0 < ∞) = 1, where τ 0 := inf{s ∈ R + : Y s = 0}, and P(Y t = 0 for all t τ 0 ) = 1.

Proof. For the existence of a pathwise unique non-negative strong solution satisfying P(Y 0 = η 0 ) = 1 and P(Y t ∈ R + for all t ∈ R + ) = 1, see Fu and Li [START_REF] Fu | Stochastic equations of non-negative processes with jumps[END_REF]Corollary 6.3], which yields (i).

Further, Theorem 6.2 in Dawson and Li [START_REF] Dawson | Skew convolution semigroups and affine Markov processes[END_REF] together with

∞ 0 (z ∧ z 2 )C α z -1-α dz = C α 1 0 z 1-α dz + C α ∞ 1 z -α dz = C α 1 2 -α + 1 α -1 < ∞ and ∞ 0 (e -zx -1 + zx)C α x -1-α dx = 1 α α(α -1) Γ(2 -α) ∞ 0 (e -zx -1 + zx)x -1-α dx = z α α (2.6)
for z ∈ R + (see, e.g., Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]Example 1.9]) imply that Y is a CBI process having branching and immigration mechanisms given in (ii).

For formula (2.2) and, in case of b ∈ R + , formula (2.4) see Li [25, formula (3.29)

- t 0 F (v s (λ)) ds = - vt(λ) λ F (z) ∂ ∂s v s ( v z (λ)) dz = vt(λ) λ F (z) R(v s ( v z (λ))) dz
and hence (2.7), where (v 0 (λ) ∧ v t (λ), v 0 (λ) ∨ v t (λ)) ∋ z → v z (λ) denotes the inverse of (0, t) ∋ s → v s (λ). By Li [25, Proposition 3.1], the function R + ∋ λ → v s (λ) ∈ R + is strictly increasing for all s ∈ R + . We have v s (θ 0 ) = θ 0 for all s ∈ R + , since R(θ 0 ) = 0 yields that this constant function is the unique locally bounded solution to the differential equation (2.3) with initial value θ 0 . If b ∈ R + , then θ 0 = 0, thus λ ∈ R ++ implies v s (λ) > v s (0) = 0 for all s ∈ R + . In this case, using the differential equation (2.3) and the inequality R(z) > 0 for all z ∈ R ++ , we obtain

∂ ∂s v s (λ) = -R(v s (λ)) < 0 for all s ∈ R + , hence the function (0, t) ∋ s → v s (λ) is strictly decreasing, thus we conclude (2.7) for b ∈ R + . If b ∈ R --, then θ 0 ∈ R ++ . Consequently, in case of b ∈ R -- and λ ∈ (0, θ 0 ) we have v s (λ) < v s (θ 0 ) = θ 0 for all s ∈ R + .
In this case, using the differential equation (2.3) and the inequality R(z) < 0 for all z ∈ (0, θ 0 ), we obtain ∂ ∂s v s (λ) = -R(v s (λ)) > 0 for all s ∈ R + , hence the function (0, t) ∋ s → v s (λ) is strictly increasing, thus we conclude (2.7) for b ∈ R --and λ ∈ (0, θ 0 ). In a similar way, in case of b ∈ R --and λ ∈ (θ 0 , ∞) we have v s (λ) > v s (θ 0 ) = θ 0 for all s ∈ R + . In this case, using the differential equation (2.3) and the inequality R(z) > 0 for all z ∈ (θ 0 , ∞), we obtain ∂ ∂s v s (λ) = -R(v s (λ)) < 0 for all s ∈ R + , hence the function (0, t) ∋ s → v s (λ) is strictly decreasing, thus we conclude (2.7) for b ∈ R --and λ ∈ (θ 0 , ∞) as well.

The form of the infinitesimal generator (2.5) can be checked similarly as in the proof of Theorem 2.1 of Barczy et al. [START_REF] Barczy | Stationarity and ergodicity for an affine two factor model[END_REF], implying (iv).

For (v), let us fix t ∈ R ++ and put

A t := ω ∈ Ω : [0, t] ∋ s → Y s (ω) is càdlàg and Y s (ω) ∈ R + for all s ∈ [0, t] .
Then, by (i), P(A t ) = 1 and for all ω ∈ A t , t 0 Y s (ω) ds = 0 if and only if Y s (ω) = 0 for all s ∈ [0, t). By (1.1),

Y s = Y 0 + as -b s 0 Y u du + σ s 0 Y u dW u + δ s 0 α Y u-dL u , s ∈ R + ,
holds P-almost surely. The stochastic integrals on the right hand side can be approximated as

sup s∈[0,t] ⌊ns⌋ i=1 Y i-1 n (W i n -W i-1 n ) - s 0 Y u dW u P -→ 0 as n → ∞, sup s∈[0,t] ⌊ns⌋ i=1 α Y i-1 n -(L i n -L i-1 n ) - s 0 α Y u-dL u P -→ 0 as n → ∞,
⌊n k s⌋ i=1 Y i-1 n k (W i n k -W i-1 n k ) - s 0 Y u dW u → 0 as k → ∞, sup s∈[0,t] ⌊n k s⌋ i=1 α Y i-1 n k -(L i n k -L i-1 n k ) - s 0 α Y u-dL u → 0 as k → ∞
hold P-almost surely. Let us denote by Ãt the event on which the above two P-almost sure convergences hold. Consequently, with the notation Ãt := ω ∈ Ω :

t 0 Y s (ω) ds = 0 , we have Ãt ∩ Ãt ∩ A t ⊂ Ãt ∩ ω ∈ Ω : s 0 Y u dW u (ω) = 0, s 0 α Y u-dL u (ω) = 0 for all s ∈ [0, t) ⊂ Ãt ∩ ω ∈ Ω : Y s (ω) = Y 0 (ω) + as for all s ∈ [0, t) ⊂ Ãt ∩ ω ∈ Ω : s 0 (Y 0 (ω) + au) du = 0 for all s ∈ [0, t) ⊂ Ãt ∩ ω ∈ Ω : Y 0 (ω)s + as 2 2 = 0 for all s ∈ [0, t) ⊂ Ãt ∩ ω ∈ Ω : Y 0 (ω) = - as 2 for all s ∈ [0, t) ,
where the last event has probability 0, implying P t 0 Y s (ω) ds = 0 = 0. Thus P t 0 Y s (ω) ds ∈ R ++ = 0, and hence we have (v).

For (vi), see Proposition 3.7 in Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF].

Finally, we prove part (vii). First note that in case of a = 0, (Y t ) t∈R + is a continuous time branching process (without immigration). If b ∈ R + , then by Corollary 3.9 in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF], P(τ 0 < ∞ | Y 0 = y 0 ) = 1 for all y 0 ∈ R ++ , since Condition 3.6 in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF] holds for all θ > 0 due to

∞ θ 1 R(z) dz ∞ θ 2 σ 2 z 2 dz < ∞.
The last statement follows from the fact that in case of a = 0 and P(Y 0 = 0) = 1, the pathwise unique non-negative strong solution of the SDE (

∈ R + ) = 1 and E(η 0 ) < ∞. For a ∈ R + , b ∈ R, σ ∈ R + , δ ∈ R ++
Y t = Y 0 + t 0 (a -bY u + γδ α Y u ) du + t 0 R (h(zδ α Y u ) -δ α Y u h(z)) m(dz) du + σ t 0 Y u dW u + t 0 R h(zδ α Y u-) µ L (du, dz) + t 0 R (zδ α Y u--h(zδ α Y u-)) µ L (du, dz) for t ∈ R + , where h : R → [-1, 1], h(z) := z½ [-1,1] (z), z ∈ R.
Proof. Using (2.1) and Proposition II.1.30 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], we obtain

Y t = Y 0 + t 0 (a -bY u ) du + t 0 σ Y u dW u + δ t 0 α Y u-dL u = Y 0 + t 0 (a -bY u ) du + t 0 σ Y u dW u + γδ t 0 α Y u-du + δ t 0 R α Y u-h(z) µ L (du, dz) + δ t 0 R α Y u-(z -h(z)) µ L (du, dz)
for t ∈ R + . In order to prove the statement, it is enough to show

δ t 0 R α Y u-h(z) µ L (du, dz) -du m(dz) = I 1 -I 2 , (2.9) δ t 0 R α Y u-(z -h(z)) µ L (du, dz) = I 3 + I 4 , (2.10)
with

I 1 := t 0 R h(zδ α Y u-) µ L (du, dz) -du m(dz) , I 2 := t 0 R (h(zδ α Y u-) -δ α Y u-h(z)) µ L (du, dz) -du m(dz) , I 3 := t 0 R (zδ α Y u--h(zδ α Y u-)) µ L (du, dz), I 4 := t 0 R (h(zδ α Y u-) -δ α Y u-h(z)) µ L (du, dz),
and the equality (2.11)

I 4 -I 2 = I 5 with I 5 := t 0 R (h(zδ α Y u ) -δ α Y u h(z)) m(dz) du.
For the equations (2.9), (2.10) and (2.11), it suffices to check the existence of I 2 , I 3 and I 5 .

First note that for every s ∈ (0, ∞) we have

h(sz) -sh(z) =        sz½ {1<|z| 1 s } if s ∈ (0, 1), z ∈ R, 0 if s = 1, z ∈ R, -sz½ { 1 s <|z| 1} if s ∈ (1, ∞), z ∈ R.
(2.12)

The existence of I 2 will be a consequence of

I 2 = I 2,1 -I 2,2 -I 2,3 with I 2,1 := t 0 R δ α Y u-z½ {1<|z| 1 δ α √ Y u- } ½ {δ α √ Y u-∈(0,1)} µ L (du, dz), I 2,2 := t 0 R δ α Y u-z½ {1<|z| 1 δ α √ Y u- } ½ {δ α √ Y u-∈(0,1)} du m(dz), I 2,3 := t 0 R δ α Y u-z½ { 1 δ α √ Y u- <|z| 1} ½ {δ α √ Y u-∈(1,∞)} µ L (du, dz) -du m(dz) , since on the set {Y u-= 0}, the integrand h(zδ α √ Y u-) -δ α √ Y u-h(z) in I 2 takes value 0.
Here we have

|I 2,1 | t 0 R |δ α Y u-z|½ {1<|z| 1 δ α √ Y u- } ½ {δ α √ Y u-∈(0,1)} µ L (du, dz) t 0 R ½ {1<|z|} µ L (du,
|I 2,2 | t 0 R |δ α Y u-z|½ {1<|z| 1 δ α √ Y u- } ½ {δ α √ Y u-∈(0,1)} du m(dz) t 0 R ½ {1<|z|} du m(dz) = tm({z ∈ R : |z| > 1}) < ∞.
Further, the function

Ω × R + × R ∋ (ω, t, z) → h(z) belongs to G loc (µ L ),
√ Y u- <|z| 1} ½ {δ α √ Y u-∈(1,∞)} | |h(z)|, hence, by the definition of G loc (µ L ), the function Ω × R + × R ∋ (ω, t, z) → z½ { 1 δ α √ Y u- <|z| 1} ½ {δ α √ Y u-∈(1,∞)}
also belongs to G loc (µ L ). By Jacod and Shiryaev [18, Proposition II.1.30], we conclude that the function

Ω × R + × R ∋ (ω, t, z) → δ α √ Y u-z½ { 1 δ α √ Y u- <|z| 1} ½ {δ α √ Y u-∈(1,∞)
} also belongs to G loc (µ L ), thus the integral I 2,3 exists, and hence we obtain the existence of I 2 , and hence that of I 1 .

Next observe that for the process

ζ t := δ t 0 α √ Y u-dL u , t ∈ R + , we have ∆ζ t = δ α √ Y t-∆L t , t ∈ R + ,
I 3 = t 0 R zδ α Y u-½ {|zδ α √ Y u-|>1} µ L (du, dz) = u∈[0,t] ∆L u δ α Y u-½ {|∆Luδ α √ Y u-|>1} = u∈[0,t] ∆ζ u ½ {|∆ζu|>1}
is a finite sum, since the process (ζ t ) t∈[0,∞) admits càdlàg trajectories, hence there can be at most finitely many points u ∈ [0, t] at which the absolute value |∆ζ u | of the jump size ∆ζ u exceeds 1, see, e.g., Billingsley [7, page 122]. Thus we obtain the existence of I 3 , and hence that of I 4 .

Finally, we have

|I 5 | t 0 R |δ α Y u z|½ {1<|z| 1 δ α √ Yu } ½ {δ α √ Yu∈(0,1)} m(dz) du + t 0 R |δ α Y u z|½ { 1 δ α √ Yu <|z| 1} ½ {δ α √ Yu∈(1,∞)} m(dz) du t 0 R ½ {1<|z|} m(dz) du + t 0 R |δ α Y u z| 2 ½ {|z| 1} m(dz) du = tm({z ∈ R : |z| > 1}) + t 0 δ 2 Y 2 α u du 1 -1 |z| 2 m(dz) < ∞, since 1 -1 |z| 2 m(dz) = 1 0 z 2 C α z -1-α dz = Cα 2-α ∈ R ++ , hence we conclude the existence of I 5 . ✷
Next we present a result about the first moment of (Y t ) t∈R + .

2.3 Proposition. Let a ∈ R + , b ∈ R, σ ∈ R + , δ ∈ R ++ , and α ∈ (1, 2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 ∈ R + ) = 1 and E(Y 0 ) < ∞. Then E(Y t ) =    e -bt E(Y 0 ) -a b + a b if b = 0, E(Y 0 ) + at if b = 0, t ∈ R + . (2.13) Consequently, if b ∈ R ++ , then (2.14) lim t→∞ E(Y t ) = a b , if b = 0, then lim t→∞ t -1 E(Y t ) = a, if b ∈ R --, then lim t→∞ e bt E(Y t ) = E(Y 0 ) - a b .
Proof. By Proposition 2.1, (Y t ) t∈R + is CBI process with an infinitesimal generator given in (2.5). By the notations of Barczy et al. [START_REF] Barczy | Stochastic differential equation with jumps for multitype continuous state and continuous time branching processes with immigration[END_REF], this CBI process has parameters (d, c, β, B, ν, µ), where

d = 1, c = 1 2 σ 2 , β = a, B = -b - ∞ 0 (z -1) + µ(dz), ν = 0 and µ = δ α m.
Since E(Y 0 ) < ∞ and the moment condition R\{0} |z|½ {|z| 1} ν(dz) < ∞ trivially holds, we may apply formula (3.1.11) in Li [START_REF] Li | Continuous-state branching processes[END_REF] or Lemma 3.4 and (2.14) in Barczy et al. [START_REF] Barczy | Stochastic differential equation with jumps for multitype continuous state and continuous time branching processes with immigration[END_REF] with

B = B + ∞ 0 (z -1) + µ(dz) = -b and β = β + R\{0} z ν(dz) = a yielding that E(Y t ) = e t B E(Y 0 ) + t 0 e u B du β.
This implies (2.13) and the other parts of the assertion. ✷

Based on the asymptotic behavior of the expectations E(Y t ) as t → ∞, we introduce a classification of the stable CIR model given by the SDE (1.1).

Definition. Let

a ∈ R + , b ∈ R, σ ∈ R + , δ ∈ R ++ , and α ∈ (1, 2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 ∈ R + ) = 1 and E(Y 0 ) < ∞. We call (Y t ) t∈R + subcritical, critical or supercritical if b ∈ R ++ , b = 0 or b ∈ R --, respectively.
The following result states the existence of a unique stationary distribution for the process (Y t ) t∈R + in the subcritical and critical cases, and the exponential ergodicity in the subcritical case.

Theorem. Let a

∈ R + , b ∈ R + , σ ∈ R + , δ ∈ R ++ , and α ∈ (1, 2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 ∈ R + ) = 1 and E(Y 0 ) < ∞.
(i) Then (Y t ) t∈R + converges in law to its unique stationary distribution π having Laplace transform

(2.15) ∞ 0 e -λy π(dy) = exp - λ 0 F (x) R(x) dx = exp - λ 0 ax σ 2 2 x 2 + δ α α x α + bx dx for λ ∈ R + .
Especially, in case of b = 0 and σ = 0, π is a strictly (2α)-stable distribution with no negative jumps. Moreover, the expectation of π is given by

∞ 0 y π(dy) =        0 if a = 0 and b = 0, a b ∈ R + if b ∈ R ++ , +∞ if a ∈ R ++ and b = 0.
(2.16)

(ii) If, in addition, a ∈ R ++ and b ∈ R ++ , then the process (Y t ) t∈R + is exponentially ergodic, i.e., there exist constants C ∈ R ++ and D ∈ R ++ such that P Yt | Y 0 =y -π TV C(y + 1)e -Dt , t ∈ R + , y ∈ R + ,
where µ TV denotes the total-variation norm of a signed measure µ on R + defined by µ TV := sup A∈B(R + ) |µ(A)|, and P Yt | Y 0 =y is the conditional distribution of Y t with respect to the condition Y 0 = y. As a consequence, for all Borel measurable functions f :

R + → R with ∞ 0 |f (y)| π(dy) < ∞, we have (2.17) 1 T T 0 f (Y s ) ds a.s. -→ ∞ 0 f (y) π(dy) as T → ∞.
Proof. The weak convergence of Y t towards π as t → ∞, and the fact that π is a stationary distribution for (Y t ) t∈R + follow immediately from Li [25, Theorem 3.20 and the paragraph after

Corollary 3.21], since R(z) = σ 2 2 z 2 + δ α α z α + bz ∈ R ++ , z ∈ R ++ , and condition (3.30) in Li [25] is satisfied. Indeed, for all λ ∈ R ++ , λ 0 F (z) R(z) dz = λ 0 az σ 2 2 z 2 + δ α α z α + bz dz aα δ α λ 0 z 1-α dz = aαλ 2-α δ α (2 -α) < ∞.
We note that Li and Ma [26, Proposition 2.2] contains the above considerations in case of b ∈ R ++ .

The uniqueness of a stationary distribution in (i) follows from, e.g., page 80 in Keller-Ressel [START_REF] Keller-Ressel | Moment explosions and long-term behavior of affine stochastic volatility models[END_REF].

Namely, let us assume that there exists another stationary distribution π ′ for (Y t ) t∈R + , and let

(Y ′ t ) t∈R + be the unique strong solution of the SDE (1.1) with a ∈ R + , b ∈ R + , σ ∈ R + , and δ ∈ R ++ satisfying L(Y ′ 0 ) = π ′ , where L(Y ′ 0 ) denotes the law of Y ′ 0 . Then, by part (iii) of Proposition 2.1, for all λ ∈ R + , lim t→∞ E(e -λY ′ t ) = lim t→∞ E(E(e -λY ′ t | Y ′ 0 )) = lim t→∞ E exp -Y ′ 0 v t (λ) + vt(λ) λ F (z) R(z) dz = E exp - λ 0 F (z) R(z) dz = exp - λ 0 F (z) R(z) dz ,
where the last but one step follows by the dominated convergence theorem and the fact that lim t→∞ v t (λ) = 0, λ ∈ R + (see, e.g., the proof of Theorem 3.20 in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]). Since L(

Y ′ t ) = π ′ , t ∈ R + , we have ∞ 0 e -λz π ′ (dz) = exp - λ 0 F (z) R(z) dz , λ ∈ R + , yielding that ∞ 0 e -λz π ′ (dz) = ∞ 0 e -λz π(dz), λ ∈ R + .
By the uniqueness of Laplace transform, we get π = π ′ , as desired.

Further, in case of b = 0 and σ = 0, by (2.15), we have

∞ 0 e -λy π(dy) = exp - λ 0 ax δ α α x α dx = exp - aα (2 -α)δ α λ 2-α , λ ∈ R + ,
so π is a strictly (2α)-stable distribution with no negative jumps. Finally, again by (2.15),

∞ 0 y π(dy) = lim λ↓0 aλ σ 2 2 λ 2 + δ α α λ α + bλ =        0 if a = 0 and b = 0, a b ∈ R + if b ∈ R ++ , +∞ if a ∈ R ++ and b = 0.
For part (ii), we can use Theorem 2.5 in Li and Ma [START_REF] Li | Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model[END_REF]. We only have to check Condition 2.1 of Li and Ma [START_REF] Li | Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model[END_REF], namely, we have to show the existence of some constant

θ ∈ R ++ such that R(z) ∈ R ++ for z > θ and ∞ θ 1 R(z) dz < ∞. Here R(z) ∈ R ++ for all z ∈ R ++ (due to b ∈ R ++ )
, and, e.g., with θ = 1,

∞ 1 1 R(z) dz α δ α ∞ 1 1 z α dz = α (α -1)δ α < ∞.
In case of σ = 0, the exponential ergodicity of (Y t ) t∈R + also follows by Theorem 6.1 in Jin et al. [START_REF] Jin | Exponential ergodicity of an affine two-factor model based on the α-root process[END_REF]. Convergence (2.17) follows, e.g., from the discussion after Proposition 2.5 in Bhattacharya [START_REF] Bhattacharya | On the functional central limit theorem and the law of the iterated logarithm for Markov processes[END_REF]. ✷

Remark.

In what follows, in case of σ ∈ R ++ , we present another (and more detailed) proof for the convergence Y t D -→ π as t → ∞ in Theorem 2.5 giving more insight as well. It is enough to consider the case of P(Y 0 = y 0 ) = 1 with some y 0 ∈ R + . Using (2.2), we have (2.3). First, we show that lim t→∞ v t (λ) = 0. The proof is based on the following version of the comparison theorem (see, e.g., Lemma C.3. in Filipović et al. [START_REF] Filipović | Density approximations for multivariate affine jump-diffusion processes[END_REF] or Amann [START_REF] Amann | Ordinary Differential Equations. An Introduction to Nonlinear Analysis[END_REF]Lemma 16.4]): if S : R + × R → R is a continuous function which is locally Lipschitz continuous in its second variable and p, q : R + → R are differentiable functions satisfying

E(e -λYt ) = exp -y 0 v t (λ) -a t 0 v s (λ)ds for t ∈ R + and λ ∈ R + , where the function R + ∋ λ → v t (λ) ∈ R + is given by
p ′ (s) S(s, p(s)), s ∈ R + , q ′ (s) = S(s, q(s)), s ∈ R + , p(0) q(0), then p(s) q(s) for all s ∈ R + . By choosing S : R + × R → R, S(s, x) := -σ 2 2 x 2 , (s, x) ∈ R + × R, the comparison theorem yields that 0 v s (λ) f (s), s ∈ R + , (2.18)
where f : R + → R + is the unique locally bounded solution to the differential equation

f ′ (s) = S(s, f (s)) = - σ 2 2 f (s) 2 , s ∈ R + with f (0) = λ.
The solution of this separable differential equation takes the form

f (s) = λ 1 + σ 2 λ 2 s , s ∈ R + . (2.19)
Hence, using σ ∈ R ++ , we readily have lim t→∞ f (t) = 0, which, by (2.18), yields that lim t→∞ v t (λ) = 0 for all λ ∈ R + , as desired. Further, by (2.4), we have

lim t→∞ t 0 v s (λ) ds = -lim t→∞ vt(λ) λ z σ 2 2 z 2 + δ α α z α + bz dz, λ ∈ R + .
Then, by the continuity of the integral upper limit function, we have

lim t→∞ t 0 v s (λ) ds = λ 0 z σ 2 2 z 2 + δ α α z α + bz dz, λ ∈ R + ,
where the integral on the right hand side is well-defined, since

λ 0 z σ 2 2 z 2 + δ α α z α + bz dz λ 0 z δ α α z α dz = α δ α λ 0 1 z α-1 dz = αλ 2-α δ α (2 -α) < ∞.
Consequently, by continuity theorem, we have (2.15) in case of σ ∈ R ++ . ✷

Next we give a statistic for σ 2 using continuous time observations (Y t ) t∈[0,T ] with an arbitrary T ∈ R ++ . Due to this result we do not consider the estimation of the parameter σ, it is supposed to be known. The parameter σ is a parameter for the diffusion part related to W , and, in general, the estimation of this kind of parameter is possible using an arbitrarily short (continuous time) observation of the underlying process (at least theoretically), and that's why authors suppose this to be known. In the forthcoming Remark 2.7, we demonstrate that this also holds for our model.

2.7 Remark. Let a ∈ R + , b ∈ R, σ ∈ R + , δ ∈ R ++ , and α ∈ (1, 2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 ∈ R + ) = 1 and E(Y 0 ) < ∞. The Grigelionis representation given in (2.8) implies that the continuous martingale part Y cont of Y is Y cont t = σ t 0 √ Y u dW u , t ∈ R + ,
Y cont is Y cont t = σ 2 t 0 Y u du, t ∈ R + . Suppose that we have P(Y 0 ∈ R ++ ) = 1 or a ∈ R ++ . Then for all T ∈ R ++ , we have σ 2 = Y cont T T 0 Y u du =: σ 2 T , since, due to (v) of Proposition 2.1, P T 0 Y u du ∈ R ++ = 1.
We note that σ 2 T is a statistic, i.e., there exists a measurable function Ξ :

D([0, T ], R) → R such that σ 2 T = Ξ((Y u ) u∈[0,T ] )
, where D([0, T ], R) denotes the space of real-valued càdlàg functions defined on [0, T ], since (2.20) 1

1 n ⌊nT ⌋ i=1 Y i-1 n ⌊nT ⌋ i=1 Y i n -Y i-1 n 2 - u∈[0,T ] (∆Y u ) 2 P -→ σ 2 T as n → ∞,
where the convergence in (2.20) holds P-almost surely along a suitable subsequence, the members of the sequence in (2.20) are measurable functions of (Y u ) u∈[0,T ] , and one can use Theorems 4.2.2 and 4.2.8 in Dudley [START_REF] Dudley | Real Analysis and Probability[END_REF]. Next we prove (2.20). By Theorem I.4.47 a) in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF],

⌊nT ⌋ i=1 Y i n -Y i-1 n 2 P -→ [Y ] T as n → ∞, T ∈ R + ,
where ([Y ] t ) t∈R + denotes the quadratic variation process of the semimartingale Y . By Theorem I.4.52 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF],

[Y ] T = Y cont T + u∈[0,T ] (∆Y u ) 2 , T ∈ R + .
Consequently, for all T ∈ R + , we have

⌊nT ⌋ i=1 Y i n -Y i-1 n 2 - u∈[0,T ] (∆Y u ) 2 P -→ Y cont T as n → ∞.
Moreover, for all T ∈ R + , we have

1 n ⌊nT ⌋ i=1 Y i-1 n P -→ T 0 Y u du as n → ∞,
see Proposition I.4.44 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]. Hence (2.20) follows by the fact that convergence in probability is closed under multiplication. Finally, we note that T is fixed above, and it is enough to know any observation (Y t ) t∈[0,T ] to carry out the above calculations, where T > 0 can be arbitrarily small. ✷

3 Joint Laplace transform of Y t and t 0 Y s ds Using Theorem 4.10 in Keller-Ressel [START_REF] Keller-Ressel | Affine Processes -Theory and Applications in Finance[END_REF] we derive a formula for the joint Laplace transform of Y t and t 0 Y s ds, where t ∈ R + . We note that this form of the joint Laplace transform in question is a consequence of Theorem 5.3 in Filipović [START_REF] Filipović | A general characterization of one factor affine term structure models[END_REF] and a special case of Proposition 3.3 in Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF] as well.

3.1 Theorem. Let a ∈ R + , b ∈ R, σ ∈ R + , δ ∈ R ++ , and α ∈ (1, 2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 = y 0 ) = 1 with some y 0 ∈ R + . Then for all u, v ∈ R -, E exp uY t + v t 0 Y s ds = exp y 0 ψ u,v (t) + a t 0 ψ u,v (s) ds , t ∈ R + ,
where the function ψ u,v : R + → R -is the unique locally bounded solution to the differential equation

ψ ′ u,v (t) = σ 2 2 ψ u,v (t) 2 + δ α α (-ψ u,v (t)) α -bψ u,v (t) + v, t ∈ R + , ψ u,v (0) = u. (3.1) Further, if (u, v) = (0, 0), then ψ u,v (t) ∈ R --, t ∈ R ++ , and if (u, v) = (0, 0), then ψ u,v (t) = 0, t ∈ R + .
Proof. By Theorem 4.10 in Keller-Ressel [START_REF] Keller-Ressel | Affine Processes -Theory and Applications in Finance[END_REF]

, Y t , t 0 Y s ds t∈R + is a 2-dimensional CBI process with branching mechanism R(z 1 , z 2 ) = ( R 1 (z 1 , z 2 ), R 2 (z 1 , z 2 )), z 1 , z 2 ∈ R + , with R 1 (z 1 , z 2 ) = R(z 1 ) -z 2 , R 2 (z 1 , z 2 ) = 0, z 1 , z 2 ∈ R + ,
and with immigration mechanism

F (z 1 , z 2 ) = F (z 1 ), z 1 , z 2 ∈ R + ,
E exp uY t + v t 0 Y s ds = exp y 0 ψ u,v (t) - t 0 F (-ψ u,v (s), -ϕ u,v (s)) ds = exp y 0 ψ u,v (t) + a t 0 ψ u,v (s) ds , t ∈ R + ,
where the function (ψ u,v , ϕ u,v ) : R + → R 2 -is the unique locally bounded solution to the system of differential equations

ψ ′ u,v (t) = R 1 (-ψ u,v (t), -ϕ u,v (t)) = σ 2 2 ψ u,v (t) 2 + δ α α (-ψ u,v (t)) α -bψ u,v (t) + ϕ u,v (t), t ∈ R + , ϕ ′ u,v (t) = R 2 (-ψ u,v (t), -ϕ u,v (t)) = 0, t ∈ R + , with initial values ψ u,v (0) = u, ϕ u,v (0) = v. Clearly, ϕ u,v (t) = v, t ∈ R + , hence we obtain ψ ′ u,v (t) = σ 2 2 ψ u,v (t) 2 + δ α α (-ψ u,v (t)) α -bψ u,v (t) + v, t ∈ R + , ψ u,v (0) = u,
as desired.

If (u, v) = (0, 0), then, since the identically zero function is a (locally bounded) solution to (3.1), by the unicity of such a solution, we have ψ 0,0 (t) = 0, t ∈ R + . If (u, v) = (0, 0), then, on the contrary, let us suppose that there exists an t 0 ∈ R ++ such that ψ u,v (t 0 ) = 0. Let

t * := inf{t ∈ R ++ : ψ u,v (t) = 0}. Then t * < ∞, ψ u,v (t) < 0 for all t ∈ [0, t * ), and ψ u,v (t * ) = 0. If t * = 0, then 0 = ψ u,v (t * ) = ψ u,v (0) = u, and hence v ∈ R --. Further, there exists a sequence (t n ) n∈N such that t n ∈ R ++ , n ∈ N, t n ↓ 0 = t * as n → ∞, and ψ 0,v (t n ) = 0, n ∈ N.
Consequently, using that a locally bounded solution to (3.1) is unique,

ψ 0,v (kt n ) = ψ 0,v (0) = 0, k, n ∈ N. Since t n ↓ 0 as n → ∞, for all t ∈ R ++ , there exists a sequence (k n ) n∈N such that k n ∈ N, n ∈ N, and k n t n → t as n → ∞ (one can choose k n := ⌊ t tn ⌋, n ∈ N). Since ψ 0,v is continuous, we have ψ 0,v (t) = 0, t ∈ R + , yielding us to a contradiction (due to v ∈ R --). So, if t * = 0, then ψ u,v (
t) < 0, t > 0, as desired. In the sequel, let us assume that t * > 0. On the one hand, ψ ′ u,v (t * ) 0, since

ψ ′ u,v (t * ) = lim h↑0 ψ u,v (t * + h) -ψ u,v (t * ) h = lim h↑0 1 h ψ u,v (t * + h),
where h < 0 and ψ u,v (t * + h) < 0 yield 1 h ψ u,v (t * + h) > 0. On the other hand, by (3.1),

ψ ′ u,v (t * ) = v 0, yielding that v = 0. Consequently, if t * > 0, then (3.1) takes the form ψ ′ u,v (t) = σ 2 2 ψ u,v (t) 2 + δ α α (-ψ u,v (t)) α -bψ u,v (t), t ∈ R + ,
where ψ u,v (0) = u and ψ u,v (t * ) = 0. By the uniqueness of a locally bounded solution of (3.1), ψ u,v (t) = 0 for all t t * , and hence ψ u,v (t) = 0 for all t 0. Indeed, let ψ u,v (τ ) := ψ u,v (-τ ), τ 0, and τ * := -t * . Then

ψ ′ u,v (τ ) = - σ 2 2 ψ u,v (τ ) 2 - δ α α (-ψ u,v (τ )) α + b ψ u,v (τ ), τ 0,
where ψ u,v (0) = u, ψ u,v (τ * ) = 0, and, by the uniqueness of a locally bounded solution of the differential equation above,

ψ u,v (τ ) = 0 for τ ∈ [τ * , 0], i.e., ψ u,v (-τ ) = 0 for τ ∈ [-t * , 0], i.e., ψ u,v (t) = 0 for t ∈ [0,
∈ R + , σ, δ ∈ R ++ , α ∈ (1, 2)
, and a known deterministic initial value Y 0 = y 0 ∈ R + , and we will consider b ∈ R as an unknown parameter.

Let P b denote the probability measure induced by (Y t ) t∈R + on the measurable space . Then for all T ∈ R ++ , the probability measures P b,T and P b,T are absolutely continuous with respect to each other, and

(D(R + , R), D(R + , R)) endowed with the natural filtration (D t (R + , R)) t∈R + , see Appendix A.
log d P b,T d P b,T ( Y ) = - b -b σ 2 Y T -y 0 -aT -δ T 0 α Y u-dL u - b 2 -b 2 2σ 2 T 0 Y u du (4.1)
holds P-almost surely, where Y is the α-stable CIR process corresponding to the parameter b.

Proof.

In what follows, we will apply Theorem III.5.34 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] (see also Appendix A). We will work on the canonical space (D(R + , R), D(R + , R)). Let (η t ) t∈R + denote the canonical process η t (ω) := ω(t), ω ∈ D(R + , R), t ∈ R + . Recall that the stable CIR process (1.1) can be written in the form (2.8). By Proposition 2.1, the SDE (1.1) has a pathwise unique strong solution (with the given deterministic initial value y 0 ∈ R + ), and hence, by Theorem III.2.26 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], under the probability measure P b , the canonical process (η t ) t∈R + is a semimartingale with semimartingale characteristics (B (b) , C, ν) associated with the truncation function h (introduced in Proposition 2.2), where

B (b) t = t 0 a -bη u + γδ α √ η u + R (h(zδ α √ η u ) -h(z)δ α √ η u ) m(dz) du, t ∈ R + , C t = t 0 (σ √ η u ) 2 du = σ 2 t 0 η u du, t ∈ R + , ν(dt, dy) = K(η t , dy) dt with the Borel transition kernel K from R + × R into R given by K(y, R) := R ½ R\{0} (zδ α √ y) m(dz) for y ∈ R + and R ∈ B(R) with m(dz) = C α z -1-α ½ (0,∞) (z) dz.
The aim of the following discussion is to check the set of sufficient conditions presented in Appendix A (of which the notations will be used) in order to have right to apply Theorem III.5.34 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]. First note that (C t ) t∈R + and ν(dt, dy)

do not depend on the unknown parameter b, and hence V ( b,b) is identically one and then (A.1) and (A.2) readily hold. We also have

P b ν({t} × R) = 0 = P b {t} K(η s , R) ds = 0 = 1, t ∈ R + , b ∈ R.
Further, (C t ) t∈R + can be represented as

C t = t 0 c u dF u , t ∈ R + ,
where the stochastic processes (c t ) t∈R + and (F t ) t∈R + are given by c t := σ 2 η t , t ∈ R + , and

F t = t, t ∈ R + . Consequently, for all b, b ∈ R, B (b) t -B ( b) t = -(b -b) t 0 η u du =
) t∈R + is given by β ( b,b) t = - b -b σ 2 , t ∈ R + ,
which yields (A.3). Next we check (A.4), i.e.,

P b t 0 β ( b,b) u 2 c u dF u < ∞ = 1, t ∈ R + . (4.2)
We have

t 0 β ( b,b) u 2 c u dF u = (b -b) 2 σ 2 t 0 η u du, t ∈ R + .
Since for each ω ∈ D(R + , R), the trajectory [0, t] ∋ u → η u (ω) is càdlàg, hence bounded (see, e.g., Billingsley [7, (12.5)]), we have t 0 η u (ω) du < ∞, hence we obtain (4.2). Next, we check that, under the probability measure P b , local uniqueness holds for the martingale problem on the canonical space corresponding to the triplet (B (b) , C, ν) with the given initial value y 0 with P b as its unique solution (for the definition of local uniqueness in question, see Definition III.2.27 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]). By Proposition 2.1, the SDE (1.1) has a pathwise unique strong solution (with the given deterministic initial value y 0 ∈ R + ), and hence Theorem III.2.26 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] yields that the set of all solutions to the martingale problem on the canonical space corresponding to (B (b) , C, ν) has only one element (P b ) yielding the desired local uniqueness. We also mention that Theorem III.4.29 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] implies that under the probability measure P b , all local martingales have the integral representation property relative to η. By Theorem III.5.34 in Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] (see also Appendix A), P b,T and P b,T are equivalent (one can change the roles of b and b), and we have Proof. Due to (v) of Proposition 2.1, P T 0 Y s ds ∈ R ++ = 1 for all T ∈ R ++ , and hence the right hand side of (4.3) is well-defined P-almost surely. The aim of the following discussion is to show that the right hand side of (4.3) is a measurable function of (Y u ) u∈[0,T ] (i.e., a statistic). By Proposition 4.1, for all b, b ∈ R and T ∈ R ++ , the probability measures P b,T and P b,T are absolutely continuous with respect to each other, and

d P b,T d P b,T (η) = exp T 0 β ( b,b) u d(η cont ) ( b) u - 1 2 T 0 β ( b,b) u 2 c u du , T ∈ R ++ holds P b -
d Y cont t = d Y t -(a -b Y t ) dt -δ α Y t-dL t , t ∈ R + . Hence log d P b,T d P b,T ( Y ) = T 0 - b -b σ 2 d Y u -δ α Y u-dL u - T 0 - b -b σ 2 (a -b Y u ) du - 1 2 T 0 - b -b σ 2 2 σ 2 Y u du = - b -b σ 2 T 0 (d Y u -δ α Y u-dL u ) + b -b σ 2 T 0 a du - b 2 -b 2
log d P b,T d P b,T (Y ) = - b -b σ 2 Y T -y 0 -aT -δ T 0 α Y u-dL u - b 2 -b 2 2σ 2 T 0 Y u du holds P-almost surely.
The left-hand side of the above equality is measurable with respect to (Y t ) t∈[0,T ] (see, e.g., Jacod and Shiryaev [18, Theorem III.3.4]), and hence its right hand-side is also measurable, which yields the measurability of 2 , we prove that L t is a measurable function of (Y u ) u∈[0,T ] for all t ∈ [0, T ], where T ∈ R ++ , which (in the special case a σ 2

T 0 α √ Y u-
∂ ∂b log(Λ T (b, b)) = - 1 σ 2 Y T -y 0 -aT -δ T 0 α Y u-dL u - b σ 2 T 0 Y u du, ∂ 2 ∂b 2 log(Λ T (b, b)) = - 1 σ 2 T 0 Y u du.
2 ) gives another proof for the fact that the right hand side of (4.3) is a statistic. Recalling the notation

ζ t = δ t 0 α √ Y u-dL u , t ∈ R + ,
½ {ε<|∆Lu| 1} ∆L u - (0,t] {ε<|z| 1} z du m(dz) = u∈[0,t] ½ ε< |∆Yu| δ α √ Y u- 1 ∆Y u δ α √ Y u- -t {ε<|z| 1} z m(dz),
which is a measurable function of (Y t ) t∈[0,T ] . Similarly, for all t ∈ [0, T ],

(0,t] {|z|>1} z µ L (du, dz) = u∈[0,t] ½ {|∆Lu|>1} ∆L u = u∈[0,t] ½ |∆Yu| δ α √ Y u- >1 ∆Y u δ α √ Y u- ,
which is a measurable function of (Y u ) u∈[0,T ] as well. Hence, using (2.1), for all t ∈ [0, T ],

u∈[0,t] ½ |∆Yu| δ α √ Y u- >ε ∆Y u δ α √ Y u- -t {ε<|z| 1} z m(dz) + γt P -→ L t as ε ↓ 0, yielding that L t is a measurable function of (Y t ) t∈[0,T ] . Finally, note also that if Y t-= 0, t ∈ R + , then, using ∆Y t = δ α √ Y t-∆L t , t ∈ R + , we have ∆Y t = 0 yielding Y t = Y t-= 0 (irrespective of
the size of the jump of L at t). ✷ 4.4 Remark. In the literature there is another way of deriving an MLE. Sørensen [START_REF] Sørensen | Likelihood methods for diffusions with jumps[END_REF] defined an MLE of b as a solution of the equation ΛT (b) = 0, where ΛT (b) is the so-called score vector given in formula (3.3) in Sørensen [START_REF] Sørensen | Likelihood methods for diffusions with jumps[END_REF]. Luschgy [START_REF] Luschgy | Local asymptotic mixed normality for semimartingale experiments[END_REF], [START_REF] Luschgy | Asymptotic inference for semimartingale models with singular parameter points[END_REF] called this equation as an estimating equation.

With the notations of the proof of Proposition 4.1, taking into account of the form of β ( b,b) and the fact that V ( b,b) is identically one, we have -→ b as T → ∞, and

ΛT (b) := T 0 - 1 σ 2 dY cont u = - 1 σ 2 T 0 (dY u -(a -bY u ) du -δ α Y u-dL u ) = - 1 σ 2 Y T -y 0 -aT + b T 0 Y u du -δ T 0 α Y u-
√ T ( b T -b) D -→ N 0, σ 2 b a as T → ∞. (5.2)
With a random scaling,

1 σ T 0 Y s ds 1/2 ( b T -b) D -→ N (0, 1)
as T → ∞. -→ b as T → ∞. Moreover, using Theorem B.2 with η := ∞ 0 y π(dy)

Proof

1/2 and Slutsky's lemma, we have

√ T ( b T -b) = -σ 1 √ T T 0 √ Y s dW s 1 T T 0 Y s ds D -→ -σ ∞ 0 y π(dy) 1/2 N (0, 1) ∞ 0 y π(dy) = N 0, σ 2 ∞ 0 y π(dy)
as T → ∞, hence, by (2.16), we obtain (5.2). Further, Slutsky's lemma yields

1 σ T 0 Y s ds 1/2 ( b T -b) = 1 σ 1 T T 0 Y s ds 1/2 √ T ( b T -b) D -→ 1 σ ∞ 0 y π(dy) 1/2 N 0, σ 2 ∞ 0 y π(dy) = N (0, 1)
as T → ∞. ✷

Consistency of the MLE in the critical case

First, we describe the asymptotic behavior of the solution ψ u,v of the differential equation (3.1) as t → ∞ in case of b = 0 using a so-called separator technique.

Note that, for b, σ ∈ R + , δ ∈ R ++ and α ∈ (1, 2), the function

R -∋ x → R(x) := R(-x) = σ 2 2 x 2 + δ α α (-x) α -bx is strictly monotone decreasing, continuous, convex and lim x→-∞ R(x) = +∞. Indeed, R ′ (x) = σ 2 x -δ α (-x) α-1 -b < 0, x ∈ R --, and R ′′ (x) = σ 2 + δ α (α -1)(-x) α-2 > 0, x ∈ R --. Hence, for all v ∈ R --, the equation R(-x) = σ 2 2 x 2 + δ α α (-x) α -bx = -v, x ∈ R -,
has a unique negative solution, denoted by c v ∈ R --. Further, if x ∈ (c v , 0], then R(-x) < -v, i.e., R(-x) + v < 0; and if x ∈ (-∞, c v ), then R(-x) > -v, i.e., R(-x) + v > 0.

6.1 Proposition. Let a ∈ R + , b = 0, σ ∈ R + , δ ∈ R ++ , and α ∈ (1, 2). Then for all u ∈ R - and v ∈ R --, the unique locally bounded solution ψ u,v of the differential equation (3.1) satisfies

lim t→∞ ψ u,v (t) = c v . Further, ψ u,v (t) ∈ (c v , 0], t ∈ R + if c v < u 0; ψ u,v (t) ∈ (-∞, c v ), t ∈ R + if u < c v ; and ψ u,v (t) = c v , t ∈ R + if u = c v . Proof. Let v ∈ R --be fixed. By Theorem 3.1, ψ u,v (t) ∈ R --for all t ∈ R ++ . Let Q(x) := R(-x) + v = σ 2 2 x 2 + δ α α (-x) α + v, x ∈ R -. Note that Q is continuously differentiable on R --. Further, Q(x) = 0, x ∈ R -, holds if and only if x = c v , and Q(x) < 0 for x ∈ (c v , 0] and Q(x) > 0 for x ∈ (-∞, c v ). If ψ u,v (0) = u = c v ,
u,v (t) = c v , t ∈ R + , since in this case ψ ′ u,v (t) = 0, t ∈ R + , and σ 2 2 (ψ u,v (t)) 2 + δ α α (-ψ u,v (t)) α + v = σ 2 2 c 2 v + δ α α (-c v ) α + v = Q(c v ) = 0, t ∈ R + , and hence (3.1) holds. If ψ u,v (0) = u > c v , then ψ u,v (t) > c v
for all t ∈ R + . Indeed, on the contrary, let us suppose that there exists t 0 ∈ R ++ such that ψ u,v (t 0 ) = c v (which can be supposed due to the continuity of ψ u,v ). Then ψ u,v (t) = c v would hold for all t ∈ R + , since it is known that if two maximal solutions of an autonomous ordinary differential equation (with a continuously differentiable function on the right hand side) coincide at some points, then their ranges coincide, see, e.g., Arnol'd [3, Corollary on page 118], and the identically c v function is a solution of (3.1) (without the initial value). Since ψ u,v (0) > c v , this leads us to a contradiction. Consequently, by (

3.1), ψ ′ u,v (t) = Q(ψ u,v (t)) < 0, t ∈ R + , yielding that ψ u,v is monotone decreasing. Since ψ u,v is bounded below by c v , there exists an c v ∈ R -such that c v c v and lim t→∞ ψ u,v (t) = c v . It remains to check that c v = c v . We show that Q( c v ) = 0, yielding c v = c v , since c v is the only root of Q(x) = 0, x ∈ R -. On the contrary, let us suppose that Q( c v ) > 0 (the case Q( c v ) < 0 can be handled similarly). Since Q is continuous at c v , there exists κ > 0 such that Q(x) > Q( cv) 2 for all x ∈ R -satisfying |x -c v | < κ. Since lim t→∞ ψ u,v (t) = c v , there exists T > 0 such that |ψ u,v (t) -c v | < κ for t T . Hence ψ ′ u,v (t) = Q(ψ u,v (t)) > Q( c v ) 2 , t T.
Integrating over [T, t], we have

ψ u,v (t) -ψ u,v (T ) Q( c v ) 2 (t -T ), t T.
Since, by assumption, Q( c v ) > 0, taking the limit t → ∞, we get lim t→∞ ψ u,v (t) = ∞, yielding us to a contradiction.

The case ψ u,v (0) = u < c v can be handled similarly, and the other parts of the proposition follow as well. ✷ -→ ξ as t → ∞, and consequently, by the dominated convergence theorem, lim t→∞ E exp v t 0 Y s ds = E[e vξ ] for any v ∈ R -. By Theorem 3.1 with b = 0, we have

E exp v t 0 Y s ds = exp y 0 ψ 0,v (t) + a t 0 ψ 0,v (s)ds , t ∈ R + , v ∈ R -.
First we check that (6.1)

t 0 ψ 0,v (s) ds = ψ 0,v (t) 0 x σ 2 2 x 2 + δ α α (-x) α + v dx for all t ∈ R ++ and v ∈ R --
, where the function ψ 0,v : R + → R -is given by (3.1). Recall that, for all v ∈ R --, c v ∈ R --denotes the unique negative solution of the equation

σ 2 2 x 2 + δ α α (-x) α + v = 0, we have σ 2 2 x 2 + δ α α (-x) α + v < 0 for all x ∈ (c v , 0]
, and, by Proposition 6.1, ψ 0,v (t) ∈ (c v , 0] for all t ∈ R + . Consequently, by (3.1), the function [0, t] ∋ s → ψ 0,v (s) ∈ (c v , 0] is strictly decreasing and continuously differentiable, hence, for all t ∈ R ++ , by the substitution x = ψ 0,v (s) we obtain

t 0 ψ 0,v (s) ds = ψ 0,v (t) 0 x ψ ′ 0,v (ψ -1 0,v (x))
dx and hence (6.1), where ψ -1 0,v denotes the inverse of ψ 0,v . By (6.1), we have

E exp v t 0 Y s ds = exp y 0 ψ 0,v (t) + a ψ 0,v (t) 0 x σ 2 2 x 2 + δ α α (-x) α + v dx for t ∈ R ++ and v ∈ R --.
Then, by Proposition 6.1, lim t→∞ ψ 0,v (t) = c v , and hence

lim t→∞ ψ 0,v (t) 0 x σ 2 2 x 2 + δ α α (-x) α + v dx = cv 0 x σ 2 2 x 2 + δ α α (-x) α + v dx = -∞.
Here the last step can be checked as follows. We have

lim x→cv σ 2 2 x 2 + δ α α (-x) α + v x -c v = lim x→cv (R(-x) + v) -(R(-c v ) + v) x -c v = -R ′ (-c v ) = σ 2 c v -δ α (-c v ) α-1 < 0, thus there exists x 0 ∈ (c v , 0) such that σ 2 2 x 2 + δ α α (-x) α +v x-cv σ 2 cv-δ α (-cv) α-1 2 for all x ∈ (c v , x 0 ). Hence cv 0 x σ 2 2 x 2 + δ α α (-x) α + v dx cv x 0 x σ 2 2 x 2 + δ α α (-x) α + v dx 2x 0 σ 2 c v -δ α (-c v ) α-1 cv x 0 1 x -c v dx = -∞,
as desired. Hence, since a ∈ R ++ , we have

lim t→∞ E exp v t 0 Y s ds = exp {y 0 c v + a(-∞)} = 0, v ∈ R --, which yields that E(e vξ ) = 0, v ∈ R --. Since, for all v ∈ R --, E(e vξ ) = E(e vξ | ξ = ∞) P(ξ = ∞) + E(e vξ | ξ < ∞) P(ξ < ∞) = 0 • P(ξ = ∞) + E(e vξ | ξ < ∞) P(ξ < ∞),
we have 0 = E(e vξ | ξ < ∞) P(ξ < ∞), yielding that P(ξ < ∞) = 0, i.e., P(ξ = ∞) = 1. That is, we have proved that

t 0 Y s ds a.s.
-→ ∞ as t → ∞. Since the quadratic variation process of the square integrable martingale 7 Asymptotic behavior of the MLE in the supercritical case -→ -V b as t → ∞.

7.1 Theorem. Let a ∈ R + , b ∈ R --, σ ∈ R + , δ ∈ R ++
(ii) Moreover, the Laplace transform of V takes the form

E(e uV ) = exp y 0 ψ * u + -ψ * u 0 F (z) R(z) dz , u ∈ R -, (7.1)
where the functions F and R are given in Proposition 2.1, and ψ * u := lim t→∞ ψ ue bt ,0 (t), where the function ψ ue bt ,0 : R + → R -is given by (3.1).

(iii) Further, ψ * 0 = 0 and ψ * u = -K -1 (-u) for all u ∈ R --, where K -1 denotes the inverse of the strictly increasing function K : (0, θ 0 ) → R ++ given by

K(λ) := λ exp λ 0 b R(z) - 1 z dz , λ ∈ (0, θ 0 ),
where

θ 0 = inf{z ∈ R ++ : R(z) ∈ R + } ∈ R ++ . (iv) If, in addition, a ∈ R ++ , then P(V ∈ R ++ ) = 1.
In the next remark we present more properties of ψ * u , u ∈ R --.

7.2 Remark. (i) For all λ, θ ∈ (0, θ 0 ), we have

K(θ) K(λ) = θ λ exp θ λ b R(z) - 1 z dz = θ λ exp θ λ b R(z) dz -(log(θ) -log(λ)) = exp θ λ b R(z) dz , hence θ λ b R(z) dz = log K(θ) K(λ) .
Consequently, for all λ ∈ (0, θ 0 ) and u ∈ R --, we conclude

-ψ * u λ b R(z) dz = log K(-ψ * u ) K(λ) = log K(K -1 (-u)) K(λ) = log - u K(λ)
.

(ii) Using the formula for the derivative of an inverse function, we have

d du ψ * u = 1 K ′ (K -1 (-u)) = 1 K ′ (-ψ * u ) = R(-ψ * u ) bK(-ψ * u ) , u ∈ R --. ✷
Proof of Theorem 7.1. First we check that the function K is well-defined and strictly increasing on (0, θ 0 ). Observe that R(z) = σ 2 2 z 2 + δ α α z α + bz ∈ R --for all z ∈ (0, θ 0 ). Moreover, we have

(7.2) b R(z) - 1 z = - σ 2 2 z + δ α α z α-1 R(z) ∈ R ++ , z ∈ (0, θ 0 ). Further, lim z↓0 b R(z) -1 z z 2-α = -δ α bα ∈ R ++ , thus there exists z 1 ∈ (0, θ 0 ) such that b R(z) - 1 z z 2-α -2δ α
bα for all z ∈ (0, z 1 ). Hence

z 1 0 b R(z) -1 z dz -2δ α bα z 1 0 z α-2 dz < ∞. The function (0, θ 0 ) ∋ z → b R(z) -1 z ∈ R ++ is continuous, thus the integral λ z 1 b R(z) -1
z dz exists and finite for all λ ∈ (0, θ 0 ), and hence the function K is well-defined. Note that the existence and finiteness of the integral

λ 0 b R(z) -1
z dz follows also from Proposition 3.14 and its proof in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF], since

(dz) = δ α C α z -1-α ½ R ++ (z) dz.
Indeed, by partial integration,

(7.3) ∞ 1 z log(z) m(dz) = δ α C α ∞ 1 log(z) z α dz = δ α C α 1 1 -α lim z→∞ log(z) z α-1 - ∞ 1 z -α 1 -α dz = δ α C α (1 -α) 2 < ∞.
The function K is strictly increasing on (0, θ 0 ), since b R(z) -1 z ∈ R ++ for all z ∈ (0, θ 0 ), and we have lim λ↓0 K(λ) = 0 and lim λ↑θ 0 K(λ) = +∞, yielding that the range of the function K is R ++ , hence the inverse K -1 is defined on R ++ . Indeed, lim z↑θ 0

R(z) z-θ 0 = lim z↑θ 0 R(z)-R(θ 0 ) z-θ 0 = R ′ (θ 0 ). We have R ′ (θ 0 ) ∈ R ++ , since
R(0) = 0 and R(θ 0 ) = 0 yields the existence of θ 1 ∈ (0, θ 0 ) with R ′ (θ 1 ) = 0, and the function R ′ is strictly increasing on R + . Thus there exists z 2 ∈ (0, θ 0 ) such that R(z) z-θ 0 2R ′ (θ 0 ) for all z ∈ (z 2 , θ 0 ). Hence, by (7.2), for all λ ∈ (z 2 , θ 0 ), we have

λ 0 b R(z) - 1 z dz - σ 2 2 z 2 + δ α α z α-1 2 2R ′ (θ 0 ) λ z 2 1 z -θ 0 dz → +∞ as λ ↑ θ 0 .
(i) We prove the existence of an appropriate non-negative random variable V . We check that

E(Y t | F Y s ) = E(Y t | Y s ) = e -b(t-s) Y s + ae bs t s e -bx dx
for all s, t ∈ R + with 0 s t, where

F Y t := σ(Y s , s ∈ [0, t]), t ∈ R + .
The first equality follows from the Markov property of the process (Y t ) t∈R + . The second equality is a consequence of the time-homogeneity of the Markov process Y and the fact that -→ -V b as t → ∞. (7.5) First proof of (ii), (iii) and (iv). We readily have (7.1) for u = 0, since then the unique locally bounded solution to the differential equation (3.1) is ψ 0,0 (t) = 0, t ∈ R + , implying ψ * 0 = 0. Convergence e bt Y t a.s.

E(Y t | Y s = y 0 ) = E(Y t-s | Y 0 = y 0 ) = e -b(t-s) y 0 + a t-s 0 e -bx dx, t ∈ R + , y 0 ∈ R + ,
-→ V as t → ∞ implies e bt Y t D -→ V as t → ∞, and hence, by the continuity theorem, lim t→∞ E(exp{ue bt Y t }) = E(e uV ) for all u ∈ R -. By Theorem 3.1, we have exists. Note that the functions ψ ue bt ,0 , u ∈ R -, do not depend on the values of a and y 0 . Consequently, with a = 0 and with some y 0 ∈ R ++ , we obtain that the limit lim t→∞ exp ψ ue bt ,0 (t) exists, and hence the limit lim t→∞ ψ ue bt ,0 (t) = ψ * u exists as well. Using (2.7) and ψ u,0 (s) = -v s (-u) for all s ∈ R + and u ∈ R -(see part (i) of Remark 3.2), we obtain

a t 0 ψ ue bt ,0 (s) ds = -ψ ue bt ,0 (t) -ue bt F (z) R(z)
dz for all t ∈ R + and u ∈ R --satisfying -ue bt ∈ (0, θ 0 ), which, together with (7.6), leads to (7.1).

If t ∈ R + and u ∈ R --satisfying -ue bt ∈ (0, θ 0 ), then, by the proof of part (iii) of Proposition 2.1, -ψ ue bt ,0 (t) ∈ (0, θ 0 ), hence, by Proposition 3.3 in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF],

-ψ ue bt ,0 (t)

-ue bt 1 R(z) dz = -t.
It yields that (7.7)

-ψ ue bt ,0 (t)

-ue bt b R(z) - 1 z dz = -bt - -ψ ue bt ,0 (t) -ue bt 1 z dz.
The right hand side of (7.7) is

(7.8) -bt - -ψ ue bt ,0 (t) -ue bt 1 z dz = -bt -(log(-ψ ue bt ,0 (t)) -log(-ue bt )) = log(-u) -log(-ψ ue bt ,0 (t)).
We have already proved that -ψ ue bt ,0 (s) ∈ (0, θ 0 ) for all s ∈ R + yielding -ψ * u ∈ [0, θ 0 ]. Letting t → ∞ in (7.7), we conclude -ψ * u ∈ (0, θ 0 ). Indeed, ψ * u = 0 is not possible, since then the left hand side of (7.7) would tend to 0 and the right hand side of (7.7) would tend to +∞ (see (7.8)). Moreover, ψ * u = -θ 0 is not possible, since then the left hand side of (7.7) would tend to

θ 0 0 b R(z) -1
z dz = +∞ (see the beginning of the proof), and the right hand side of (7.7) would tend to log(-u)log(θ 0 ) (see (7.8)). Thus -ψ * u ∈ (0, θ 0 ), and letting t → ∞ in (7.7), we obtain

-ψ * u 0 b R(z) - 1 z dz = log(-u) -log(-ψ * u )
for all u ∈ R --. This can be written in the form

u = ψ * u exp -ψ * u 0 b R(z) - 1 z dz .
Consequently, u = -K(-ψ * u ) for all u ∈ R --. The function K is strictly increasing on (0, θ 0 ), see the beginning of the proof, hence we conclude ψ * u = -K -1 (-u) for all u ∈ R --. Next, we check that if, in addition, a ∈ R ++ , then P(V ∈ R ++ ) = 1.

The monotone convergence theorem yields E(e uV ) ↓ E(½ {V =0} ) = P(V = 0) as u → -∞. We have lim u→-∞ ψ * u =lim u→-∞ K -1 (-u) = -θ 0 , since lim λ↑θ 0 K(λ) = +∞, see the beginning of the proof. Consequently, by (7.1), we obtain

lim u→-∞ E(e uV ) = exp -y 0 θ 0 + θ 0 0 F (z) R(z) dz = 0 and hence P(V ∈ R ++ ) = 1, since θ 0 0 F (z) R(z) dz = -∞.
Indeed, as at the beginning of the proof, there exists

z 2 ∈ (0, θ 0 ) such that R(z) z-θ 0 2R ′ (θ 0 ) for all z ∈ (z 2 , θ 0 ) with R ′ (θ 0 ) ∈ R ++ , hence θ 0 0 F (z) R(z) dz θ 0 z 2 F (z) R(z) dz az 2 2R ′ (θ 0 ) θ 0 z 2 1 z-θ 0 dz = -∞.
Second proof of (ii), (iii) and (iv). The idea of this proof is due to Clément Foucart. First, we need to introduce some notations based on Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]. For all t ∈ R + , let v t := lim λ→∞ v t (λ), which exists in (0, ∞], see Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]Theorem 3.5], where v t (λ), t, λ ∈ R + , is given in (iii) of Proposition 2.1. Let v := lim t→∞ v t , which exists in R + , and it is known that v is the largest root of the equation R(z) = 0, z ∈ R + , see Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]Theorem 3.8]. Indeed, Condition 3.6 in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF] holds in our case, since

R(z) bz + δ α α z α δ α 2α z α > 0 for z > -2bα δ α 1 α-1 =: θ > 0, and ∞ θ 1 R(z) dz < 2α δ α ∞ θ z -α dz = 2α θ 1-α δ α (α -1) < ∞.
Further, by Li [25, page 63], v = θ 0 and θ 0

-2bα δ α 1 α-1 . Since R ′ (0) = b < 0, R ′′ (z) = σ 2 + δ α (α -1)z α-2
0, z ∈ R ++ , and lim z→∞ R(z) = ∞, we get there is a positive root of R yielding that v = θ 0 > 0. For all t ∈ R + , let [0, v t ) ∋ q → η t (q) be the inverse function of R + ∋ λ → v t (λ), which exists and is strictly monotone increasing, due to the fact that R + ∋ λ → v t (λ) is strictly monotone increasing, see Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]Proposition 3.1]. By Li [25, Proposition 3.14], we have

lim t→∞ η t (λ) e bt = K(λ) ∈ R ++ , λ ∈ (0, θ 0 ). (7.9)
Indeed, b ∈ R --, and by (7.3), ∞ 1 z log(z) m(dz) < ∞, where the measure m takes the form

m(dz) = δ α C α z -1-α ½ R ++ (z) dz. The form of the limit K(λ) is K(λ) = λ exp λ 0 b R(z) - 1 z dz , λ ∈ (0, θ 0 ),
which follows by the proof of Proposition 3.14 in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]. Using (7.4) and (7.9), we have for all λ ∈ (0, θ 0 ),

η t (λ)Y t a.s. -→ K(λ)V =: U λ as t → ∞. (7.10)
Using the same ideas as in the proofs of Theorems 3.13 and 3.15 in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF], we show that

E(e uU λ ) = exp -y 0 f (-u, λ) + f (-u,λ) 0 F (z) R(z)
dz , u ∈ R -, λ ∈ (0, θ 0 ), (7.11) where f (-u, λ) := lim t→∞ v t (-uη t (λ)) ∈ [0, θ 0 ) and, in case of u ∈ R --, f (-u, λ) ∈ (0, θ 0 ) and it satisfies

f (-u,λ) λ 1 R(z) dz = log(-u) b . (7.12)
The case u = 0 is trivial, since in this case f (-u, λ) = f (0, λ) = lim t→∞ v t (0) = 0, because of v t (0) = 0, t ∈ R + for u = 0. So we can assume that u ∈ R --.

Note that the integral f (-u,λ) 0

F (z) R(z) dz is well-defined. First we check that (7.13) 0 λ 1 R(z) dz = +∞, θ 0 λ 1 R(z) dz = -∞, λ ∈ (0, θ 0 ). Indeed, we have lim z↓0 R(z) z = b ∈ R --, thus there exists z 3 ∈ (0, λ) such that R(z) z b
2 for all z ∈ (0, z 3 ). Hence

0 λ 1 R(z) dz 0 z 3 2
bz dz = +∞. Moreover, as at the beginning of the proof, there exists

z 2 ∈ (λ, θ 0 ) such that R(z) z-θ 0 2R ′ (θ 0 ) for all z ∈ (z 2 , θ 0 ) with R ′ (θ 0 ) ∈ R ++ . Hence θ 0 λ 1 R(z) dz 1 2R ′ (θ 0 ) θ 0 z 2 1 z-θ 0 dz = -∞. Consequently, f (-u, λ) can not be a root of the equation R(z) = 0, z ∈ R + , i.e., f (-u, λ) /
∈ {0, θ 0 }, since otherwise, by (7.13), the left hand side of (7.12) would be ±∞, but the right hand side of (7.12) is a real number. Hence, using the same argument as in the proof of (2.7), we have f (-u, λ) ∈ (0, θ 0 ). The integrand F R is continuous on [0, f (-u, λ)], since in case of a = 0 the integrand is zero, and in case of a ∈ R ++ , by the definition of θ 0 , we have R(z) < 0 for all z ∈ (0, θ 0 ) and hence for all (0, f (-u, λ)], and lim z↓0

F (z) R(z) = a b ∈ R --.
Next, by (7.10), we have η t (λ)Y t D -→ U λ as t → ∞ for all λ ∈ (0, θ 0 ), and, by continuity theorem and (2.4), for all u ∈ R -and λ ∈ (0, θ 0 ), we get

E(e uU λ ) = lim t→∞ E (exp {uη t (λ)Y t }) = lim t→∞ exp -y 0 v t (-uη t (λ)) + vt(-uηt(λ)) -uηt(λ) F (z) R(z)
dz , (7.14) since η t (λ) ↓ 0 as t → ∞ (see Li [25, proof of Proposition 3.14]), and hence for all u ∈ R --, we have -uη t (λ) ∈ (0, v) = (0, θ 0 ) for sufficiently large t. Recall that, by formula (3.23) in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF], if η t (λ) and -uη t (λ) belong to (0, θ 0 ) = (0, v), where λ ∈ (0, v), then

vt(-uηt(λ)) λ 1 R(z) dz = -uηt(λ) ηt(λ) 1 R(z) dz.
Since η t (λ) ↓ 0 as t → ∞, for all u ∈ R --, we have η t (λ), -uη t (λ) ∈ (0, v) for sufficiently large t, and hence for all u ∈ R --,

lim t→∞ vt(-uηt(λ)) λ 1 R(z) dz = lim t→∞ -uηt(λ) ηt(λ) 1 R(z) dz = lim t→∞ -uηt(λ) ηt(λ) 1 bz dz = lim t→∞ log(-u) b = log(-u) b ,
where we used that lim z↓0

R(z) bz = 1. The function (0, v) ∋ x → x λ 1 R(z) dz =: G(x)
is continuous and strictly decreasing, hence its inverse is also continuous and strictly decreasing, implying that for all u ∈ R --and λ ∈ (0, v), the limit

f (-u, λ) = lim t→∞ v t (-uη t (λ)) = lim t→∞ G -1 (G(v t (-uη t (λ)))) = G -1 log(-u) b
exists and it satisfies (7.12), where G -1 denotes the inverse of G, since, by (7.13), the range of G is R. Consequently, using the continuity of the integral upper limit function, the fact that η t (λ) ↓ 0 as t → ∞, and (7.14), we have (7.11), as desired. Using that K(λ) ∈ R ++ and V = U λ /K(λ), we have E(e uV ) = E(e uU λ /K(λ) ), u ∈ R -, and then (7.11) yields (7.1).

We point out that, in the second proof of (7.1), we were not able to use directly (7.4), and that's why the usage of η t (λ) in the argument above is really essential for us.

Next, we check that if, in addition, a ∈ R ++ , then P(V ∈ R ++ ) = 1. By the law of total probability, E(e uU λ ) = 1 • P(e -U λ = 1) + E(e uU λ | e -U λ = 1) P(e -U λ = 1) , u ∈ R -, and hence, by the dominated convergence theorem and (7.11),

P(e -U λ = 1) = lim u→-∞ E(e uU λ ) = lim u→-∞ exp -y 0 f (-u, λ) + f (-u,λ) 0 F (z) R(z) dz = exp -y 0 θ 0 + θ 0 0 F (z) R(z) dz ,
where we used that, by (7.13 Li [25,proof of Theorem 3.15]). In case of a ∈ R ++ , we have

), lim u→-∞ f (-u, λ) = lim u→-∞ G -1 log(-u) b = lim y→-∞ G -1 (y) = θ 0 = v (see also
θ 0 0 F (z) R(z) dz = -∞
(see the end of the first proof of (iv)), and hence P(e -U λ = 1) = 0. This yields that, in case of a ∈ R ++ , P(U λ = 0) = 0 and, since K(λ) ∈ R ++ , we have P(V = 0) = 0, i.e., P(V ∈ R ++ ) = 1.

✷

In the next remark we specialize Theorem 7.1 to the case σ = 0.

7.3 Remark. Under the conditions of Theorem 7.1, in case of σ = 0, using part (ii) of Theorem 7.1, we derive the Laplace transform of V , which results in an explicit expression. Recall that the function ψ ue bt ,0 : R + → R -is the unique locally bounded solution to the differential equation

ψ ′ ue bt ,0 (s) = δ α α (-ψ ue bt ,0 (s)) α -bψ ue bt ,0 (s), s ∈ R + , ψ ue bt ,0 (0) = ue bt . (7.15)
We have to determine the limit lim t→∞ ψ ue bt ,0 (t). If u = 0, then the unique locally bounded solution of the differential equation (7.15) is ψ ue bt ,0 (s) = 0, s ∈ R + , and hence in this case lim t→∞ ψ ue bt ,0 (t) = 0. In what follows, let us suppose that u ∈ R --. The unique solution of the differential equation (7.15) (which can be transformed into a Bernoulli differential equation) is

ψ ue bt ,0 (s) = - (-ue bt ) 1-α + δ α bα e b(α-1)s - δ α bα 1 1-α , s ∈ R + ,
and consequently, by part (ii) of Theorem 7.1,

ψ * u = lim t→∞ ψ ue bt ,0 (t) = -(-u) 1-α - δ α bα 1 1-α , u ∈ R --,
and hence

E(e uV ) = exp -y 0 (-u) 1-α - δ α bα 1 1-α + a (-u) 1-α -δ α bα 1 1-α 0 1 δ α α z α-1 + b dz , u ∈ R --.
We can derive the above formula for ψ * u , u ∈ R --, using part (iii) of Theorem 7.1 as well. We have θ 0 = -bα δ α 1 α-1 and, by (7.2)

, λ 0 b R(z) - 1 z dz = - λ 0 δ α α z α-2 δ α α z α-1 + b dz = - 1 α -1 log δ α α λ α-1 + b -log(b) = log δ α α λ α-1 + b b -1 α-1
for all λ ∈ (0, θ 0 ). Consequently, -→ b as T → ∞, and

K(λ) = λ δ α bα λ α-1 + 1 -1 α-1 = δ α bα + λ 1-α 1 1-α , λ ∈ (0, θ 0 ), thus ψ * u = -K -1 (-u) = -(-u) 1-α - δ α bα 1 1-α , u ∈ R --. ✷ 7.4 Theorem. Let a ∈ R ++ , b ∈ R --, σ, δ ∈ R ++ ,
e -bT /2 ( b T -b) D -→ σZ - V b -1/2 as T → ∞,
where V is a positive random variable having Laplace transform given in (7.1), and Z is a standard normally distributed random variable, independent of V .

With a random scaling, we have

1 σ T 0 Y s ds 1/2 ( b T -b) D -→ N (0, 1)
as T → ∞.

Proof. By Proposition 4.2, there exists a unique MLE b T of b for all T ∈ R ++ which takes the form given in (4.3). By Theorem 7.1, e bt t 0 Y s ds a.s.

-→ -V b as t → ∞, where P(V ∈ R ++ ) = 1, and hence

-→ ∞ • - V b = ∞ as t → ∞.
Since the quadratic variation process of the square integrable martingale Applying again (7.16) and the continuous mapping theorem, we obtain

1 σ T 0 Y s ds 1/2 ( b T -b) = -e bT T 0 Y s ds -1/2 e bT /2 T 0 Y s dW s D -→ -- V b -1/2 - V b 1/2 Z = -Z D = N (0,
P(V ∈ R ++ ) = 1 and P( t 0 Y s ds ∈ R ++ ) = 1, t ∈ R ++ (see (v) of Proposition 2.1), in case of a ∈ R ++ , -log Y t+1 Y t = -log e -b e b(t+1) Y t+1 e bt Y t a.s. -→ -log e -b V V = b as t → ∞, and
- Y t t 0 Y s ds = - e bt Y t e bt t 0 Y s ds a.s. -→ - V -V b = b as t → ∞.

✷

Appendices

A Likelihood-ratio process

Based on Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF], see also Jacod and Mémin [START_REF] Jacod | Caractéristiques locales et conditions de continuité absolue pour les semi-martingales Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete[END_REF], Sørensen [START_REF] Sørensen | Likelihood methods for diffusions with jumps[END_REF] and Luschgy [START_REF] Luschgy | Asymptotic inference for semimartingale models with singular parameter points[END_REF], we recall certain sufficient conditions for the absolute continuity of probability measures induced by semimartingales together with a representation of the corresponding Radon-Nikodym derivative (likelihood-ratio process).

Let D(R + , R d ) denote the space of R d -valued càdlàg functions defined on R + . Let (η t ) t∈R + denote the canonical process η t (ω

) := ω(t), ω ∈ D(R + , R d ), t ∈ R + . Put F η t := σ(η s , s ∈ [0, t]), t ∈ R + , and
D t (R + , R d ) := ε∈R ++ F η t+ε , t ∈ R + , D(R + , R d ) := σ t∈R + F η t .
Let Ψ ⊂ R k be an arbitrary non-empty set, and let P ψ , ψ ∈ Ψ, are probability measures on the canonical space (D(R + , R d ), D(R + , R d )). Suppose that for each ψ ∈ Ψ, under P ψ , the canonical process (η t ) t∈R + is a semimartingale with semimartingale characteristics (B (ψ) , C, ν

ω ∈ D(R + , R d ),
where ε (t,x) denotes the Dirac measure at the point (t, x) ∈ R + × R d , and ∆η t := η tη t-, t ∈ R ++ , ∆η 0 := 0, and B (ψ) is the predictable process (with values in R d having finite variation over each finite interval [0, t], t ∈ R + ) appearing in the canonical decomposition

η t = η 0 + M (ψ) t + B (ψ) t , t ∈ R + ,
of the special semimartingale ( η t ) t∈R + under P ψ given by

η t := η t - s∈[0,t] (η s -h(∆η s )), t ∈ R + ,
where (M (ψ) t

) t∈R + is a local martingale with M (ψ) 0 = 0. We call the attention that, by our assumption, the process C = (η cont ) (ψ) does not depend on ψ, although (η cont ) (ψ) might depend on ψ. In addition, assume that P ψ (ν (ψ) ({t} × R d ) = 0) = 1 for every ψ ∈ Ψ, t ∈ R + , and P ψ (η 0 = x 0 ) = 1 with some x 0 ∈ R d for every ψ ∈ Ψ. Note that we have the semimartingale representation

η t = x 0 + B (ψ) t + (η cont ) (ψ) t + t 0 R d h(x) (µ η -ν (ψ) )(ds, dx) + t 0 R d (x -h(x)) µ η (
(R + , R d )× R + .
Assume also that for every ψ, ψ ∈ Ψ, there exist a P ⊗ B(R d )-measurable function

V ( ψ,ψ) : D(R + , R d ) × R + × R d → R ++ and a predictable R d -valued process β ( ψ,ψ) satisfying ν (ψ) (dt, dx) = V ( ψ,ψ) (t, x)ν ( ψ) (dt, dx), (A.1) t 0 R d V ( ψ,ψ) (s, x) -1 2 ν ( ψ) (ds, dx) < ∞, (A.2) B (ψ) t = B ( ψ) t + t 0 c (ψ) s β ( ψ,ψ) s dF (ψ) s + t 0 R d (V ( ψ,ψ) (s, x) -1)h(x) ν ( ψ) (ds, dx), (A.3) t 0 (β ( ψ,ψ) s ) ⊤ c (ψ) s β ( ψ,ψ) s dF (ψ)
s < ∞, (A.4) P ψ -almost sure for every t ∈ R + . Further, assume that for each ψ ∈ Ψ, local uniqueness holds for the martingale problem on the canonical space corresponding to the triplet (B (ψ) , C, ν (ψ) ) with the given initial value x 0 with P ψ as its unique solution. Then for each T ∈ R + , P ψ,T is absolutely continuous with respect to P ψ,T , where P ψ,T := P ψ | D T (R + ,R d ) denotes the restriction of P ψ to D T (R + , R d ) (similarly for P ψ,T ), and, under P ψ,T , the corresponding likelihood-ratio process takes the form

log d P ψ,T d P ψ,T (η) = T 0 (β ( ψ,ψ) s ) ⊤ d(η cont ) ( ψ) s - 1 2 T 0 (β ( ψ,ψ) s ) ⊤ c (ψ) s β ( ψ,ψ) s dF (ψ) s + T 0 R d (V ( ψ,ψ) (s, x) -1) (µ η -ν ( ψ) )(ds, dx) + T 0 R d (log(V ( ψ,ψ) (s, x)) -V ( ψ,ψ) (s, x) + 1) µ η (ds, dx) (A.5)
for all T ∈ R ++ , see Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]Theorem III.5.34]. A detailed proof of (A.5) using Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]

B Limit theorems for continuous local martingales

In what follows we recall some limit theorems for continuous local martingales. We use these limit theorems for studying the asymptotic behavior of the MLE of b. First we recall a strong law of large numbers for continuous local martingales.

B.1 Theorem. (Liptser and Shiryaev [START_REF] Liptser | Statistics of Random Processes II. Applications[END_REF]Lemma 17.4]) Let Ω, F, (F t ) t∈R + , P be a filtered probability space satisfying the usual conditions. Let (M t ) t∈R + be a square-integrable continuous local martingale with respect to the filtration (F t ) t∈R + such that P(M 0 = 0) = 1. Let (ξ t ) t∈R + be a progressively measurable process such that

P t 0 ξ 2 u d M u < ∞ = 1, t ∈ R + , and t 0 ξ 2 u d M u a.s. -→ ∞ as t → ∞, (B.1)
where ( M t ) t∈R + denotes the quadratic variation process of M . Then

t 0 ξ u dM u t 0 ξ 2 u d M u a.s. -→ 0 as t → ∞. (B.2)
If (M t ) t∈R + is a standard Wiener process, the progressive measurability of (ξ t ) t∈R + can be relaxed to measurability and adaptedness to the filtration (F t ) t∈R + .

The next theorem is about the asymptotic behavior of continuous multivariate local martingales, see van Zanten [START_REF] Van Zanten | A multivariate central limit theorem for continuous local martingales[END_REF]Theorem 4.1].

B.2 Theorem. (van Zanten [START_REF] Van Zanten | A multivariate central limit theorem for continuous local martingales[END_REF]Theorem 4.1]) Let Ω, F, (F t ) t∈R + , P be a filtered probability space satisfying the usual conditions. Let (M t ) t∈R + be a d-dimensional square-integrable continuous local martingale with respect to the filtration (F t ) t∈R + such that P(M 0 = 0) = 1. Suppose that there exists a function Q : [t 0 , ∞) → R d×d with some t 0 ∈ R + such that Q(t) is an invertible (non-random) matrix for all t ∈ [t 0 , ∞), lim t→∞ Q(t) = 0 and

Q(t) M t Q(t) ⊤ P -→ ηη ⊤ as t → ∞,
where η is a d× d random matrix. Then, for each R k -valued random vector v defined on (Ω, F, P), we have

(Q(t)M t , v) D -→ (ηZ, v) as t → ∞,
where Z is a d-dimensional standard normally distributed random vector independent of (η, v).

C Some explicit formulae in case of α = 3 2 First, in the special case of α = 3 2 , we make explicit the Laplace transform of the stationary distribution in the subcritical and critical cases given in Theorem 2.5 by evaluating the integral in its expression.

C.1 Example. We calculate the Laplace transform of the unique stationary distribution π given in Theorem 2.5 in case of α = First we consider the case of b ∈ R ++ and σ ∈ R ++ . Then we can write 2y

σ 2 2 y 2 + 2δ 3 2 3 y + b = 2 σ 2 • 2y + 4δ 3 2 3σ 2 y 2 + 4δ 3 2 3σ 2 y + 2b σ 2 - 8δ 3 2 3σ 4 • 1 y 2 + 4δ 3 2 3σ 2 y + 2b σ 2 .
We have

(-u) 1 2 0 2y + 4δ 3 2 3σ 2 y 2 + 4δ 3 2 3σ 2 y + 2b σ 2 dy = log y 2 + 4δ 3 2 3σ 2 y + 2b σ 2 y=(-u) 1 2 y=0 = log σ 2 2b (-u) + 2δ 3 2 3b (-u) 1 2 + 1 .
Moreover, using

y 2 + 4δ 3 2 3σ 2 y + 2b σ 2 = y + 2δ 3 2 3σ 2 2 + 2b σ 2 - 4δ 3 9σ 4 ,
we obtain 1

y 2 + 4δ 3 2 3σ 2 y + 2b σ 2 dy =                      1 2b σ 2 -4δ 3 9σ 4 arctan y+ 2δ 3 2 3σ 2 2b σ 2 -4δ 3 9σ 4 + C if b ∈ 2δ 3 9σ 2 , ∞ , -1 y+ 2δ 3 2 3σ 2 + C if b = 2δ 3 9σ 2 , 1 4δ 3 9σ 4 -2b σ 2 log y+ 2δ 3 2 3σ 2 -4δ 3 9σ 4 -2b σ 2 y+ 2δ 3 2 3σ 2 + 4δ 3 9σ 4 -2b σ 2 + C if b ∈ 0, 2δ 3 9σ 2 ,
where

C ∈ R. If b ∈ 2δ 3 9σ 2 , ∞ and σ ∈ R ++ then, applying the formula arctan(u) -arctan(v) = arctan u-v 1+uv , u, v ∈ R + , we get (-u) 1 2 0 1 y 2 + 4δ 3 2 3σ 2 y + 2b σ 2 dy = 1 2b σ 2 -4δ 3 9σ 4   arctan (-u) 1 2 + 2δ 3 2 3σ 2 2b σ 2 -4δ 3 9σ 4 -arctan 2δ 3 2 3σ 2 2b σ 2 -4δ 3 9σ 4   = 1 2b σ 2 -4δ 3 9σ 4 arctan   2b σ 2 -4δ 3 9σ 4 2b σ 2 (-u) -1 2 + 2δ 3 2 3σ 2   ,
and hence

∞ 0 e uy π(dy) = σ 2 2b (-u) + 2δ 3 2 3b (-u) 1 2 + 1 -2a σ 2 exp    8aδ 3 2 3σ 4 2b σ 2 -4δ 3 9σ 4 arctan   2b σ 2 -4δ 3 9σ 4 2b σ 2 (-u) -1 2 + 2δ 3 2 3σ 2      . If b = 2δ 3 9σ 2 and σ ∈ R ++ , then (-u) 1 2 0 1 y 2 + 4δ 3 2 3σ 2 y + 2b σ 2 dy = - 1 (-u) 1 2 + 2δ 3 2 3σ 2 + 1 2δ 3 2 3σ 2 = (-u) 1 2
(-u)

1 2 + 2δ 3 2 3σ 2 = 1 1 + 2δ 3 2 3σ 2 (-u) -1 2 ,
and hence

∞ 0 e uy π(dy) = 9σ 4 4δ 3 (-u) + 3σ 2 δ 3 2 (-u) 1 2 + 1 -2a σ 2 exp 8aδ 3 2 3σ 4 1 + 2δ 3 2 3σ 2 (-u) -1 2 . If b ∈ 0, 2δ 3 9σ 2 and σ ∈ R ++ , then (-u) 1 2 0 1 y 2 + 4δ 3 2 3σ 2 y + 2b σ 2 dy = 1 4δ 3 9σ 4 -2b σ 2   log    (-u) 1 2 + 2δ 3 2 3σ 2 -4δ 3 9σ 4 -2b σ 2 (-u) 1 2 + 2δ 3 2 3σ 2 + 4δ 3 9σ 4 -2b σ 2    -log    2δ 3 2 3σ 2 -4δ 3 9σ 4 -2b σ 2 2δ 3 2 3σ 2 + 4δ 3 9σ 4 -2b σ 2       = 1 4δ 3 9σ 4 -2b σ 2 log    (-u) 1 2 2δ 3 2 3σ 2 + 4δ 3 9σ 4 -2b σ 2 + 2b σ 2 (-u) 1 2 2δ 3 2 3σ 2 -4δ 3 9σ 4 -2b σ 2 + 2b σ 2    ,
and hence

∞ 0 e uy π(dy) = σ 2 2b (-u) + 2δ 3 2 3b (-u) 1 2 + 1 -2a σ 2    (-u) 1 2 2δ 3 2 3σ 2 + 4δ 3 9σ 4 -2b σ 2 + 2b σ 2 (-u) 1 2 2δ 3 2 3σ 2 -4δ 3 9σ 4 -2b σ 2 + 2b σ 2    8aδ 3 2 3σ 4 4δ 3 9σ 4 -2b σ 2
.

Next we consider the case of b ∈ R ++ and σ = 0. Then we can write 2y (-u)

1 2 - 9b 2δ 3 log (-u) 1 2 + 3b 2δ 3 2 -log 3b 2δ 3 2 = 3 δ 3 2 (-u) 1 2 - 9b 2δ 3 log 2δ 3 2 3b (-u) 1 2 + 1 ,

and hence

∞ 0 e uy π(dy) = exp -3a dx .

δ 3 2 (-u) 1 2 1 + 2δ 3 2 3b (-u)

By substitution

x = y 2 , -u 0 1 σ 2 2 x + 2δ 3 2 3 x 1 2 dx = (-u) 1 2 0 2y σ 2 2 y 2 + 2δ 3 2 3 y dy = (-u) 1 2 0 2 σ 2 2 y + 2δ 3 2 3
dy.

Consequently,

(-u) 1 2 0 2 σ 2 2 y + 2δ 3 2 3 dy = 4 σ 2 log y + 4δ 3 2 3σ 2 y=(-u) 1 2 y=0 = 4 σ 2 log (-u) 1 2 + 4δ 3 2 3σ 2 - 4 σ 2 log 4δ 3 2 3σ 2 = 4 σ 2 log 3σ 2 4δ 3 2 (-u) 1 2 + 1 , hence ∞ 0 e uy π(dy) = 3σ 2 4δ 3 2 (-u) 1 2 + 1 -4a σ 2
.

Finally, by the proof of Theorem 2.5, if b = 0, σ = 0 and α = 3 2 , then

∞ 0 e uy π(dy) = exp{-3a δ 3/2 (-u) 1/2 }, u ∈ R -. ✷
where C ∈ R. Moreover, using

g 2 u + 4δ 3 2 3σ 2 g u + 2b σ 2 = g u + 2δ 3 2 3σ 2 2 + 2b σ 2 - 4δ 3 9σ 4 , we obtain δ 3 2 3b g 2 u + 4δ 3 2 3σ 2 g u + 2b σ 2 dg u =                        δ 3 2 3b 2b σ 2 -4δ 3 9σ 4 arctan gu+ 2δ 3 2 3σ 2 2b σ 2 -4δ 3 9σ 4 + C if b > 2δ 3 9σ 2 , - δ 3 2 3b gu+ 2δ 3 2 3σ 2 + C if b = 2δ 3 9σ 2 , δ 3 2 3b 4δ 3 9σ 4 -2b σ 2 log gu+ 2δ 3 2 3σ 2 -4δ 3 9σ 4 -2b σ 2 gu+ 2δ 3 2 3σ 2 + 4δ 3 9σ 4 -2b σ 2 + C if b < 2δ 3 9σ 2 . (C.4) Consequently, if b > 2δ 3 9σ 2 , then - σ 2 4b log 1 + 4δ 3 2 3σ 2 g u (t) + 2b σ 2 g u (t) 2 - δ 3 2 3b 2b σ 2 -4δ 3 9σ 4 arctan   g u (t) + 2δ 3 2 3σ 2 2b σ 2 -4δ 3 9σ 4   = - σ 2 4 t + C, t ∈ R + , with some C ∈ R + . Using the initial value g u (0) = (-u) 1 2
, we obtain

C = - σ 2 4b log 1 + 4δ 3 2 3σ 2 (-u) 1 2 - 2b σ 2 u - δ 3 2 3b 2b σ 2 -4δ 3 9σ 4 arctan   (-u) 1 2 + 2δ 3 2 3σ 2 2b σ 2 -4δ 3 9σ 4   ,
and hence, by g u (t) = (-ψ u,0 (t))

1 2 , we conclude , and hence, by g u (t) = (-ψ u,0 (t)) (-ψ u,0 (t)) (-u)

σ 2 4b log     1 + 4δ 3 2 3σ 2 (-ψ u,0 (t)) 1 2 - 2b σ 2 ψ u,0 (t) 1 + 4δ 3 2 3σ 2 (-u) 1 2 -2b σ 2 u     + δ 3 2 3b 2b σ 2 -4δ 3 9σ 4   arctan   (-ψ u,0 (t)) 1 2 + 2δ 3 2 3σ 2 2b σ 2 -4δ 3 9σ 4   -arctan   (-u) 1 2 + 2δ 3
1 2 + 2δ
1 2 + 2δ 3 2 3σ 2 + 4δ 3 9σ 4 -2b σ 2    = σ 2 4 t.
Finally, if b = 0, then, by separation of variables, we have 1 x 2 + 4δ Note that an explicit formula for K -1 is not available. Next we consider the case of σ = 0. By Remark 7.3, we have θ 0 = 9b 2 4δ 3 , K(λ) = 2δ By substitution x = y 2 , for all u ∈ R --, we have for u ∈ R --.

g u + 4δ
(-u) -1 2 -2δ
Calculations based on part (ii) of Theorem 7.1. By (7.1), it is enough to know ψ * u to have an explicit formula for the Laplace transform of V . We carry out this calculation only in case of σ ∈ R ++ . By (C.6), for all t ∈ R + and u ∈ R --with ue bt = u 0 , we obtain σ 2 2b log((-ψ ue bt ,0 (t))

  , and α ∈ (1, 2), let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 = η 0 ) = 1. Then the Grigelionis form of (Y t ) t∈R + takes the form(2.8)

  Further, for all T ∈ R ++ , let P b,T := P b | D T (R + ,R) be the restriction of P b to D T (R + , R). The next result is about the form of the Radon-Nikodym derivative d P b,T d P b,T for b, b ∈ R. We will consider P b,T as a fixed reference measure, and we will derive the MLE for the parameter b based on the observations (Y t ) t∈[0,T ] . 4.1 Proposition. Let a ∈ R + , b, b ∈ R, σ, δ ∈ R ++ , and α ∈ (1, 2)

t 0 c

 0 u β ( b,b) u dF u P b -almost surely for every t ∈ R + , where the stochastic process (β ( b,b) t

2σ 2 T 0 Y

 20 u du holds P-almost surely, which yields the statement. ✷ Next, using Proposition 4.1, by considering P b,T as a fixed reference measure, we derive an MLE for the parameter b based on the observations (Y t ) t∈[0,T ] . Let us denote the right hand side of (4.1) by Λ T (b, b) replacing Y by Y . By an MLE b T of the parameter b based on the observations (Y t ) t∈[0,T ] , we mean b T := arg max b∈R Λ T (b, b), which will turn out to be not dependent on b. Our method for deriving an MLE is one of the known ones in the literature, and it turns out that these lead to the same estimator b T , see Remark 4.4. Next, we formulate a result about the unique existence of MLE b T of b for all T ∈ R ++ . 4.2 Proposition. Let a ∈ R + , b ∈ R, σ, δ ∈ R ++ , α ∈ (1, 2), and y 0 ∈ R + . If a ∈ R ++ or y 0 ∈ R ++ , then for each T ∈ R ++ , there exists a unique MLE b T of b P-almost surely having the form (4.3) b T = -Y Ty 0 -aTδ T 0 α √ Y u-dL u T 0 Y s ds , provided that T 0 Y s ds ∈ R ++ (which holds P-almost surely due to (v) of Proposition 2.1).

  dL u with respect to (Y t ) t∈[0,T ] and consequently that of b T . By Proposition 4.1, for all b, b ∈ R, we have

  Thus the MLE b T of b based on a continuous time observation (Y s ) s∈[0,T ] exists P-almost surely, and it takes the form (4.3) provided that T 0 Y s ds ∈ R ++ . ✷ 4.3 Remark. In what follows, under the assumptions of Proposition 4.2 and the additional assumption a σ 2

0 α√0

 0 dL u for b ∈ R and T ∈ R ++ . The estimating equation ΛT (b) = 0, b ∈ R, has a unique solution -Y T -y 0 -aT -δ T 0 α √ Y u-dLu T 0 Yu du provided that T 0 Y u du is strictly positive, which holds P-almost surely provided that a ∈ R ++ or y 0 ∈ R ++ (due to (v) of Proposition 2.1). Recall that this unique solution coincides with b T , see (4.3). ✷ 5 Asymptotic behavior of the MLE in the subcritical case If a ∈ R ++ or y 0 ∈ R ++ , then, using (4.3) and the SDE (1.1), we get b Tb = -Y Ty 0 -aTδ T Y u-dL u + b Y s ds ∈ R ++ , which holds P-almost surely due to (v) of Proposition 2.1. Here note that σ T 0 √ Y s dW s = Y cont T , T ∈ R + , due to part 1) of Remarks III.2.28 in Jacod and Shyriaev [18] and (2.8). 5.1 Theorem. Let a, b, σ, δ ∈ R ++ and α ∈ (1, 2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 = y 0 ) = 1 with some y 0 ∈ R + . Then the MLE b T of b is strongly consistent and asymptotically normal, i.e., b T a.s.

. 0 √

 0 By Proposition 4.2, there exists a unique MLE b T of b for all T ∈ R ++ , which has the form given in (4.3). By (i) of Theorem 2.5, (Y t ) t∈R + has a unique stationary distribution π with ∞ 0 y π(dy) = a b ∈ R ++ . By (ii) of Theorem 2.5, we have 1 T T 0 Y s ds a.s. -→ ∞ 0 y π(dy) as T → ∞, as T → ∞. Since the quadratic variation process of the square integrable martingale t Y s dW s t∈R + takes the form t 0 Y s ds t∈R + , using (5.1) and Theorem B.1, we have b T a.s.

  then the unique locally bounded solution of the differential equation (3.1) takes the form ψ

6. 2

 2 Theorem. Let a ∈ R ++ , b = 0, σ, δ ∈ R ++ , and α ∈ (1, 2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 = y 0 ) = 1 with some y 0 ∈ R + . Then the MLE of b is strongly consistent, i.e., b T a.s. -→ b as T → ∞. Proof. Since t 0 Y s ds t∈R + is monotone increasing P-almost surely, there exists an [0, ∞]-valued random variable ξ such that t 0 Y s ds a.s.

t 0 √

 0 Y s dW s t∈R + takes the form t 0 Y s ds t∈R + , using (5.1) and Theorem B.1, we have b T a.s.-→ b as T → ∞, as desired. ✷In the critical case the description of the asymptotic behavior of the MLE in question remains open.

  , and α ∈[START_REF] Amann | Ordinary Differential Equations. An Introduction to Nonlinear Analysis[END_REF][START_REF] Barczy | Asymptotic behavior of maximum likelihood estimators for a jump-type Heston model[END_REF]. Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 = y 0 ) = 1 with some y 0 ∈ R + .(i) Then there exists a random variable V with P(V ∈ R + ) = 1 such that e bt Y t a.s.-→ V and e bt t 0 Y u du a.s.

∞ 1 z

 1 log(z) m(dz) < ∞, where the measure m takes the form m

(7. 6 ) 0 ψ 0 ψ

 600 E(exp{ue bt Y t }) = exp y 0 ψ ue bt ,0 (t) + a t ue bt ,0 (s) ds , t ∈ R + , thus the limit lim t→∞ exp y 0 ψ ue bt ,0 (t) + a t ue bt ,0 (s) ds = E(e uV ) ∈ (0, 1]

  and α ∈[START_REF] Amann | Ordinary Differential Equations. An Introduction to Nonlinear Analysis[END_REF][START_REF] Barczy | Asymptotic behavior of maximum likelihood estimators for a jump-type Heston model[END_REF]. Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 = y 0 ) = 1 with some y 0 ∈ R + . Then the MLE of b is strongly consistent and asymptotically mixed normal, i.e., b T a.s.

t 0 Y

 0 s ds = e -bt e bt t 0 Y s ds a.s.

t 0 √ Y s dW s t∈R + takes the form t 0 Y

 00 s ds t∈R + , using (5.1) and Theorem B.1, we have b T a.s. -→ b as T → ∞. Further, by (5.1), e -bT /2 ( b Tb) = -σ e bT /2 T 0 √ Y s dW s e bT T 0 Y s ds , T ∈ R ++ . Again, by Theorem 7.1, e bT T 0 Y s ds a.s. -→ -V b as T → ∞, and using Theorem B.2 with η := -V b as T → ∞. (7.16) Consequently, by the continuous mapping theorem, e -bT /2 ( b Tb) D -→ -σZ -V b -1/2 as T → ∞, yielding the first assertion.

3 2 . 0 e

 20 Let u ∈ R --. By Theorem 2.5, ∞ uy π(dy) = exp -a

. 0 e

 0 Next we consider the case of b = 0 and σ ∈ R ++ . Let u ∈ R -. By Theorem 2.5, ∞ uy π(dy) = exp -a

  2δ

  t ∈ R + ,with some C ∈ R + . Using the initial value g u (0) = (-u)

3 2 3σ 2 -4δ 3 9σ 4 -2b σ 2

 3222

3 2 3σ 2 g 2 u = 9σ 4 16δ 3 g

 3223 u + 4δ

,

  First we consider the case of σ ∈ R ++ . Then we can write σ 2 x + 4δ

  u = -K -1 (-u) = -(-u) -uV ) = exp -y 0 (-u) -

  uV ) = exp -y 0 (-u) -

1 ,

 1 R(-ψ * u ) < 0. We have R(-ψ * u ) = σ 2 2 -ψ * u + 4δ

σ 2 (

 2 -ψ * u ), where -ψ * u ∈ (0, θ 0 ), hence -ψ * u + 4δ

3 2 3σ 2 (-ψ * u ) 1 2

 3221 + 2b σ 2 < 0. Moreover,

3 2 3σ 2 -4δ 3 9σ 4 -2b σ 2 (

 3222

  ∈ R + and λ ∈ R ++ \ {θ 0 }. It is enough to verify that the continuously differentiable function (0, t) ∋ s → v s (λ) is strictly monotone for all λ ∈ R ++ \ {θ 0 }, since then, by the substitution z = v s (λ), we obtain

	for all t
								and page 67].
	Next we check that
	(2.7)	-	0	t	F (v s (λ)) ds =	vt(λ) λ	F (z) R(z)	dz

  Let η 0 be a random variable independent of (W t ) t∈R + and (L t ) t∈R + satisfying P(η 0

	1.1) is Y t = 0 for all
	t ∈ R + .	✷
	Note that, by Proposition 2.1, the process (Y t ) t∈R + is a semimartingale, see, e.g., Jacod and
	Shiryaev [18, I.4.33]. Now we derive a so-called Grigelionis form for the semimartingale (Y t ) t∈R + ,
	see, e.g., Jacod and Shiryaev [18, III.2.23] or Jacod and Protter [17, Theorem 2.1.2].
	2.2 Proposition.

  see Jacod and Shiryaev [18, Definitions II.1.27, Theorem II.2.34]. We have |z½ { 1

	δ α

  see Jacod and Shiryaev [18, III.2.28 Remarks, part 1)]. Consequently, the (predictable) quadratic variation process of

  where R and F are given in Proposition 2.1. Consequently, by Theorem 2.7 of Duffie et al.[START_REF] Duffie | Affine processes and applications in finance[END_REF] (see alsoBarczy et al. [5, Theorem 2.4]), we have

  t * ]. This yields us to a contradiction taking into account the definition of t * and the fact that t * > 0. ✷ 3.2 Remark. (i) By (iii) of Proposition 2.1, ψ u,0 (t) = -v t (-u) for all t ∈ R + and u ∈ R -.

	(ii) The differential equation (3.1) is a special case of Polyanin and Zaitsev [34, Section 1.5.1-2/(27)].
	✷
	4 Existence and uniqueness of MLE

In this section, we will consider the stable CIR model (1.1) with known a

  almost surely, where ((η cont ) ) t∈R + denotes the continuous (local) martingale part of (η t ) t∈R + under P b . Using Remarks III.2.28 in Jacod and Shiryaev [18] and (2.8), the continuous (local) martingale part ( Y cont t ) t∈R + of ( Y t ) t∈R + takes the form Y cont Y u dW u , t ∈ R + , and, by (1.1), we have

	( b)
	t	= σ	t 0

t

  we have ∆ζ t = δ

α √ Y t-∆L t , t ∈ R + (following from (2.1) and Jacod and Shiryaev [18, Definitions II.1.27]) and using the SDE (1.1), we get ∆Y t = ∆ζ t = δ α √ Y t-∆L t , t ∈ R + . Hence ∆L t = ∆Yt δ α √ Y t- , t ∈ R ++ , since, by (iv) of Proposition 2.1, P Y t ∈ R ++ for all t ∈ R ++ = 1. For all t ∈ [0, T ] and ε ∈ (0, 1), (0,t] {ε<|z| 1} z µ L (du, dz) = u∈[0,t]

  Under the conditions of Theorem 7.1, in case of a ∈ R ++ ,log Y t+1 Yt , t ∈ R ++ , and -Yt t 0 Ys ds , t ∈ R ++ , are strongly consistent estimators of b as well. Indeed, by Theorem 7.1, using that

	1)	as T → ∞,
	as desired.	✷
	7.5 Remark.

  (ψ) ) associated with a fixed Borel measurable truncation function h : R d → R d , see Jacod and Shiryaev [18, Definition II.2.6 and Remark II.2.8]. Namely, C t := (η cont ) (ψ) t , t ∈ R + , where ( (η cont ) (ψ) t ) t∈R + denotes the (predictable) quadratic variation process (with values in R d×d ) of the continuous martingale part (η cont ) (ψ) of η under P ψ , ν (ψ) is the compensator of the integer-valued random measure µ η on R + × R d associated with the jumps of η under P ψ given by µ η (ω, dt, dx) :=

	s∈R +	½ {∆ηs(ω) =0} ε (s,∆ηs(ω)) (dt, dx),

  ds, dx), t ∈ R + , of η under P ψ , see Jacod and Shiryaev[START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] Theorem II.2.34]. Moreover, for each ψ ∈ Ψ, let us choose a nondecreasing, continuous, adapted process (F ) t∈R + with values in the set of all symmetric positive semidefinite d × d matrices such that -almost sure for every t ∈ R + . Due to the assumption P ψ (ν (ψ) ({t} × R d ) = 0) = 1 for every ψ ∈ Ψ, t ∈ R + , such choices of (F ) t∈R + are possible, see Jacod and Shiryaev [18, Proposition II.2.9 and Corollary II.1.19]. Let P denote the predictable σ-algebra on D

	process (c (ψ)			(ψ) t	) t∈R + with F 0 (ψ)	= 0 and a predictable
	C t =	0	t	c (ψ) s dF (ψ)
	(ψ) t	) t∈R + and (c (ψ)

t s P ψ t

  can be found in Barczy et al. [2, Appendix A].

  Consequently, (C.9) yields again that ψ * u satisfies equation (C.7), and hence, equation (C.8).

	√	3 2 3σ 2 -4δ 3 9σ 4 -2b σ 2 λ + 2δ 3 2 3σ 2 + 4δ 3 9σ 4 -2b σ 2	   -log	  	4δ 3 9σ 4 -2b σ 2 -2δ 3σ 2 3 2 4δ 3 9σ 4 -2b 3σ 2 σ 2 + 2δ 3 2	      .

2

Acknowledgements

We are grateful to Clément Foucart for providing us an idea how to derive (7.1), a formula for the Laplace transform of V in Theorem 7.1. We would like to thank Hatem Zaag for explaining us several methods that may be used for describing the asymptotic behavior of the ordinary differential equation (3.1). We would like to thank the referees for their comments that helped us to improve the paper.

This research is supported by Laboratory of Excellence MME-DII, Grant no. ANR11-LBX-0023-01 (http://labex-mme-dii.u-cergy.fr/). Mátyás Barczy was supported between September 2016 and January 2017 by the "Magyar Állami Eötvös Ösztöndíj 2016" Grant no. 75141 funded by the Tempus Public Foundation, and from September 2017 by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Ahmed Kebaier benefited from the support of the chair Risques Financiers, Fondation du Risque.

C.2 Example. Now we formulate a special case of Theorem 3.1 giving the Laplace transform of Y t in case of α = 3 2 . Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0 = y 0 ) = 1 with some y 0 ∈ R + , with a ∈ R + , b ∈ R, σ ∈ R ++ , δ ∈ R ++ and α = 3 2 . Then, by Theorem 3.1, for all u ∈ R -, E(e uYt ) = exp ψ u,0 (t)y 0 + a t 0 ψ u,0 (s) ds , t ∈ R + , (C. [START_REF] Amann | Ordinary Differential Equations. An Introduction to Nonlinear Analysis[END_REF] where the function ψ u,0 : R + → R -is the unique locally bounded solution to the differential equation

In case of u = 0, the unique locally bounded solution of (C.2) is ψ 0,0 (t) = 0, t ∈ R + . Let us consider the function g u (t) := (-ψ u,0 (t))

In case of b ∈ R + , (C.3) has a constant solution if and only if u = 0, and then g 0 (t) = ψ 0,0 (t) = 0 for all t ∈ R + . In case of b ∈ R --, (C.3) has a constant solution if and only if u = 0 or

and then g 0 (t) = ψ 0,0 (t) = 0 for all t ∈ R + or g u 0 (t) = (-u 0 )

1 2 , and hence ψ u 0 ,0 (t) = u 0 for all t ∈ R + . In the sequel, we suppose that u ∈ R --, and in case of b ∈ R --, in addition, we suppose that u = u 0 . Then, by separation of variables, we have 1

, and we have

with some C ∈ R. Using the initial value g u (0) = (-u)

1 2 , we obtain

, and hence, by g u (t) = (-ψ u,0 (t))

1 2 , we conclude

3σ 2 (-ψ u,0 (t))

C.3 Example. We derive an explicit formula for the Laplace transform of V given in Theorem 7.1 in case of α = 3 2 . In fact, we present two detailed calculations, the first one is based on the representation of ψ * u given in part (iii) of Theorem 7.1, and the second one is based on part (ii) of Theorem 7.1.

Calculations based on part (iii) of Theorem 7.1. We have

with ψ * 0 = 0 and ψ * u = -K -1 (-u) for u ∈ R --, where K -1 is the inverse of the strictly increasing function K : (0, θ 0 ) → R ++ given by

where we used (7.2) and

.

By substitution x = y 2 , for all λ ∈ (0, θ 0 ), we have where (-ψ * u )

Note that, by Theorem 7.1, (C.8) is equivalent to K(-ψ * u ) = -u. We show another way to derive this equation. By Theorem 7.1, for all sufficiently small λ ∈ R ++ , (C.9)