Mátyás Barczy
email: barczy.matyas@inf.unideb.hu

Mohamed Ben Alaya

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

Gyula Pap

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

Keywords: 2010 Mathematics Subject Classifications: 60H10, 91G70, 60F05, 62F12 jump-type Cox-Ingersoll-Ross (CIR) process, basic affine jump diffusion (BAJD), subordinator, maximum likelihood estimator

We consider a jump-type Cox-Ingersoll-Ross (CIR) process driven by a standard Wiener process and a subordinator, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate. We distinguish three cases: subcritical, critical and supercritical. In the subcritical case we prove weak consistency and asymptotic normality, and, under an additional moment assumption, strong consistency as well. In the supercritical case, we prove strong consistency and mixed normal (but non-normal) asymptotic behavior, while in the critical case, weak consistency and non-standard asymptotic behavior are described. We specialize our results to socalled basic affine jump-diffusions as well. Concerning the asymptotic behavior of the MLE in the supercritical case, we derive a stochastic representation of the limiting mixed normal distribution, where the almost sure limit of an appropriately scaled jump-type supercritical CIR process comes into play. This is a new phenomenon, compared to the critical case, where a diffusion-type critical CIR process plays a role.

Introduction

Continuous state and continuous time branching processes with immigration, especially, the Cox-Ingersoll-Ross (CIR) process (introduced by Feller [START_REF] Feller | Two singular diffusion problems[END_REF] and Cox et al. [START_REF] Cox | A theory of the term structure of interest rates[END_REF]) and its variants, play an 1 important role in stochastics, and there is a wide range of applications of these processes in biology and financial mathematics as well. In the framework of the famous Heston model, which is popular in finance, a CIR process can be interpreted as a stochastic volatility (or instantaneous variance) of a price process of an asset. In this paper, we consider a jump-type CIR process driven by a standard Wiener process and a subordinator dY t = (a -bY t ) dt + σ Y t dW t + dJ t , t ∈ [0, ∞), (1.1) with an almost surely non-negative initial value Y 0 , where a ∈ [0, ∞), b ∈ R, σ ∈ (0, ∞), (W t ) t∈[0,∞) is a 1-dimensional standard Wiener process, and (J t ) t∈[0,∞) is a subordinator (an increasing Lévy process) with zero drift and with Lévy measure m concentrating on (0, ∞) such that

∞ 0 z m(dz) ∈ [0, ∞), (1.2)
that is, (1.3) E(e uJt ) = exp t ∞ 0 (e uz -1) m(dz)

for any t ∈ [0, ∞) and for any complex number u with Re(u) ∈ (-∞, 0], see, e.g., Sato [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]proof of Theorem 24.11]. We suppose that Y 0 , (W t ) t∈[0,∞) and (J t ) t∈[0,∞) are independent. Note that the moment condition (1.2) implies that m is a Lévy measure (since min(1, z 2 ) z for z ∈ (0, ∞)).

Moreover, the subordinator J has sample paths of bounded variation on every compact time interval almost surely, see, e.g., Sato [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]Theorem 21.9]. We point out that the assumptions assure that there is a (pathwise) unique strong solution of the SDE (1.1) with P(Y t ∈ [0, ∞) for all t ∈ [0, ∞)) = 1 (see Proposition 2.1). In fact, (Y t ) t∈[0,∞) is a special continuous state and continuous time branching process with immigration (CBI process), see Proposition 2.1.

In the present paper, we focus on parameter estimation for the jump-type CIR process (1.1) in critical and supercritical cases (b = 0 and b ∈ (-∞, 0), respectively), which have not been addressed in previous research. We also study the subcritical case (b ∈ (0, ∞)) and we get results extending those of Mai [START_REF] Mai | Drift estimation for jump diffusions: time-continuous and high-frequency observations[END_REF]Theorem 4.3.1] in several aspects: we do not suppose the ergodicity of the process Y and we make explicit the expectation of the unique stationary distribution of Y in the limit law in Theorem 5.2. However, we note that some points in Mai's approach [START_REF] Mai | Drift estimation for jump diffusions: time-continuous and high-frequency observations[END_REF]Sections 3.3 and 4.3] should be corrected concerning the expressions of the likelihood ratio (Mai [40,formula (3.10)]) and the maximum likelihood estimator (MLE) of b ∈ R (Mai [40, formula (4.23)]), see our results in Propositions 4.1 and 4.2, respectively. Supposing that a ∈ [0, ∞), σ ∈ (0, ∞) and the measure m 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, ∞). It will turn out that for the calculation of the MLE of b, one does not need to know the value of the parameter σ and the measure m, see (4.3). We have restricted ourselves to studying the MLE of b supposing that a is known, since in order to describe the asymptotic behavior of the MLE of a supposing that b is known or the joint MLE of (a, b), one has to find, for instance, the limiting behavior of t 0 1 Ys ds as t → ∞, which seems to be a hard task even in the subcritical case. In general, we would need an explicit formula for the Laplace transform of t 0 1 Ys ds, t ∈ R + , which is not known up to our knowledge. This can be a topic of further research.

Studying asymptotic properties of various kinds of estimators for the drift parameters of the CIR process and its variants has a long history, but most of the existing results refer to the original (diffusion-type) CIR process. Overbeck [START_REF] Overbeck | Estimation for continuous branching processes[END_REF] studied the MLE of the drift parameters of the original CIR process based on continuous time observations, and later on, Ben Alaya and Kebaier [START_REF] Ben Alaya | Parameter estimation for the square root diffusions: ergodic and nonergodic cases[END_REF], [START_REF] Ben Alaya | Asymptotic behavior of the maximum likelihood estimator for ergodic and nonergodic square-root diffusions[END_REF] completed the results of Overbeck [START_REF] Overbeck | Estimation for continuous branching processes[END_REF] giving explicit forms of the joint Laplace transforms of the building blocks of the MLE in question as well. Another type of estimator, so-called conditional least squares estimator (LSE) has also been investigated for the drift parameters for the original CIR process, see, e.g., Overbeck and Rydén [START_REF] Overbeck | Estimation in the Cox-Ingersoll-Ross model[END_REF]. For a generalization of the original CIR process, namely, for a CIR model driven by a stable noise instead of a standard Wiener process (also called a stable CIR model) Li and Ma [START_REF] Li | Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model[END_REF] described the asymptotic behaviour of the (weighted) conditional LSE of the drift parameters of this model based on a discretely observed low frequency data set in the subcritical case. For a CBI process, being a generalization of a (stable) CIR process, Huang et al. [START_REF] Huang | Estimation for discretely observed continuous state branching processes with immigration[END_REF] studied the asymptotics of the weighted conditional LSE of the drift parameters of the model based on low frequency discrete time observations under second order moment assumptions of the branching and immigration mechanisms of the CBI process in question.

Note that we have E(J t ) = t ∞ 0 z m(dz) ∈ [0, ∞), and E(Y t ) ∈ [0, ∞) (see Proposition 2.2). Moreover, (J t ) t∈[0,∞) is a compound Poisson process if and only if m(R) = m((0, ∞)) ∈ [0, ∞), see, e.g., Sato [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]Examples 8.5]. If m((0, ∞)) = ∞, then its jump intensity is infinity, i.e., almost surely, the jump times are infinitely many, countable and dense in [0, ∞), and if m((0, ∞)) ∈ (0, ∞), then, almost surely, there are finitely many jump times on every compact intervals yielding that the jump times are infinitely many and countable in increasing order, and the jump intensity is m((0, ∞)), i.e., the first jump time has an exponential distribution with mean 1/m((0, ∞)), and the distribution of the jump size is m(dz)/m((0, ∞)) (see, e.g., Sato [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]Theorem 21.3]). The case m((0, ∞)) = 0 corresponds to the usual CIR process. Our forthcoming results will cover both cases m((0, ∞)) ∈ [0, ∞) and m((0, ∞)) = ∞.

In case of b ∈ (0, ∞), Y t converges in law as t → ∞ to its unique stationary distribution π (see Theorem 2.4). This follows from a general result for CBI processes which has been announced without proof in Pinsky [START_REF] Pinsky | Limit theorems for continuous state branching processes with immigration[END_REF] and a proof has been given in Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]Theorem 3.20 and the paragraph after Corollary 3.21], see also Keller-Ressel and Steiner [START_REF] Keller-Ressel | Yield curve shapes and the asymptotic short rate distribution in affine one-factor models[END_REF], Keller-Ressel [START_REF] Keller-Ressel | Moment explosions and long-term behavior of affine stochastic volatility models[END_REF], and Keller-Ressel and Mijatović [START_REF] Keller-Ressel | On the limit distributions of continuous-state branching processes with immigration[END_REF]Theorem 2.6]. The mean ∞ 0 y π(dy) ∈ [0, ∞) of the unique stationary distribution is the so-called long variance (long run average price variance, i.e., the limit of E(Y t ) as t → ∞, see (2.6) and (2.7)), b is the rate at which E(Y t ) reverts to

∞ 0 y π(dy) as t → ∞ (speed of adjustment, since E(Y t ) = ∞ 0 y π(dy) + e -bt E(Y 0 ) - ∞ 0 y π(dy) for all t ∈ [0, ∞), see (2.5) and
(2.7)). Under a ∈ (0, ∞), the moment condition (1.2) and the extra moment condition (1.4)

1 0 z log 1 z m(dz) < ∞,
Jin et al. [START_REF] Jin | Exponential ergodicity of the jump-diffusion CIR process[END_REF] established an explicit positive lower bound of the transition densities of (Y t ) t∈[0,∞) , and based on this result, they showed the existence of a Foster-Lyapunov function and derived exponential ergodicity for (Y t ) t∈[0,∞) (see Theorem 2.4). Comparing the moment conditions (1.2) and (1.4), note that the integrability of z log 1 z on the interval (0, e -1 ) yields that of z on the same interval.

In case of b = 0, if a + ∞ 0 z m(dz) ∈ (0, ∞), then lim t→∞ E(Y t ) = ∞ such that lim t→∞ t -1 E(Y t ) ∈ (0, ∞), and, in case of b ∈ (-∞, 0), if E(Y 0 ) ∈ (0, ∞) or a+ ∞ 0 z m(dz) ∈ (0, ∞)
(which rule out the case that Y is identically zero), then lim t→∞ E(Y t ) = ∞ such that lim t→∞ e bt E(Y t ) ∈ (0, ∞), hence the parameter b can always be interpreted as the growth rate, see Proposition 2.2.

The jump-type CIR process in (1.1) includes the so-called basic affine jump-diffusion (BAJD) as a special case, in which the drift takes the form κ(θ -Y t ) with some κ ∈ (0, ∞) and θ ∈ [0, ∞), and the Lévy process (J t ) t∈[0,∞) is a compound Poisson process with exponentially distributed jump sizes, namely,

m(dz) = cλe -λz ½ (0,∞) (z) dz (1.5)
with some constants c ∈ [0, ∞) and λ ∈ (0, ∞). Note that the measure m given by (1.5) satisfies (1.2) and (1.4), and, for the compound Poisson process in question, the first jump time has an exponential distribution with parameter c and the distribution of the jump size is exponential with parameter λ. Indeed, in this special case

∞ 0 z m(dz) = c ∞ 0 zλe -λz dz = c λ ∈ [0, ∞), and
1 0 z log 1 z m(dz) cλ 1 0 z log 1 z dz = cλ ∞ 0 ue -2u du = cλ 4 ∈ [0, ∞).
For describing the dynamics of default intensity, the BAJD was introduced by Duffie and Gârleanu [START_REF] Gârleanu | Risk and valuation of collateralized debt obligations[END_REF]. Filipović [START_REF] Filipović | A general characterization of one factor affine term structure models[END_REF] and Keller-Ressel and Steiner [START_REF] Keller-Ressel | Yield curve shapes and the asymptotic short rate distribution in affine one-factor models[END_REF] used the BAJD as a short-rate model.

The paper is organized as follows. In Section 2, we prove that the SDE (1.1) has a pathwise unique strong solution (under some appropriate conditions), see Proposition 2.1. We describe the asymptotic behaviour of the first moment of (Y t ) t∈[0,∞) , and, based on it, we introduce a classification of jumptype CIR processes given by the SDE (1.1), see Proposition 2.2 and Definition 2.3. Namely, we call (Y t ) t∈[0,∞) subcritical, critical or supercritical if b ∈ (0, ∞), b = 0, or b ∈ (-∞, 0), respectively. We recall a result about the existence of a unique stationary distribution and exponential ergodicity for the process (Y t ) t∈[0,∞) given by the equation (1.1), see Theorem 2.4. In Remark 2.5, we derive a Grigelionis representation for the process (Y t ) t∈[0,∞) . Further, we explain why we do not estimate the parameter σ, see Remark 2. [START_REF] Keller-Ressel | Affine Processes -Theory and Applications in Finance[END_REF]Theorem 4.10] derived a formula for the joint Laplace transform of a regular affine process and its integrated process containing the solutions of Riccati-type differential equations, and Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF]Proposition 4.3] derived a formula for that of a general CBI process and its integrated process. We point out that our proof of technique of Theorem 3.1 is different from those of Keller-Ressel [26, Theorem 4.10] and Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF]Proposition 4.3], and we make the solutions of Riccati-type differential equations explicit in case of (Y t ) t∈[0,∞) . Section 4 is devoted to study the existence and uniqueness of the MLE b T of b based on observations (Y t ) t∈[0,T ] with T ∈ (0, ∞). We derive an explicit formula for b T as well, see (4.3). Sections 5, 6 and 7 are devoted to study asymptotic behaviour of the MLE of b for subcritical, critical and supercritical jump-type CIR models, respectively. In Section 5, we show that in the subcritical case, the MLE of b is asymptotically normal with the usual square root scaling T 1/2 (especially, it is weakly consistent), but unfortunately, the asymptotic variance depends on the unknown parameters a and m, as well. To get around this problem, we also replace the deterministic scaling T 1/2 by the random scaling

1 σ T 0 Y s ds 1/2 with
the advantage that the MLE of b with this scaling is asymptotically standard normal. Under the extra moment condition (1.4), we prove strong consistency as well. In Section 6, we describe the (nonnormal) asymptotic behaviour of the MLE of b in the critical case both with the deterministic scaling T and with the random scaling 1 σ T 0 Y s ds 1/2 . In Section 7, for the supercritical case, we prove that the MLE of b is strongly consistent, and it is 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 .
We close the paper with Appendices, where we prove a comparison theorem for the SDE (1.1) in the jump process J, 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 B) and some limit theorems for continuous local martingales (Appendix C).

Finally, we summarize the novelties of the paper. We point out that only few results are available for parameter estimation for jump-type CIR processes, see Mai [START_REF] Mai | Drift estimation for jump diffusions: time-continuous and high-frequency observations[END_REF]Section 4.3] (MLE for subcritical case), Huang, Ma and Zhu [START_REF] Huang | Estimation for discretely observed continuous state branching processes with immigration[END_REF] and Li and Ma [START_REF] Li | Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model[END_REF] (conditional LSE). Concerning the asymptotic behavior of the MLE in the subcritical case, we use an explicit formula for the Laplace transform of t 0 Y s ds to derive stochastic convergence of 1 t t 0 Y s ds as t → ∞, and we prove asymptotic normality avoiding ergodicity, see Theorem 5.2. Further, in the supercritical case, we derive a stochastic representation in Theorem 7.1 of the limiting mixed normal distribution given in Theorem 7.3, where the almost sure limit of an appropriately scaled jump-type supercritical CIR process comes into play. This is a new phenomenon, compared to the critical case in Theorem 6.3, where a diffusion-type critical CIR process plays a role. We remark that for all b ∈ R,

1 σ T 0 Y s ds 1/2 ( b T -b) converges in distribution as
T → ∞, and the limit distribution is standard normal for the non-critical cases, while it is non-normal for the critical case (given explicitly in Theorem 6.3). Hence we have a kind of unified theory.

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 equality in distribution and almost surely by

P -→, D -→, a.s.
-→, D = and a.s.

= , respectively.

Let Ω, F, P be a probability space. 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.

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.

2.1 Proposition. Let η 0 be a random variable independent of (W t ) t∈R + and (J t ) t∈R + satisfying P(η 0 ∈ R + ) = 1 and E(η 0 ) < ∞. Then for all a ∈ R + , b ∈ R, σ ∈ R ++ and Lévy measure m on R ++ satisfying (1.2), there is 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. Moreover, (Y t ) t∈R + is a CBI process having branching mechanism

R(u) = σ 2 2 u 2 -bu, u ∈ C with Re(u) 0,
and immigration mechanism

F (u) = au + ∞ 0 (e uz -1) m(dz), u ∈ C with Re(u) 0.
Further, the infinitesimal generator of Y takes the form

(Af )(y) = (a -by)f ′ (y) + 1 2 yσ 2 f ′′ (y) + ∞ 0 (f (y + z) -f (y)) m(dz), (2.1)
where y ∈ R + , f ∈ C 2 c (R + , R)
, and f ′ and f ′′ denote the first and second order partial derivatives of f . If, in addition,

P(η 0 ∈ R ++ ) = 1 or a ∈ R ++ , then P t 0 Y s ds ∈ R ++ = 1 for all t ∈ R ++ .
Proof. The Lévy-Itô's representation of J takes the form

J t = t 0 ∞ 0 z µ J (ds, dz), t ∈ R + , (2.2)
where µ J (ds, dz) := u∈R + ½ {∆Ju =0} ε (u,∆Ju) (ds, dz) is the integer-valued Poisson random measure on R 2 ++ associated with the jumps ∆J u := J u -J u-, u ∈ R ++ , ∆J 0 := 0, of the process J, and ε (u,x) denotes the Dirac measure at the point (u, x) ∈ R 2 + , see, e.g., Sato [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]Theorem 19.2]. Consequently, the SDE (1.1) can be rewritten in the form

Y t = Y 0 + t 0 (a -bY s ) ds + t 0 σ Y s dW s + J t = Y 0 + t 0 (a -bY s ) ds + t 0 σ Y s dW s + t 0 ∞ 0 z µ J (ds, dz), t ∈ R + .
(2.3)

Equation (2.
3) is a special case of the equation (6.6) in Dawson and Li [START_REF] Dawson | Skew convolution semigroups and affine Markov processes[END_REF], and Theorem 6.2 in Dawson and Li [START_REF] Dawson | Skew convolution semigroups and affine Markov processes[END_REF] implies that for any initial value η 0 with P(η 0 ∈ R + ) = 1 and E(η 0 ) < ∞, there exists a pathwise unique non-negative strong solution satisfying P(Y 0 = η 0 ) = 1 and

P(Y t ∈ R + for all t ∈ R + ) = 1. Let (Y ′ t ) t∈R + be a pathwise unique non-negative strong solution of the SDE dY ′ t = (a -bY ′ t ) dt + σ Y ′ t dW t , t ∈ R + , such that P(Y ′ 0 = η 0 ) = 1.
Applying the comparison Theorem A.1, we obtain

P(Y t Y ′ t for all t ∈ R + ) = 1. (2.4) Further, if P(η 0 ∈ R ++ ) = 1 or a ∈ R ++ , then P t 0 Y ′ s ds ∈ R ++ = 1 for all t ∈ R ++ . Indeed, if ω ∈ Ω is such that [0, t] ∋ u → Y ′ u (ω) is continuous and Y ′ v (ω) ∈ R + for all v ∈ R + , then we have t 0 Y ′ s (ω) ds = 0 if and only if Y ′ s (ω) = 0 for all s ∈ [0, t].
Using the method of the proof of Theorem 3.1 in Barczy et. al [START_REF] Barczy | On parameter estimation for critical affine processes[END_REF], we get P t 0 Y ′ s ds = 0 = 0, t ∈ R + , as desired. Since (Y s ) s∈[0,t] has càdlàg, hence bounded sample paths almost surely (see, e.g., Billingsley [8, (12.5)]), using (2.4), we conclude

P t 0 Y s ds ∈ R ++ = 1 for all t ∈ R ++ .
The form of the infinitesimal generator (2.1) readily follows by (6.5) in Dawson and Li [START_REF] Dawson | Skew convolution semigroups and affine Markov processes[END_REF]. Further, Theorem 6.2 in Dawson and Li [START_REF] Dawson | Skew convolution semigroups and affine Markov processes[END_REF]

∈ R + ) = 1 and E(Y 0 ) < ∞. Then E(Y t ) =    e -bt E(Y 0 ) + a + ∞ 0 z m(dz) 1-e -bt b if b = 0, E(Y 0 ) + a + ∞ 0 z m(dz) t if b = 0, t ∈ R + . (2.5) Consequently, if b ∈ R ++ , then (2.6) lim t→∞ E(Y t ) = a + ∞ 0 z m(dz) 1 b , if b = 0, then lim t→∞ t -1 E(Y t ) = a + ∞ 0 z m(dz), if b ∈ R --, then lim t→∞ e bt E(Y t ) = E(Y 0 ) -a + ∞ 0 z m(dz) 1 b .
Proof. By Proposition 2.1, (Y t ) t∈R + is CBI process with an infinitesimal generator given in (2.1). 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, ν = m and µ = 0. Since E(Y 0 ) < ∞ and the moment condition R\{0} |z|½ {|z| 1} ν(dz) = ∞ 1 z m(dz) < ∞ holds (due to (1.2)), we may apply formula (3.1.11) in Li [START_REF] Li | Continuous-state branching processes[END_REF] or Lemma 3.4 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 the choices B = B = -b and

β = β + R\{0} z ν(dz) = β + ∞ 0 z m(dz) ∈ R + , yielding that E(Y t ) = e t B E(Y 0 ) + t 0 e u B du β.
This implies (2.5) 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 jump-type CIR model driven by a subordinator given by the SDE (1.1).

Definition

. Let a ∈ R + , b ∈ R, σ ∈ R ++ , and let m be a Lévy measure on R ++ satisfying (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.
In the subcritical case, the following result states the existence of a unique stationary distribution and the exponential ergodicity for the process (Y t ) t∈R + , see Pinsky [START_REF] Pinsky | Limit theorems for continuous state branching processes with immigration[END_REF], Li [START_REF] Li | Measure-Valued Branching Markov Processes[END_REF]Theorem 3.20 and the paragraph after Corollary 3.21], Keller-Ressel and Steiner [START_REF] Keller-Ressel | Yield curve shapes and the asymptotic short rate distribution in affine one-factor models[END_REF], Keller-Ressel [START_REF] Keller-Ressel | Moment explosions and long-term behavior of affine stochastic volatility models[END_REF], Keller-Ressel and Mijatović [28, Theorem 2.6] and Jin et al. [START_REF] Jin | Exponential ergodicity of the jump-diffusion CIR process[END_REF]Theorem 1]. As a consequence, according to 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], one also obtains a strong law of large numbers for (Y t ) t∈R + .

2.4 Theorem. Let a ∈ R + , b ∈ R ++ , σ ∈ R ++ , and let m be a Lévy measure on R ++ satisfying (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 given by ∞ 0 e uy π(dy) = exp 0 u F (v) R(v) dv = exp 0 u av + ∞ 0 (e vz -1) m(dz) σ 2 2 v 2 -bv dv for u ∈ R -.
Moreover, π has a finite expectation given by

∞ 0 y π(dy) = a + ∞ 0 z m(dz) 1 b ∈ R + . (2.7) (ii) If, in addition, a ∈ R ++
and the extra moment condition (1.4) holds, then the process (Y t ) t∈R + is exponentially ergodic, namely, there exist constants β ∈ (0, 1) and C ∈ R ++ such that

P Yt | Y 0 =y -π TV C(y + 1)β t , 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. Moreover, for all Borel measurable functions f :

R + → R with ∞ 0 |f (y)| π(dy) < ∞, we have (2.8) 1 T T 0 f (Y s ) ds a.s. -→ ∞ 0 f (y) π(dy) as T → ∞. 2.5 Remark. Let a ∈ R + , b ∈ R, σ ∈ R ++ ,
E(e iθJt ) = exp itθ 1 0 z m(dz) + t ∞ 0 e iθz -1 -iθzh(z) m(dz) for θ ∈ R and t ∈ R + , where h(z) := z½ [-1,1] (z), z ∈ R.
Using again (1.1) and the Lévy-Itô's representation (2.2) of J, we can write the process (Y t ) t∈R + in the form (2.9)

Y t = Y 0 + t 0 (a -bY u ) du + t 1 0 z m(dz) + σ t 0 Y u dW u + t 0 R h(z) µ J (du, dz) + t 0 R (z -h(z)) µ J (du, dz), t ∈ R + ,
where µ J (ds, dz) := µ J (ds, dz)ds m(dz). In fact, (2.9) is a so-called Grigelionis form for the semimartingale (Y Next we give a statistic for σ 2 using continuous time observations (Y t ) t∈[0,T ] with some T > 0. Due to this result we do not consider the estimation of the parameter σ, it is supposed to be known.

2.6 Remark. Let a ∈ R + , b ∈ R, σ ∈ R ++ ,
and let m be a Lévy measure on R ++ satisfying (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 (2.9) 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 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 (2.10) 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 (2.10) holds almost surely along a suitable subsequence, the members of the sequence in (2.10) 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 (2.10). 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 (2.10) follows by the fact that convergence in probability is closed under multiplication. ✷

3 Joint Laplace transform of Y t and t 0 Y s ds

We study the joint Laplace transform of Y t and t 0 Y s ds, since it plays a crucial role in deriving the asymptotic behavior of the MLE of b given in (4.3). Our formula for the joint Laplace transform in question given in Theorem 3.1 is in accordance with the corresponding one obtained in Keller-Ressel [START_REF] Keller-Ressel | Affine Processes -Theory and Applications in Finance[END_REF]Theorem 4.10] in case of a regular affine process and with the one in Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF]Proposition 4.3] in case of a general CBI process. Here, our contribution is to give a new proof for this joint Laplace transform and to make the solutions of the Riccati-type differential equations appearing in the formulas of Keller-Ressel [26, Theorem 4.10] and Jiao et al. [START_REF] Jiao | Alpha-CIR model with branching processes in sovereign interest rate modelling[END_REF]Proposition 4.3] explicit in case of (Y t ) t∈R + , which turns out to be crucial for our forthcoming statistical study. For all b ∈ R and v ∈ R -, let us introduce the notation

γ v := √ b 2 -2σ 2 v. 3.1 Theorem. Let a ∈ R + , b ∈ R, σ ∈ R ++ ,
(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 ψ u,v (t)y 0 + t 0 aψ u,v (s) + ∞ 0 e zψu,v(s) -1 m(dz) ds for t ∈ R + , where the function ψ u,v : R + → R -takes the form ψ u,v (t) =      uγv cosh( γv t 2 )+(-ub+2v) sinh( γv t 2 ) γv cosh( γv t 2 )+(-σ 2 u+b) sinh( γv t 2 ) if v ∈ R --or b = 0 (i.e., if γ v ∈ R ++ ), u 1-σ 2 u 2 t if v = 0 and b = 0 (i.e., if γ v = 0), t ∈ R + .
(3.1)

3.2 Remark. (i) If v ∈ R --, then γ v > |b|, hence γ v cosh γ v t 2 + (-σ 2 u + b) sinh γ v t 2 (γ v + b) sinh γ v t 2 ∈ R ++ , t ∈ R + . If v = 0 and b = 0, then γ v = |b| ∈ R ++ , hence γ v cosh γ v t 2 + (-σ 2 u + b) sinh γ v t 2 |b| cosh γ v t 2 + b |b| sinh γ v t 2 ∈ R ++ , t ∈ R + . Consequently, if γ v ∈ R ++ , then γ v cosh γvt 2 + (-σ 2 u + b) sinh γvt 2 ∈ R ++ , t ∈ R + , hence the function ψ u,v in (3.1) is well-defined. (ii) In Theorem 3.1, we have t 0 ψ u,v (s) ds = b σ 2 t -2 σ 2 log cosh γv t 2 + -σ 2 u+b γv sinh γvt 2 , if v ∈ R --or b = 0, -2 σ 2 log 1 -σ 2 u 2 t , if v = 0 and b = 0, (3.2)
for all t ∈ R + , see, e.g., Lamberton

Y t Z t = Y 0 0 + t 0 a 0 + -b 0 1 0 Y s Z s ds + t 0 σ 2 Y s 1 0 1 0 ⊤ dW s d W s + t 0 0 • Z s 0 1 0 1 ⊤ dW s d W s + t 0 R 2 + \{0} r M (ds, dr), t ∈ R + ,
where (W t ) t∈R + and ( W t ) t∈R + are independent standard Wiener processes, and M is a Poisson random measure on R + × (R 2 + \ {0}) with intensity measure ds ν(dr), where the measure ν on R 2 + \ {0} is given by ν(B) :

= ∞ 0 ½ B (z, 0) m(dz), B ∈ B(R 2 + \ {0}), hence R 2 + \{0} (1 ∧ r ) ν(dr) ∞ 0 z m(dz) < ∞. Put c := c 1 c 2 := 1 2 σ 2 0 ∈ R 2 + , β := a 0 ∈ R 2 + , B := -b 0 1 0 , µ := (µ 1 , µ 2 ) := (0, 0).
Then B is an essentially non-negative matrix (i.e., its off-diagonal entries are non-negative), and, due to (1.2),

R 2 + \{0} r ½ { r 1} ν(dr) R 2 + \{0} r ν(dr) = ∞ 0 z m(dz) < ∞ yielding that condition (2.
) t∈R + is R(u, v) = (R 1 (u, v), R 2 (u, v)), u, v ∈ R -, with R 1 (u, v) = c 1 u 2 + B 1 0 , u v = σ 2 2 u 2 -bu + v, R 2 (u, v) = c 2 v 2 + B 0 1 , u v = 0,
and the immigration mechanism of (Y t , Z t ) t∈R + is

F (u, v) = β, u v - R 2 + \{0} exp u v , r -1 ν(dr) = au + ∞ 0 (e uz -1) m(dz),
see, e.g., Theorem 2.4 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].

Note that R(u, v) = (R(u) + v, 0), u, v ∈ R -, and
F (u, v) = F (u), u, v ∈ R -, where R(u), u ∈ R -, and
F (u), u ∈ R -,
(3.3) E exp uY t + v t 0 Y s ds = exp ψ u,v (t)y 0 + t 0 F (ψ u,v (s), ϕ u,v (s)) ds for 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 -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 -bψ u,v (t) + v, t ∈ R + , ψ u,v (0) = u.
The solution of this differential equation is (3.1). Indeed, in case of γ v > 0, we can refer to, e.g., Lamberton and Lapeyre [32, Chapter 6, Proposition 2.5], and in case of γ v = 0, this is a simple separable ODE taking the form ψ ′ u,0 (t) = σ 2 2 ψ u,0 (t) 2 , t ∈ R + , with initial condition ψ u,0 (0) = u. Hence, by (3.3), we obtain the statement. ✷ 3.3 Example. Now we formulate a special case of Theorem 3.1 in the critical case (b = 0) supposing that the Lévy measure m takes the form given in (1.5), i.e., in the case of a critical BAJD process. For all u, v ∈ R -, let us introduce the notations

α (1) u,v := uγ v + 2v, α (2) u,v := uγ v -2v, β (1) u,v := λ(-σ 2 u + γ v ) -α (1) u,v , β (2) u,v := λ(σ 2 u + γ v ) -α (2) u,v ,
where

γ v = √ -2σ 2 v (since now b = 0)
. 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 b = 0 and m being a Lévy measure on R ++ satisfying (1.5). Then we check that for all u, v ∈ R -,

E exp uY t + v t 0 Y s ds = exp ψ u,v (t)y 0 + φ u,v (t) , t ∈ R + , (3.4)
where the function ψ u,v : R + → R -is given by (3.1) with

γ v = √ -2σ 2 v (since now b = 0), and if v ∈ R --(i.e., if γ v ∈ R ++ ) and v / ∈ {-σ 2 u 2 2 , -σ 2 λ 2 2 } (i.e., if β (2)
u,v = 0), then

φ u,v (t) = - 2a σ 2 log cosh γ v t 2 - σ 2 u γ v sinh γ v t 2 + c α (2) u,v β (2) u,v t + 1 γ v α (1) u,v β (1) u,v - α (2) u,v β (2) u,v log β (1) u,v e γvt + β (2) u,v β (1)
u,v + β (2) u,v , t ∈ R + , (3.5
)

and if v ∈ {-σ 2 u 2 2 , -σ 2 λ 2 2 } and v ∈ R --(i.e., if v ∈ R --and β (2) u,v = 0), then φ u,v (t) = - 2a σ 2 log cosh γ v t 2 - σ 2 u γ v sinh γ v t 2 + c β (1) u,v α (1) u,v t + α (2) u,v γ v (1 -e -γv t ) , t ∈ R + , (3.6)
and if v = 0 (i.e., if

γ v = 0), then φ u,v (t) = - 2a σ 2 log 1 - σ 2 u 2 t - 2c σ 2 λ log 1 - σ 2 λu 2(λ -u) t , t ∈ R + . (3.7)
Especially, for all u ∈ R -,

E(e uYt ) = exp uy 0 1 -σ 2 u 2 t 1 - σ 2 u 2 t -2a σ 2 1 - σ 2 λu 2(λ -u) t -2c σ 2 λ , t ∈ R + .
First, we check that the function

φ u,v in (3.5) is well-defined. If γ v ∈ R ++ , then β (1)
u,v ∈ R ++ (due to α (1) u,v ∈ R -). If γ v = 0, then α (1) u,v = α (2) u,v = 0, β (1)
u,v = -σ 2 λu and β

(2) u,v = σ 2 λu. Further, β (1) u,v e γvt + β (2) u,v β (1) u,v + β (2) u,v ∈ R + , and β (1) u,v + β (2) u,v = 0 holds if and only if γ v = 0 (i.e., v = 0). Indeed, since β (1)
u,v ∈ R + , we have β (1) u,v e γv t + β (2) u,v

β (1) u,v + β (2) u,v = 2γ v (λ -u) ∈ R + ,
and β

(1)

u,v + β (2)
u,v = 0 holds if and only if

γ v = 0 (since λ -u ∈ R ++ due to λ ∈ R ++ and u ∈ R -). Consequently, if γ v ∈ R ++ , then β (1) u,v ∈ R ++ and β (1) u,v + β (2) u,v ∈ R ++ , yielding that the function φ u,v in (3.5) is well-defined.
Next, we check that, provided that v ∈ R --(i.e., γ v ∈ R ++ ), we have β

(2) u,v = 0 if and only if v ∈ {-σ 2 u 2 2 , -σ 2 λ 2 2 }. Indeed, β (2)
u,v = 0 holds if and only if λ(σ

2 u + γ v ) = γ v u -2v, i.e.
, by some algebraic transformations, if and only if

( √ -2v) 2 + σ(u -λ) √ -2v -λσ 2 u = 0.
Solving this quadratic equation with respect to √ -2v, we get

√ -2v ∈ {-σu, σλ}, yielding that v ∈ {-σ 2 u 2
2 , -σ 2 λ 2 2 }, as desired. Finally, we check (3.4). Using Theorem 3.1 and part (ii) of Remark 3.2, it remains to calculate

t 0 ∞ 0 (e zψu,v(s) -1) m(dz) ds for u, v ∈ R -. For all u, v ∈ R -, we have ∞ 0 (e zψu,v(s) -1) m(dz) = ∞ 0 (e zψu,v(s) -1)cλe -λz dz = cλ ∞ 0 e (ψu,v(s)-λ)z dz -c = cλ ψ u,v (s) -λ lim z→∞ e (ψu,v(s)-λ)z -1 -c = - cψ u,v (s) ψ u,v (s) -λ , s ∈ R + , (3.8)
where we used that ψ u,v (s)λ ∈ R --, s ∈ R + , following from λ ∈ R ++ and ψ u,v (s) ∈ R -, s ∈ R + . By some algebraic transformations,

cψ u,v (t) λ -ψ u,v (t)
= c uγ v (1 + e γv t ) + 2v(e γv t -1) -λσ 2 u(e γv t -1) + λγ v + λγ v e γv tuγ v (1 + e γv t ) -2v(e γv t -1) = c α

u,v e γv t + α

(2) u,v β (1)
u,v e γv t + β

(2) u,v , t ∈ R + . As we have seen, if v ∈ R --(i.e., if γ v ∈ R ++ ), then β (1)
u,v ∈ R ++ , and β

(2)

u,v = 0 if and only if v ∈ {-σ 2 u 2 2 , -σ 2 λ 2 2 }. Hence cψ u,v (t) λ -ψ u,v (t) =            c α (2) u,v β (2) u,v + α (1) u,v - β (1) u,v α (2) u,v β (2) u,v e γv t β (1) u,v e γv t +β (2) u,v if v ∈ R --, v / ∈ {-σ 2 u 2 2 , -σ 2 λ 2 2 }, c β (1) u,v (α (1) u,v + α (2) u,v e -γvt ) if v ∈ {-σ 2 u 2 2 , -σ 2 λ 2 2 }, v ∈ R --, 2cu 2(λ-u)-λuσ 2 t if v = 0, for t ∈ R + .
By integration, we get (3.5), (3.6)

= y 0 ) = 1 with some y 0 ∈ R + . Then for all u ∈ R -, E e uYt = exp ψ u,0 (t)y 0 + t 0 aψ u,0 (s) + ∞ 0 e zψ u,0 (s) -1 m(dz) ds , t ∈ R + ,
where the function ψ u,0 : R + → R -takes the form

ψ u,0 (t) =    2ube -bt σ 2 u(e -bt -1)+2b if b = 0, u 1-σ 2 u 2 t if b = 0, t ∈ R + . (3.9)
Proof. We have to substitute v = 0 in (3.1). The case of b = 0 is trivial. In case of b = 0 we have where the function ψ u,0 : R + → R -is given by (3.9), and if u = 2b σ 2 , then

γ v = γ 0 = |b|, hence ψ u,0 (t) = u|b| cosh |b|t 2 -ub sinh |b|t 2 |b| cosh |b|t 2 + (-σ 2 u + b) sinh |b|t 2 , t ∈ R + , yielding (3
φ u,0 (t) = - 2a σ 2 log 1 + σ 2 u 2b (e -bt -1) + 2c -σ 2 λ + 2b log (-σ 2 λ + 2b)ue -bt + (σ 2 u -2b)λ 2b(u -λ) (3.11) for t ∈ R + , and if u = 2b σ 2 , then φ u,0 (t) = 2b(2ab -cσ 2 -aσ 2 λ) σ 2 (-σ 2 λ + 2b) t, t ∈ R + . (3.12)
To check (3.10), using Corollary 3.4, it remains to calculate t 0 aψ u,0 (s) + ∞ 0 e zψ u,0 (s) -1 m(dz) ds. Here, by (3.9),

t 0 ψ u,0 (s) ds = - 2 σ 2 log 1 + σ 2 u 2b (e -bt -1) , t ∈ R + .
Similarly to (3.8), for all u ∈ R -,

∞ 0 (e zψ u,0 (s) -1) m(dz) = - cψ u,0 (s) ψ u,0 (s) -λ , s ∈ R + , (3.13)
By some algebraic transformations,

- cψ u,0 (s) ψ u,0 (s) -λ = -2bcue -bt (-σ 2 λ + 2b)ue -bt + (σ 2 u -2b)λ , t ∈ R + ,
where (-σ 2 λ+2b)ue -bt +σ 2 λu-2bλ (-σ 2 λ+2b)u+σ 2 λu-2bλ = 2b(u-λ) > 0, since -σ 2 λ+2b < 0 due to b < 0 and λ > 0. If u = 2b σ 2 (i.e., if σ 2 u -2b = 0), then

- t 0 cψ u,0 (s) ψ u,0 (s) -λ ds = -2bc -σ 2 λ + 2b t, t ∈ R + ,
and hence

φ u,0 (t) = - 2 σ 2 log 1 + σ 2 u 2b (e -bt -1) - 2bc -σ 2 λ + 2b t = 2ab σ 2 t - 2bc -σ 2 λ + 2b t = 2b(2ab -cσ 2 -aσ 2 λ) σ 2 (-σ 2 λ + 2b) t, t ∈ R + , yielding (3.12). If u = 2b σ 2 (i.e., if σ 2 u -2b = 0), then - t 0 cψ u,0 (s) ψ u,0 (s) -λ ds = 2c -σ 2 λ + 2b log (-σ 2 λ + 2b)ue -bt + (σ 2 u -2b)λ 2b(u -λ) , t ∈ R + , yielding (3.

11). ✷

Substituting u = 0 in (3.1), we obtain the following corollary. where the function ψ 0,v : R + → R -takes the form

ψ 0,v (t) =    2v sinh( γv t 2 ) γv cosh( γv t 2 )+b sinh( γv t 2 ) if v ∈ R --or b = 0, 0 if v = 0 and b = 0, t ∈ R + . (3.14)

Existence and uniqueness of MLE

In this section, we will consider the jump-type CIR model (1.1) with known a ∈ R + , σ ∈ R ++ , Lévy measure m satisfying (1.2), and a known deterministic initial value Y 0 = y 0 ∈ R + , and we will consider b ∈ R as an unknown parameter. Then for all T ∈ R ++ , the probability measures P b,T and P b,T are absolutely continuous with respect to each other, and, under P,

log d P b,T d P b,T ( Y ) = - b -b σ 2 ( Y T -y 0 -aT -J T ) - b 2 -b 2 2σ 2 T 0 Y s ds, (4.1)
where Y is the 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 B). 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 + . The jump-type CIR process (1.1) can be written in the form (2.9). 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, where

B (b) t = t 0 a -bη u + 1 0 z 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 = dt m(dy)
with the Borel transition kernel

K from R + × R into R given by K(y, R) := R ½ R\{0} (z) m(dz) = m(R)
for y ∈ R + and R ∈ B(R).

The aim of the following discussion is to check the set of sufficient conditions presented in Appendix B (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 (B.1) and (B.2) readily hold. We also have

P b ν({t} × R) = 0 = P b (0 • m(R) = 0) = 1, t ∈ R + , b ∈ R, since m(R) = m(R + ) ∈ R + due to (1.
2). 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 (B.3).

Next we check (B.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 [8, (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. 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 B), P b,T and P b,T are equivalent (one can change the roles of b and b), and under the probability measure P b , we have

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 ++ ,
= d Y t -(a -b Y t ) dt -dJ t , t ∈ R + .
Hence, under P, Proof. Due to 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 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 the Lévy-Itô's representation (2.2) of J t , t ∈ R + , we obtain (4.4)

log d P b,T d P b,T ( Y ) = T 0 - b -b σ 2 (d Y u -dJ 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 -dJ u ) + b -b σ 2 T 0 a du - b 2 -b 2 2σ 2 T 0 Y u du,
J t = s∈[0,t] ∆J s , t ∈ R + , since 1 -1 |z| m(dz) = 1 0 z m(dz) < ∞,
J t = s∈[0,t] ∆Y s , t ∈ R + .
Consequently, for all t ∈ [0, T ], J t is a measurable function of (Y u ) u∈[0,T ] , yielding that the right hand side of (4.3) is a measurable function (i.e., a statistic) of (Y u ) u∈[0,T ] , as desired. By Proposition 4.1, for all b, b ∈ R, we have

∂ ∂b Λ T (b, b) = - 1 σ 2 (Y T -y 0 -aT -J T ) - b σ 2 T 0 Y s ds, ∂ 2 ∂b 2 Λ T (b, b) = - 1 σ 2 T 0 Y s ds.
Thus the MLE b T of b based on a sample (Y s ) s∈[0,T ] exists almost surely, and it takes the form (4.3) provided that

T 0 Y s ds ∈ R ++ . ✷
In fact, it turned out that for the calculation of the MLE of b, one does not need to know the value of the parameter σ ∈ R ++ or the measure m.

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 ψ as a solution of the equation ΛT (ψ) = 0, where ΛT (ψ) 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]

T 0 - 1 σ 2 dY cont u = - 1 σ 2 T 0 (dY u -(a -bY u ) du -dJ u ) = - 1 σ 2 Y T -y 0 -aT + b T 0 Y u du -J T for b ∈ R and T ∈ (0, ∞).
1 t t 0 Y s ds P -→ 1 b a + ∞ 0 z m(dz) ∈ R + as t → ∞. (5.1)
Proof. Using Corollary 3.6, we have

E exp v t t 0 Y s ds = exp ψ 0, v t (t)y 0 + t 0 aψ 0, v t (s) + ∞ 0 e zψ 0, v t (s) -1 m(dz) ds for t ∈ R ++ and v ∈ R -,
where the function ψ 0,v : R + → R -is given in (3.14). We check that

E exp v t t 0 Y s ds → exp v b a + ∞ 0 z m(dz) as t → ∞
for all v ∈ R -. The case of v = 0 is trivial, so we may and do suppose that v ∈ R --. Then we have

ψ 0, v t (t) = 1 t 2v(1 -e -tγ v t ) γ v t (1 + e -tγ v t ) + b(1 -e -tγ v t ) → 0 as t → ∞, since γ v t = b 2 -2σ 2 v t → b ∈ R ++ as t → ∞.
Moreover, by the dominated convergence theorem,

t 0 ψ 0, v t (s) ds = 1 t t 0 2v(1 -e -sγ v t ) γ v t (1 + e -sγ v t ) + b(1 -e -sγ v t ) ds = 1 0 2v(1 -e -xtγ v t ) γ v t (1 + e -xtγ v t ) + b(1 -e -xtγ v t ) dx → v b as t → ∞.
Indeed, the integrand in the last integral is dominated by 2|v| b . Finally, by the dominated convergence theorem, we obtain

t 0 ∞ 0 e zψ 0, v t (s) -1 m(dz) ds = t 0 ∞ 0 exp z t • 2v(1 -e -sγ v t ) γ v t (1 + e -sγ v t ) + b(1 -e -sγ v t ) -1 m(dz) ds = 1 0 ∞ 0 t exp z t • 2v(1 -e -xtγ v t ) γ v t (1 + e -xtγ v t ) + b(1 -e -xtγ v t ) -1 m(dz) dx → 1 0 ∞ 0 z v b m(dz) dx = v b ∞ 0 z m(dz)
as t → ∞ for all z ∈ R + . Indeed,

A t (z, x) := z • 2v(1 -e -xtγ v t ) γ v t (1 + e -xtγ v t ) + b(1 -e -xtγ v t ) → z v b as t → ∞, where A t (z, x) ∈ R -for all t ∈ R ++ , z ∈ R + and x ∈ [0, 1]. Hence t(e A t (z,x) t -1) → z v b as t → ∞ for all z ∈ R + and x ∈ [0, 1], and |t(e A t (z,x) t -1)| |A t (z, x)| 2z|v| b for t ∈ R ++ , z ∈ R + and x ∈ [0, 1]
, where 2z|v| b is integrable with respect to the measure m(dz) dx on R ++ × [0, 1]. By the continuity theorem, we obtain

1 t t 0 Y s ds D -→ 1 b a + ∞ 0 z m(dz) ∈ R + as t → ∞,
which implies (5.1). ✷

If a ∈ R ++ or y 0 ∈ R ++ , then, using (4.3) and the SDE (1.1), we get

b T -b = - Y T -y 0 -aT -J T + b T 0 Y s ds T 0 Y s ds = - σ T 0 √ Y s dW s T 0 Y s ds (5.2)
provided that T 0 Y s ds ∈ R ++ , which holds almost surely due to Proposition 2.1. Here note that σ

T 0 √ Y s dW s = Y cont T , T ∈ R + , see Remark 2.6.
Despite the mistake in the formula (4.23) in Mai [START_REF] Mai | Drift estimation for jump diffusions: time-continuous and high-frequency observations[END_REF] for the MLE b T of b, Mai [40, Theorem 4.3.1] formulated the right asymptotic behavior of b T as T → ∞, namely, asymptotic normality assuming ergodicity of the process Y . We give a correct proof of this result, and in fact, we extend it as well, since we do not assume the ergodicity of Y .

D -→ N 0, σ 2 b a + ∞ 0 z m(dz) = N 0, σ 2 ∞ 0 y π(dy) as T → ∞, (5.3)
where π denotes the unique stationary distribution of (Y t ) t∈R + (see part (i) of Theorem 2.4).

Especially, b T is weakly consistent, i.e., b T P -→ b as T → ∞. With a random scaling,

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

Under the additional moment condition 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 we obtain (5.3). 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 → ∞.

Under the additional moment condition (1.4), by (ii) of Theorem 2.4, we have

1 T T 0 Y s ds a.s.
-→ ∞ 0 y π(dy) as T → ∞, implying also

T 0 Y s ds a.s.
-→ ∞ as T → ∞. Using (5.2) and Theorem C.1, we have b T a.s.

-→ b as T → ∞. ✷

We note that the moment condition (1.4) does not imply the moment condition (1.2) in general, and, since the moment condition (1.2) is already needed for us for the existence of a pathwise unique strong solution of the SDE (1.1) (see Proposition 2.1), in order to get our strong consistency result in Theorem 5.2 we need to assume both (1.2) and (1.4).

6 Asymptotic behaviour of the MLE in the critical case 6.1 Theorem. Let a ∈ R + , b = 0, σ ∈ R ++ , and let m be a Lévy measure on R ++ satisfying (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

Y t t , 1 t 2 t 0 Y s ds D -→ Y 1 , 1 0 Y s ds as t → ∞, (6.1)
where (Y t ) t∈R + is the unique strong solution of a critical (diffusion type) CIR model

dY t = a + ∞ 0 z m(dz) dt + σ Y t dW t , t ∈ R + , (6.2)
with initial condition Y 0 = 0, where (W t ) t∈R + is a 1-dimensional standard Wiener process. Moreover, the Laplace transform of (Y 1 ,

1 0 Y s ds) takes the form E e uY 1 +v 1 0 Ys ds =        cosh γv 2 -σ 2 u γv sinh γv 2 -2 σ 2 (a+ ∞ 0 z m(dz)) if v ∈ R --, 1 -σ 2 u 2 -2 σ 2 (a+ ∞ 0 z m(dz)) if v = 0, (6.3)
for all u, v ∈ R -, where

γ v = √ -2σ 2 v, v ∈ R -(since now b = 0).
6.2 Remark. Under the conditions of Theorem 6.1, in the special case of m = 0, i.e., in the diffusion case, we get back part 1 of Theorem 1 in Ben Alaya and Kebaier [START_REF] Ben Alaya | Asymptotic behavior of the maximum likelihood estimator for ergodic and nonergodic square-root diffusions[END_REF]. ✷

Proof of Theorem 6.1. Using Theorem 3.1, we have

E exp u Y t t + v 1 t 2 t 0 Y s ds = exp ψ u t , v t 2 (t)y 0 + t 0 aψ u t , v t 2 (s) + ∞ 0 e zψ u t , v t 2
(s) -1 m(dz) ds for t ∈ R + and u, v ∈ R -, where the function ψ u,v : R + → R -is given in (3.1). According to Theorem 3.1, we have to distinguish two cases: v < 0 (Case I) and v = 0 (Case II).

Case I: If v < 0, then

ψ u t , v t 2 (t) = 1 t uγ v cosh γv 2 + 2v sinh γv 2 γ v cosh γv 2 -σ 2 u sinh γv 2 → 0 as t → ∞,
and, for each

t ∈ R + , t 0 ψ u t , v t 2 (s) ds = 1 t t 0 uγ v cosh γvs 2t + 2v sinh γv s 2t γ v cosh γvs 2t -σ 2 u sinh γv s 2t ds = 1 0 uγ v cosh γvx 2 + 2v sinh γvx 2 γ v cosh γvx 2 -σ 2 u sinh γvx 2 dx = 1 0 ψ u,v (x) dx = - 2 σ 2 log cosh γ v 2 - σ 2 u γ v sinh γ v 2 =: I, (6.4)
where we used (3.2). By the monotone convergence theorem we obtain

t 0 ∞ 0 e zψ u t , v t 2 (s) -1 m(dz) ds = t 0 ∞ 0 exp z t • uγ v cosh γv s 2t + 2v sinh γvs 2t γ v cosh γv s 2t -σ 2 u sinh γvs 2t -1 m(dz) ds = 1 0 ∞ 0 t exp z t • uγ v cosh γvx 2 + 2v sinh γvx 2 γ v cosh γvx 2 -σ 2 u sinh γvx 2 -1 m(dz) dx = 1 0 ∞ 0 t(e z t ψu,v(x) -1) m(dz) dx → I ∞ 0 z m(dz)
as t → ∞ for all z ∈ R + , where we used that ψ u,v (t) ∈ R -for all t ∈ R + and u, v ∈ R -, and the function R ++ ∋ t → t(e A t -1) is monotone decreasing for all A ∈ R -. The equality (6.3) is a consequence of Theorem 3.1 and (6.4), and hence continuity theorem yields (6.1).

Case II: If v = 0, then

ψ u t , v t 2 (t) = 1 t u 1 -σ 2 u 2 → 0 as t → ∞,
and, for each

t ∈ R + , t 0 ψ u t , v t 2 (s) ds = 1 t t 0 u 1 -σ 2 us 2t ds = 1 0 u 1 -σ 2 ux 2 dx = - 2 σ 2 log 1 - σ 2 u 2 . (6.5)
Moreover, by the monotone convergence theorem,

t 0 ∞ 0 e zψ u t , v t 2 (s) -1 m(dz) ds = t 0 ∞ 0 exp z t • u 1 -σ 2 us 2t -1 m(dz) ds = 1 0 ∞ 0 t exp z t • u 1 -σ 2 ux 2 -1 m(dz) dx → - 2 σ 2 log 1 - σ 2 u 2 ∞ 0 z m(dz) as t → ∞.
The equality (6.3) is a consequence of Theorem 3.1 and (6.5), and hence continuity theorem yields (6.1). ✷

6.3 Theorem. Let a ∈ R + , b = 0, σ ∈ R ++ ,
(Y 0 = y 0 ) = 1 with some y 0 ∈ R + . Suppose that a ∈ R ++ or a = 0, y 0 ∈ R ++ , ∞ 0 z m(dz) ∈ R ++ . Then T ( b T -b) = T b T D -→ a + ∞ 0 z m(dz) -Y 1 1 0 Y s ds as T → ∞, (6.6)
where (Y t ) t∈R + is the unique strong solution of the SDE (6.2). As a consequence, the MLE b T of b is weakly consistent, i.e., b T converges to b in probability as T → ∞.

With a random scaling, we have

1 σ T 0 Y s ds 1/2 ( b T -b) = 1 σ T 0 Y s ds 1/2 b T D -→ a + ∞ 0 z m(dz) -Y 1 σ 1 0 Y s ds 1/2 as T → ∞.
Proof. 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 (4.3), we have

T b T = - Y T T -y 0 T -a -J T T 1 T 2 T 0 Y s ds , T ∈ R + .
Here, by strong law of large numbers for the Lévy process (J t ) t∈R + (see, e.g., Kyprianou [31, Exercise 7.2]),

P lim T →∞ J T T = E(J 1 ) = ∞ 0 z m(dz) = 1. (6.7)
Hence, using Slutsky's lemma, Theorem 6.1 and the continuous mapping theorem together with the fact that P( Applying Slutsky's lemma, Theorem 6.1 and the continuous mapping theorem, we obtain

1 0 Y s ds ∈ R ++ ) = 1 (following from Proposition 2.1, since under the conditions of Theorem 6.3, a + ∞ 0 z m(dz) ∈ R ++ ), we have T b T D -→ - Y 1 -a - ∞ 0 z m(dz)
1 σ T 0 Y s ds 1/2 b T = - 1 σ 1 T 2 T 0 Y s ds -1/2 Y T T - y 0 T -a - J T T D -→ - 1 σ 1 0 Y s ds -1/2 Y 1 -a - ∞ 0 z m(dz) = a + ∞ 0 z m(dz) -Y 1 σ 1 0 Y s ds 1/2
as T → ∞, as desired. ✷

7 Asymptotic behaviour of the MLE in the supercritical case -→ -V b as t → ∞.

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

E(e uV ) = exp uy 0 1 + σ 2 u 2b 1 + σ 2 u 2b -2a σ 2 exp ∞ 0 ∞ 0 exp zue by 1 + σ 2 u 2b e by -1 m(dz) dy (7.1)
for all u ∈ R -, and consequently,

V D = V + V,
where V and V are independent random variables such that e bt Y t a.s.

-→ V and e bt Y t a.s.

-→ V as t → ∞, where ( Y t ) t∈R + and ( Y t ) t∈R + are the pathwise unique strong solutions of the supercritical CIR models

d Y t = (a -b Y t ) dt + σ Y t d W t , t ∈ R + , with Y 0 = y 0 , (7.2) and d Y t = -b Y t dt + σ Y t d W t + dJ t , t ∈ R + , with Y 0 = 0, (7.3)
respectively, where ( W t ) t∈R + and ( W t ) t∈R + are independent 1-dimensional standard Wiener processes (yielding that ( Y t ) t∈R + and ( Y t ) t∈R + are independent). Further,

V D = Z -1 b ,
where (Z t ) t∈R + is the pathwise unique strong solutions of the critical CIR model

dZ t = a dt + σ Z t dW t , t ∈ R + , with Z 0 = y 0 , (7.4)
where (W t ) t∈R + is a 1-dimensional standard Wiener process.

Moreover, P(V ∈ R ++ ) = 1 if and only if a ∈ R ++ or m = 0.
If, in addition, a ∈ R ++ , then V is absolutely continuous.

V D = X -1 b + X -1 b 1-2b σ 2 λ , (7.5)
where ( X t ) t∈R + and ( X t ) t∈R + are the pathwise unique strong solutions of the critical CIR models

d X t = a dt + σ X t d W t , t ∈ R + , with X 0 = y 0 , and
d X t = c λ -2b σ 2 dt + σ X t d W t , t ∈ R + , with X 0 = 0,
∈ R ++ , then X -1 b 1-2b σ 2 λ is absolutely continuous. The independence of X -1 b and X -1 b 1-2b σ 2 λ
implies that V is absolutely continuous in both cases.

(ii) Under the conditions of Theorem 7.1, in the special case m = 0, i.e., in the diffusion case, we get back part 2 of Theorem 3 in Ben Alaya and Kebaier [START_REF] Ben Alaya | Asymptotic behavior of the maximum likelihood estimator for ergodic and nonergodic square-root diffusions[END_REF]. Indeed, if m = 0, then the pathwise unique strong solution of the SDE (7.3) is the identically zero process. ✷ Proof of Theorem 7.1. First, we prove the existence of an appropriate non-negative random variable V . We check that -→ -V b as t → ∞. -→ b as T → ∞. Further,

E(Y t | F Y s ) = E(Y t | Y s ) = e -b(
E(Y t | Y 0 = y 0 ) = e -bt y 0 + a + ∞ 0 z m(dz) t 0 e -bu du, t ∈ R + , y 0 ∈ 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 Applying again (7.8) and the continuous mapping theorem, we obtain

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

Appendices

A A comparison theorem in the jump process

Next we prove a comparison theorem for the SDE (1.1) in the jump process J.

A.1 Proposition. Let a ∈ R + , b ∈ R and σ ∈ R ++ . Suppose that the Lévy measure m on R ++ satisfies (1.2). Let η 0 and η ′ 0 be random variables independent of (W t ) t∈R + and (J t ) t∈R + satisfying P(η 0 ∈ R + ) = 1 and P(η ′ 0 ∈ R + ) = 1. Let (Y t ) t∈R + be a pathwise unique strong solution of the SDE (1.1) such that P(Y 0 = η 0 ) = 1. Let (Y ′ t ) t∈R + be a pathwise unique strong solution of the SDE

(A.1) dY ′ t = (a -bY ′ t ) dt + σ Y ′ t dW t , t ∈ R + , such that P(Y ′ 0 = η ′ 0 ) = 1. Then P(η 0 η ′ 0 ) = 1 implies P(Y t Y ′ t for all t ∈ R + ) = 1.
Proof. We follow the ideas of the proof of Theorem 3.1 of Ma [START_REF] Ma | Stochastic equations for two-type continuous-state branching processes with immigration[END_REF], which is an adaptation of that of Theorem 5.5 of Fu and Li [START_REF] Fu | Stochastic equations of non-negative processes with jumps[END_REF]. There is a sequence φ k : R → R + , k ∈ N, of twice continuously differentiable functions such that

(i) φ k (z) ↑ z + := max(z, 0) as k → ∞; (ii) φ ′ k (z) ∈ [0, 1] for all z ∈ R + and k ∈ N; (iii) φ ′ k (z) = φ k (z) = 0 whenever z ∈ R -and k ∈ N; (iv) φ ′′ k (x -y)( √ x - √ y) 2 2/k for all x, y ∈ R + and k ∈ N.
For a construction of such functions, see, e.g., the proof of Theorem 3.1 of Ma [START_REF] Ma | Stochastic equations for two-type continuous-state branching processes with immigration[END_REF]. Let Z t := Y ′ t -Y t for all t ∈ R + . By (1.1), (2.2) and (A.1), we have

Z t = Z 0 -b t 0 Z s ds + σ t 0 Y ′ s -Y s dW s - t 0 ∞ 0 z µ J (ds, dz), t ∈ R + .
For each j ∈ N, put τ j := inf t ∈ R + : max{Y t , Y ′ t } j . By Itô's formula (which can be used since Y and Y ′ are adapted to the augmented filtration corresponding to (W t ) t∈R + and (J t ) t∈R + constructed as in Section 3 of Barczy et al. [START_REF] Barczy | Yamada-Watanabe results for stochastic differential equations with jumps[END_REF]), we obtain

φ k (Z t∧τ j ) = φ k (Z 0 ) + 4 ℓ=1 I j,k,ℓ (t), t ∈ R + , with I j,k,1 (t) := -b t∧τ j 0 φ ′ k (Z s )Z s ds, I j,k,2 (t) := σ t∧τ j 0 φ ′ k (Z s ) Y ′ s -Y s dW s , I j,k,3 (t) := σ 2 2 t∧τ j 0 φ ′′ k (Z s ) Y ′ s -Y s 2 ds, I j,k,4 (t) := t∧τ j 0 ∞ 0 [φ k (Z s--z) -φ k (Z s-)] µ J (ds, dz). 29
Using formula (3.8) in Chapter II in Ikeda and Watanabe [START_REF] Ikeda | Stochastic Differential Equations and Diffusion Processes[END_REF], the last integral can be written as I j,k,4 (t) = I j,k,4,1 (t) + I j,k,4,2 (t), where

I j,k,4,1 (t) := t∧τ j 0 ∞ 0 [φ k (Z s--z) -φ k (Z s-)] µ J (ds, dz), I j,k,4,2 (t) := t∧τ j 0 ∞ 0 [φ k (Z s--z) -φ k (Z s-)] ds m(dz),
where µ J (ds, dz) := µ J (ds, dz)ds m(dz), since

(A.2) E t∧τ j 0 ∞ 0 |φ k (Z s--z) -φ k (Z s-)| ds m(dz) E t∧τ j 0 ∞ 0 z ds m(dz) t ∞ 0 z m(dz) < ∞,
where we used that, by properties (ii) and (iii) of the function φ k , we have φ ′ k (u) ∈ [0, 1] for all u ∈ R, and hence, by mean value theorem,

(A.3) 0 φ k (y) -φ k (y -z) z, y ∈ R, z ∈ R + , k ∈ N.
One can check that the process (I j,k,2 (t) + I j,k,4,1 (t)) t∈R + is a martingale. Indeed, by properties (ii) and (iii) of the function φ k and the definition of τ j , By (iv),

E t∧τ j 0 φ ′ k (Z s ) Y ′ s -Y s 2 ds 2 E t∧τ j 0 (Y ′ s + Y s ) ds 4jt < ∞,
× R ++ × Ω ∋ (s, z, ω) → φ k (Z s-(ω) -z) -φ k (Z s-(ω))1 (-∞,τ j ) (
I j,k,3 (t) (t ∧ τ j ) σ 2 k σ 2 t k .
By (A.3), we obtain I j,k,4,2 (t) 0.

Summarizing, we have

φ k (Z t∧τ j ) φ k (Z 0 ) + |b| t∧τ j 0 Z + s ds + σ 2 t k + I j,k,2 (t) + I j,k,4,1 (t), t ∈ R + . (A.4)
By (iii), we obtain P(φ k (Z 0 ) = 0) = 1. By (i), the non-negativeness of φ k and monotone convergence theorem yield E(φ k (Z t∧τ j )) → E(Z + t∧τ j ) as k → ∞ for all t ∈ R + and j ∈ N. We have t∧τ j 0 Z + s ds t 0 Z + s∧τ j ds, hence taking the expectation of (A.4) and letting k → ∞, we obtain

E Z + t∧τ j |b| t 0 E Z + s∧τ j ds.
By Gronwall's inequality, we conclude E Z + t∧τ j = 0 for all t ∈ R + and j ∈ N. Hence P(Y ′ t∧τ j Y t∧τ j ) = 1 for all t ∈ R + and j ∈ N, and then P(Y ′ t∧τ j Y t∧τ j for all j ∈ N) = 1 for all t ∈ R + . Since Y and Y ′ have càdlàg trajectories, these trajectories are bounded almost surely on [0, T ] for all T ∈ R + , hence τ j a.s.

-→ ∞ as j → ∞. This yields P(Y ′ t Y t ) = 1 for all t ∈ R + . Since the set of non-negative rational numbers Q + is countable, we obtain P(Y ′ t Y t for all t ∈ Q + ) = 1. Using again that Y and Y ′ have càdlàg trajectories almost surely, we get P(Y ′ t Y t for all t ∈ R + ) = 1. ✷

B 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[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). This appendix (together with proofs) already appears in Barczy et al. [START_REF] Barczy | Asymptotic behavior of maximum likelihood estimators for a jump-type Heston model[END_REF], we decided to present it here as well for better readability and being self-contained.

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 = 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 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 (B.5) using Jacod and Shiryaev [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]

η t = x 0 + B (ψ) t + (η cont ) (ψ) t + t 0 R d h(x) (µ η -ν (ψ) )(

C 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 behaviour of the MLE of b. First we recall a strong law of large numbers for continuous local martingales.

C.1 Theorem. (Liptser and Shiryaev [36,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 -→ ∞ as t → ∞, (C. [START_REF] Alfonsi | On the discretization schemes for the CIR (and Bessel squared) processes[END_REF] where ( M t ) t∈R + denotes the quadratic variation process of M . Then -→ 0 as t → ∞. (C.2)

P t 0 ξ 2 u d M u < ∞ = 1, t ∈ R + ,
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 behaviour 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].

C.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 : R + → R d×d such that Q(t) is an invertible (non-random) matrix for all t ∈ R + , 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).

We note that Theorem C.2 remains true if the function Q is defined only on an interval [t 0 , ∞) with some t 0 ∈ R ++ .

  .9). ✷ 3.5 Example. Now we formulate a special case of Corollary 3.4 in the supercritical case (b ∈ R --) supposing that the Lévy measure m takes the form given in (1.5), i.e., in the case of a supercritical BAJD process. 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 b ∈ R --and m being a Lévy measure on R ++ satisfying(1.5). Then for all u ∈ R -, E(e uYt ) = exp ψ u,0 (t)y 0 + φ u,0 (t) , t ∈ R + , (3.10)

3. 6 e

 6 Corollary. Let a ∈ R + , b ∈ R, σ ∈ R ++ ,and let m be a Lévy measure on R ++ satisfying (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 v ∈ R -, zψ 0,v (s) -1 m(dz) ds

Let P b

  denote the probability measure induced by (Y t ) t∈R + on the measurable space (D(R + , R), D(R + , R)) endowed with the natural filtration (D t (R + , R)) t∈R + , see Appendix B. 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 proposition 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 MLE for the parameter b based on the observations (Y t ) t∈[0,T ] . 4.1 Proposition. Let b, b ∈ R.

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

  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 ] . 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 4.3. 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. Next, we formulate a lemma about the unique existence of MLE b T of b for all T ∈ R ++ . 4.2 Proposition. Let a ∈ R + , b ∈ R, σ ∈ R ++ , y 0 ∈ R + , and let m be a Lévy measure on R ++ satisfying (1.2). If a ∈ R ++ or y 0 ∈ R ++ , then for each T ∈ R ++ , there exists a unique MLE b T of b almost surely having the form (4.3) b T = -Y Ty 0 -aT -J T T 0 Y s ds , provided that T 0 Y s ds ∈ R ++ (which holds almost surely due to Proposition 2.1).

  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 ΛT (b) :=

T 0

 0 The estimating equation ΛT (b) = 0, b ∈ R, has a unique solution -Y T -y 0 -aT -J T Yu du provided that T 0 Y u du is strictly positive, which holds almost surely. Recall that this unique solution coincides with b T , see (4.3). ✷ 5 Asymptotic behaviour of the MLE in the subcritical case 5.1 Theorem. Let a ∈ R + , b ∈ R ++ , σ ∈ R ++ , and let m be a Lévy measure on R ++ satisfying (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

5. 2

 2 Theorem. Let a ∈ R ++ , b ∈ R ++ , σ ∈ R ++ , and let m be a Lévy measure on R ++ satisfying (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 asymptotically normal, i.e., √ T ( b Tb)

( 1 . 4 )

 14 , b T is strongly consistent, i.e., b T a.s. -→ b as T → ∞. Proof. 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 2.4, (Y t ) t∈R + has a unique stationary distribution π with ∞ 0 y π(dy) = a + ∞ 0 z m(dz) 1 b ∈ R ++ . By Theorem 5.1, we have 1 T T 0 Y s ds P -→ ∞ 0 y π(dy) as T → ∞. Hence, since the quadratic variation process of the square integrable martingale t 0 √ Y s dW s t∈R + takes the form t 0 Y s ds t∈R + , by Theorem C.2 with η :=

1 0 1 T

 11 Y s ds as T → ∞, as desired. Consequently, by Slutsky's lemma, weak consistency follows by the decomposition b T = T b T , T ∈ R ++ .

7. 2

 2 Remark. (i) Under the conditions of Theorem 7.1 in the special case of m having the form (1.5), Corollary 3.4 yields another representation of the law of V in Theorem 7.1, namely,

(7. 7 ) 0 e 1 + σ 2 u 2b as t → ∞. By the monotone convergence theorem we obtain t 0 ψ 0 ∞ 0 eσ 2

 7010002 Now we turn to calculate the Laplace transform of V . Since P(lim t→∞ e bt Y t = V ) = 1, we also have e bt Y t D -→ V as t → ∞, and, by continuity theorem, lim t→∞ E(exp{ue bt Y t }) = e uV for all u ∈ R -. By Corollary 3.4,E(exp{ue bt Y t }) = exp ψ ue bt ,0 (t)y 0 + t 0 aψ ue bt ,0 (s) + ∞ zψ ue bt ,0 (s) -1 m(dz) dsfor each u ∈ R -and t ∈ R + , where we haveψ ue bt ,0 (t) = 2ub σ 2 u(1e bt ) + 2b → u ue bt ,0 (s) ds = t 0 2ube b(t-s) σ 2 u(e b(t-s)e bt ) + 2b ds = t 02ube by σ 2 u(e bye bt ) + 2b dy→ ∞ 0 2ube by σ 2 ue by + 2b dy = -2 σ 2 log 1 + σ 2 u 2b as t → ∞,where we used that the function R ++ ∋ t → 2ube by σ 2 u(e by -e bt )+2b ∈ R --is monotone decreasing for all y ∈ R ++ and u, b ∈ R --. In a similar way,t zψ ue bt ,0 (s) -1 m(dz) ds = t-s) σ 2 u(e b(t-s)e bt ) + 2b -1 m(dz) σ2 u(e bye bt ) + 2b -1 m(dz) dy → ue by + 2b -1 m(dz) dy as t → ∞, yielding (7.1). 7.3 Theorem. Let a ∈ R + , b ∈ R --, σ ∈ R ++ , and let m be a Lévy measure on R ++ satisfying (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 + . Suppose that a ∈ R ++ or a = 0, m = 0 and y 0 ∈ R ++ . Then the MLE of b is strongly consistent, i.e., b T a.s.

Proof. By Proposition 4 . 2 ,

 42 there exists a unique MLE b T of b for all T ∈ R ++ which takes the form given in (4.3). Using (4.3), (6.7), Theorem 7.1 and the fact that lim T →∞ T e bT = 0, we have b T = -e bT Y T + e bT y 0 + aT e bT + T e bT J T T e bT T 0 Y s ds a.s. as T → ∞, as desired. Further, by the second equality in (5.2), e -bT /2 ( b Tb) = -σ e bT /2 T 0 √ Y s dW s e bT T 0 Y s ds , T ∈ R ++ . By Theorem 7.1, e bT T 0 Y s ds a.s. -→ -V b as T → ∞, and using Theorem C.2 with η := -By the continuous mapping theorem, e -bT /2 ( b Tb) D -→ -σZ -V b -1/2 as T → ∞, yielding the first assertion.

tV 2 ν

 2 ) t∈R + with values in the set of all symmetric positive semidefinite d × d matrices such thatC t = t 0 c (ψ)s dF (ψ) s P ψ -almost sure for every t ∈ R + . Due to the assumptionP ψ (ν (ψ) ({t} × R d ) = 0) = 1 for every ψ ∈ Ψ, t ∈ R + ,such choices of (F (ψ) t ) t∈R + and (c (ψ) t ) t∈R + are possible, see Jacod and Shiryaev [23, Proposition II.2.9 and Corollary II.1.19]. Let P denote the predictable σ-algebra on D(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), (B.1) ( ψ,ψ) (s, x) -1 ( ψ) (ds, dx) < ∞, ψ,ψ) (s, x) -1)h(x) ν ( ψ) (ds, dx),(B.3)

  6. Next we drive explicit formulas for the Laplace transform of Y t , t 0 Y s ds in Section 3, together with some examples for the BAJD process. Here we use the fact that Y t , t 0 Y s ds t∈[0,∞) is a 2-dimensional CBI process following also from Keller-Ressel [26, Theorem 4.10] or Filipović et al. [17, paragraph before Theorem 4.3]. For completeness, we note that Keller-Ressel

  also implies that Y is a continuous state and continuous time branching process with immigration having branching and immigration mechanisms given in the Proposition. ✷ Next we present a result about the first moment of (Y t ) t∈R + . 2.2 Proposition. Let a ∈ R + , b ∈ R, σ ∈ R ++ , and let m be a Lévy measure on R ++ satisfying (1.2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0

  see Jacod and Shiryaev [23, III.2.28 Remarks, part 1)]. Consequently, the (predictable) quadratic variation process of

  and let m be a Lévy measure on R ++ satisfying (1.2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P

  and Lapeyre [32, Chapter 6, Proposition 2.5]. , Z t ) t∈R + is a 2-dimensional CBI process. Using the SDE (1.1) and (2.2), this process satisfies a SDE of the form given in Barczy et al. [5, Section 5], namely,

✷

Proof of Theorem 3.1. Introducing the process Z t := t 0 Y s ds, t ∈ R + , first, we show that (Y t

  and (3.7). Let a ∈ R + , b ∈ R, σ ∈ R ++ , and let m be a Lévy measure on R ++ satisfying (1.2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P(Y 0

✷ Next, we formulate two corollaries of Theorem 3.1 giving the Laplace transforms of Y t and t 0 Y s ds, t ∈ R + , separately.

3.4 Corollary.

  ) t∈R + of ( Y t ) t∈R + takes the form Y cont

					Using
	part 1) of Remarks III.2.28 in Jacod and Shiryaev [23] and (2.9), the continuous (local) martingale
	part ( Y cont t have	t	= σ	t 0	Y u dW u , t ∈ R + , and, by (1.1), we
	d Y cont t

where ((η cont ) ( b) t ) t∈R + denotes the continuous (local) martingale part of (η t ) t∈R + under P b .

  see, e.g., Sato[START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF] Theorem 19.3]. Using the SDE (1.1), we have ∆J t = ∆Y t , t ∈ R + , and then, by (4.4), we obtain

  and let m be a Lévy measure on R ++ satisfying (1.2). Let (Y t ) t∈R + be the unique strong solution of the SDE (1.1) satisfying P

  and let m be a Lévy measure on R ++ satisfying (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 there exists a random variable V with P(V ∈ R + ) = 1 such that

	e bt Y t	a.s. -→ V	and	e bt	0	t	Y u du	a.s.

  respectively, where ( W t ) t∈R + and ( W t ) t∈R + are independent 1-dimensional standard Wiener processes (yielding that ( X t ) t∈R + and ( X t ) t∈R + are independent). We point out that in the representation (7.5) there are only diffusion-type (critical) CIR processes, while the representation of V given in Theorem 7.1 contains the almost sure limit of the appropriately scaled jump-type (supercritical) CIR process(7.3). Further, by Alfonsi [1, Proposition 1.2.11], if a ∈ R ++ , then X -1

	b	is absolutely con-
	tinuous, and, if a = 0 and c

  t-s) Y s + a + , 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

			∞	z m(dz) e bs	t	e -bu du
			0	s
	for all s, t ∈ R + with 0	s	t, where F Y t := σ(Y s

  Further, if ω ∈ Ω is such that e bt Y t (ω) → V (ω) as t → ∞, then, by the integral Toeplitz lemma (seeKüchler and Sørensen [30, Lemma B.3.2]), we have

	hence, by the submartingale convergence theorem, there exists a non-negative random variable V
	such that
	(7.6)								e bt Y t	a.s. -→ V	as t → ∞.
		1 0 e -bu du t	0	t	e -bu (e bu Y u (ω)) du → V (ω)	as t → ∞.
	Here	t 0 e -bu du = e -bt -1 -b , t ∈ R + , thus we conclude
		e bt	0	t	Y u du =	1 -e bt -b		t 0 e -bu du t 0 Y u du	a.s.
	following from Proposition 2.2. Then
		E(e bt Y t | F Y s ) = e bs Y s + a +	0	∞	z m(dz) e b(s+t)	s	t	e -bu du e bs Y s
	s submartingale with respect to the filtration (F Y t, consequently, the process (e bt Y t ) t∈R + is a non-negative for all s, t ∈ R + with 0 t ) t∈R + . Moreover,
	E(e bt Y t ) = y 0 + a +		∞	z m(dz) e bt		t	e -bu du = y 0 + a +	∞	z m(dz)	t	e bu du
			0								0		0	0
		y 0 + a +	0	∞	z m(dz)	0	∞	e bu du = y 0 -	1 b	a +	0	∞	z m(dz) < ∞,	t ∈ R + ,

  (ψ) ) associated with a fixed Borel measurable truncation function h : R d → R d , see Jacod and Shiryaev [23, 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 byµ η (ω, dt, dx) := s∈R + ½ {∆ηs(ω) =0} ε (s,∆ηs(ω)) (dt, dx), 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 ) t∈R + is a local martingale with M

	(ψ) t t ∈ R where (M + B (ψ) t , (ψ) (ψ) t 0

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

η t := η t -s∈[0,t] (η sh(∆η s )), t ∈ R + ,

  h(x)) µ η (ds, dx), t ∈ R + , of η under P ψ , see Jacod and Shiryaev [23, Theorem II.2.34]. Moreover, for each ψ ∈ Ψ, let us choose a nondecreasing, continuous, adapted process (F

					ds, dx)
		t
	+	0	R d	(x (ψ) t	) t∈R + with F

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

Acknowledgements

We would like to thank the referee for his/her 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 by the "Magyar Állami Eötvös Ösztöndíj 2016" Grant no. 75141 funded by the Tempus Public Foundation. Ahmed Kebaier benefited from the support of the chair Risques Financiers, Fondation du Risque.

Particularly, we obtain e bt Y t a.s.

-→ V and e bt Y t a.s.

-→ V as t → ∞ with E(e u V ) = exp uy 0

for all u ∈ R -, and we conclude

and hence

The monotone convergence theorem yields E(e uV ) ↓ E(½ {V =0} ) = P(V = 0) as u → -∞. We have

Moreover, for each z ∈ R + and y ∈ R + , the function

is monotone increasing, hence, again by the monotone convergence theorem, we obtain

Summarizing, by (7.1), we conclude that P(V = 0) = 0 if and only if a ∈ R ++ or m = 0. Consequently,

(and hence that of V) is absolutely continuous (see, e.g., Alfonsi [1, Proposition 1.2.11]), and consequently V is absolutely continuous. ✷