Asymptotic properties of maximum likelihood estimator
for the growth rate for a jump-type CIR process
based on continuous time observations
Mátyás Barczy∗,⋄ , Mohamed Ben Alaya∗∗ ,

arXiv:1609.05865v4 [] 10 Aug 2017

Ahmed Kebaier∗∗ and Gyula Pap∗∗∗
* MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi
vértanúk tere 1, H–6720 Szeged, Hungary.
** Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), Villetaneuse, France.
*** Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H–6720 Szeged, Hungary.
e–mails: barczy.matyas@inf.unideb.hu (M. Barczy),
mba@math.univ-paris13.fr (M. Ben Alaya),
kebaier@math.univ-paris13.fr (A. Kebaier),
papgy@math.u-szeged.hu (G. Pap).
⋄ Corresponding author.
Abstract
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.

1

Introduction

Continuous state and continuous time branching processes with immigration, especially, the Cox–
Ingersoll–Ross (CIR) process (introduced by Feller [15] and Cox et al. [10]) and its variants, play an
2010 Mathematics Subject Classifications: 60H10, 91G70, 60F05, 62F12.
Key words and phrases: jump-type Cox–Ingersoll–Ross (CIR) process, basic affine jump diffusion (BAJD), subordinator, maximum likelihood estimator.
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 Ösztöndı́j 2016”
Grant no. 75141 funded by the Tempus Public Foundation. Ahmed Kebaier benefited from the support of the chair
Risques Financiers, Fondation du Risque.

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
p
(1.1)
t ∈ [0, ∞),
dYt = (a − bYt ) dt + σ Yt dWt + dJt ,

with an almost surely non-negative initial value Y0 , where a ∈ [0, ∞), b ∈ R, σ ∈ (0, ∞), (Wt )t∈[0,∞)
is a 1-dimensional standard Wiener process, and (Jt )t∈[0,∞) is a subordinator (an increasing Lévy
process) with zero drift and with Lévy measure m concentrating on (0, ∞) such that
Z ∞
(1.2)
z m(dz) ∈ [0, ∞),
0

that is,
(1.3)

 Z
E(euJt ) = exp t

∞
0


(euz − 1) m(dz)

for any t ∈ [0, ∞) and for any complex number u with Re(u) ∈ (−∞, 0], see, e.g., Sato [44, proof
of Theorem 24.11]. We suppose that Y0 , (Wt )t∈[0,∞) and (Jt )t∈[0,∞) are independent. Note that
the moment condition (1.2) implies that m is a Lévy measure (since min(1, z 2 ) 6 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 [44, 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(Yt ∈ [0, ∞) for all t ∈ [0, ∞)) = 1 (see
Proposition 2.1). In fact, (Yt )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 [40, 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 [40, 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 (Yt )t∈[0,T ] with T ∈ (0, ∞), starting the process Y from some known non-random
initial value y0 ∈ [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,
Rt
for instance, the limiting behavior of 0 Y1s ds as t → ∞, which seems to be a hard task even in the
Rt
subcritical case. In general, we would need an explicit formula for the Laplace transform of 0 Y1s 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
2

(diffusion-type) CIR process. Overbeck [41] studied the MLE of the drift parameters of the original
CIR process based on continuous time observations, and later on, Ben Alaya and Kebaier [6], [7]
completed the results of Overbeck [41] 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 [42]. 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 [35] 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. [19]
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.
R∞
Note that we have E(Jt ) = t 0 z m(dz) ∈ [0, ∞), and E(Yt ) ∈ [0, ∞) (see Proposition 2.2).
Moreover, (Jt )t∈[0,∞) is a compound Poisson process if and only if m(R) = m((0, ∞)) ∈ [0, ∞),
see, e.g., Sato [44, 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 [44, 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, ∞), Yt 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 [43] and a proof has been given in Li [33, Theorem 3.20 and the paragraph
after Corollary 3.21], see also Keller-Ressel and Steiner [29], Keller-Ressel [27], and Keller-Ressel and
R∞
Mijatović [28, 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(Yt ) as t → ∞,
R∞
see (2.6) and (2.7)), b is the rate at which E(Yt ) reverts to 0 y π(dy) as t → ∞ (speed of

R∞
R∞
adjustment, since E(Yt ) = 0 y π(dy) + e−bt E(Y0 ) − 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
 
Z 1
1
m(dz) < ∞,
z log
(1.4)
z
0
Jin et al. [25] established an explicit positive lower bound of the transition densities of (Yt )t∈[0,∞) , and
based on this result, they showed the existence of a Foster–Lyapunov function and derived exponential
ergodicity for (Yt )t∈[0,∞) (see Theorem 2.4). Comparing the moment conditions (1.2) and (1.4), note

that the integrability of z log 1z on the interval (0, e−1 ) yields that of z on the same interval.
R∞
In case of b = 0, if a + 0 z m(dz) ∈ (0, ∞), then limt→∞ E(Yt ) = ∞ such that
R∞
limt→∞ t−1 E(Yt ) ∈ (0, ∞), and, in case of b ∈ (−∞, 0), if E(Y0 ) ∈ (0, ∞) or a+ 0 z m(dz) ∈ (0, ∞)
(which rule out the case that Y is identically zero), then limt→∞ E(Yt ) = ∞ such that
limt→∞ ebt E(Yt ) ∈ (0, ∞), hence the parameter b can always be interpreted as the growth rate,
see Proposition 2.2.
3

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 − Yt ) with some κ ∈ (0, ∞) and θ ∈ [0, ∞),
and the Lévy process (Jt )t∈[0,∞) is a compound Poisson process with exponentially distributed jump
sizes, namely,
m(dz) = cλe−λz 1(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
Z ∞
Z ∞
c
zλe−λz dz = ∈ [0, ∞),
z m(dz) = c
λ
0
0
and
Z

1
0

 
 
Z 1
Z ∞
1
1
cλ
z log
∈ [0, ∞).
z log
ue−2u du =
m(dz) 6 cλ
dz = cλ
z
z
4
0
0

For describing the dynamics of default intensity, the BAJD was introduced by Duffie and Gârleanu
[13]. Filipović [16] and Keller-Ressel and Steiner [29] 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 (Yt )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
(Yt )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 (Yt )t∈[0,∞) given by the equation (1.1), see Theorem 2.4. In Remark 2.5, we derive
a Grigelionis representation for the process (Yt )t∈[0,∞) . Further, we explain why we do not estimate
the parameter σ, see Remark 2.6. Next we drive explicit formulas for the Laplace transform of
Rt

Yt , 0 Ys ds in Section 3, together with some examples for the BAJD process. Here we use the
Rt

fact that Yt , 0 Ys 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 [26, 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. [24, 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. [24, Proposition 4.3], and we make the solutions
of Riccati-type differential equations explicit in case of (Yt )t∈[0,∞) . Section 4 is devoted to study the
existence and uniqueness of the MLE bbT of b based on observations (Yt )t∈[0,T ] with T ∈ (0, ∞).
We derive an explicit formula for b
bT 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
RT
1/2
with
problem, we also replace the deterministic scaling T 1/2 by the random scaling σ1 0 Ys ds
4

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
1/2
RT
. In Section 7, for the supercritical case, we prove that
T and with the random scaling σ1 0 Ys ds
the MLE of b is strongly consistent, and it is asymptotically mixed normal with the deterministic
RT
1/2
.
scaling e−bT /2 , and it is asymptotically standard normal with the random scaling σ1 0 Ys ds
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 [40, Section 4.3] (MLE for subcritical case),
Huang, Ma and Zhu [19] and Li and Ma [35] (conditional LSE). Concerning the asymptotic behavior
Rt
of the MLE in the subcritical case, we use an explicit formula for the Laplace transform of 0 Ys ds to
Rt
derive stochastic convergence of 1t 0 Ys 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
RT
1/2
(bbT −b) converges in distribution as
process plays a role. We remark that for all b ∈ R, σ1 0 Ys ds
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.

2

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 kxk and kAk, we denote the Euclidean norm of a vector x ∈ Rd and the induced
matrix norm of a matrix A ∈ Rd×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
P
D
a.s.
D
a.s.
distribution and almost surely by −→, −→, −→, = and = , respectively.

Let Ω, F, P be a probability space. By Cc2 (R+ , R) and Cc∞ (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 η0 be a random variable independent of (Wt )t∈R+ and (Jt )t∈R+ satisfying
P(η0 ∈ R+ ) = 1 and E(η0 ) < ∞. 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 (Yt )t∈R+ of the SDE (1.1) such
that P(Y0 = η0 ) = 1 and P(Yt ∈ R+ for all t ∈ R+ ) = 1. Moreover, (Yt )t∈R+ is a CBI process
5

having branching mechanism
R(u) =

σ2 2
u − bu,
2

u ∈ C with Re(u) 6 0,

and immigration mechanism
F (u) = au +

Z

∞
0

(euz − 1) m(dz),

u ∈ C with Re(u) 6 0.

Further, the infinitesimal generator of Y takes the form
Z ∞
1
(Af )(y) = (a − by)f ′ (y) + yσ 2 f ′′ (y) +
(f (y + z) − f (y)) m(dz),
(2.1)
2
0
where y ∈ R+ , f ∈ Cc2 (R+ , R), and f ′ and f ′′ denote the first and second order partial derivatives
of f .

Rt
If, in addition, P(η0 ∈ R++ ) = 1 or a ∈ R++ , then P 0 Ys ds ∈ R++ = 1 for all t ∈ R++ .
Proof. The Lévy–Itô’s representation of J takes the form
Z tZ ∞
Jt =
(2.2)
z µJ (ds, dz),
t ∈ R+ ,
0

0

P

where µJ (ds, dz) := u∈R+ 1{∆Ju 6=0} ε(u,∆Ju ) (ds, dz) is the integer-valued Poisson random measure
on R2++ associated with the jumps ∆Ju := Ju − Ju− , u ∈ R++ , ∆J0 := 0, of the process J,
and ε(u,x) denotes the Dirac measure at the point (u, x) ∈ R2+ , see, e.g., Sato [44, Theorem 19.2].
Consequently, the SDE (1.1) can be rewritten in the form
Z t p
Z t
σ Ys dWs + Jt
(a − bYs ) ds +
Yt = Y0 +
0

0

(2.3)

= Y0 +

Z

t

0

(a − bYs ) ds +

Z

0

t

σ

p

Ys dWs +

Z tZ
0

∞

z µJ (ds, dz),

0

t ∈ R+ .

Equation (2.3) is a special case of the equation (6.6) in Dawson and Li [11], and Theorem 6.2 in
Dawson and Li [11] implies that for any initial value η0 with P(η0 ∈ R+ ) = 1 and E(η0 ) <
∞, there exists a pathwise unique non-negative strong solution satisfying P(Y0 = η0 ) = 1 and
P(Yt ∈ R+ for all t ∈ R+ ) = 1. Let (Yt′ )t∈R+ be a pathwise unique non-negative strong solution of
the SDE
q
dYt′ = (a − bYt′ ) dt + σ Yt′ dWt ,
t ∈ R+ ,
such that P(Y0′ = η0 ) = 1. Applying the comparison Theorem A.1, we obtain

(2.4)

P(Yt > Yt′ for all t ∈ R+ ) = 1.


Rt
Further, if P(η0 ∈ R++ ) = 1 or a ∈ R++ , then P 0 Ys′ ds ∈ R++ = 1 for all t ∈ R++ . Indeed,
if ω ∈ Ω is such that [0, t] ∋ u 7→ Yu′ (ω) is continuous and Yv′ (ω) ∈ R+ for all v ∈ R+ , then we
Rt
have 0 Ys′ (ω) ds = 0 if and only if Ys′ (ω) = 0 for all s ∈ [0, t]. Using the method of the proof of

Rt
Theorem 3.1 in Barczy et. al [3], we get P 0 Ys′ ds = 0 = 0, t ∈ R+ , as desired. Since (Ys )s∈[0,t]
has càdlàg, hence bounded sample paths almost surely (see, e.g., Billingsley [8, (12.5)]), using (2.4),

Rt
we conclude P 0 Ys ds ∈ R++ = 1 for all t ∈ R++ .
6

The form of the infinitesimal generator (2.1) readily follows by (6.5) in Dawson and Li [11]. Further, Theorem 6.2 in Dawson and Li [11] 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 (Yt )t∈R+ .
2.2 Proposition. Let a ∈ R+ , b ∈ R, σ ∈ R++ , and let m
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE
E(Y0 ) < ∞. Then


R
e−bt E(Y0 ) + a + ∞ z m(dz) 1−e−bt
b
0
E(Yt ) =
(2.5)
R

∞
E(Y ) + a +
0
0 z m(dz) t
Consequently, if b ∈ R++ , then
(2.6)

lim E(Yt ) =

t→∞



a+

Z

∞
0

if b = 0, then
lim t

t→∞

if b ∈ R−− , then

−1

E(Yt ) = a +

Z

be a Lévy measure on R++ satisfying
(1.1) satisfying P(Y0 ∈ R+ ) = 1 and
if b 6= 0,
if b = 0,

t ∈ R+ .


1
z m(dz) ,
b
∞

z m(dz),

0


Z
lim e E(Yt ) = E(Y0 ) − a +

∞

bt

t→∞

0


1
z m(dz) .
b

Proof. By Proposition 2.1, (Yt )t∈R+ is CBI process with an infinitesimal generator given in (2.1).
By the notations of Barczy et al. [5], this CBI process has parameters (d, c, β, B, ν, µ), where d = 1,
c = 21 σ 2 , β = a, B = −b, ν = m and µ = 0. Since E(Y0 ) < ∞ and the moment condition
R∞
R
R\{0} |z|1{|z|>1} ν(dz) = 1 z m(dz) < ∞ holds (due to (1.2)), we may apply formula (3.1.11) in Li
e = B = −b and
[34] or Lemma 3.4 in Barczy et al. [5] with the choices B
yielding that

βe = β +

Z

z ν(dz) = β +

R\{0}

e
tB

E(Yt ) = e

E(Y0 ) +

Z

This implies (2.5) and the other parts of the assertion.

Z

∞

0

t

e
uB

e
0

z m(dz) ∈ R+ ,


e
du β.

✷

Based on the asymptotic behavior of the expectations E(Yt ) as t → ∞, we introduce a classification of jump-type CIR model driven by a subordinator given by the SDE (1.1).
2.3 Definition. Let a ∈ R+ , b ∈ R, σ ∈ R++ , and let m be a Lévy measure on R++ satisfying
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 ∈ R+ ) = 1 and
E(Y0 ) < ∞. We call (Yt )t∈R+ subcritical, critical or supercritical if b ∈ R++ , b = 0 or b ∈ R−− ,
respectively.

7

In the subcritical case, the following result states the existence of a unique stationary distribution
and the exponential ergodicity for the process (Yt )t∈R+ , see Pinsky [43], Li [33, Theorem 3.20 and
the paragraph after Corollary 3.21], Keller-Ressel and Steiner [29], Keller-Ressel [27], Keller-Ressel
and Mijatović [28, Theorem 2.6] and Jin et al. [25, Theorem 1]. As a consequence, according to the
discussion after Proposition 2.5 in Bhattacharya [9], one also obtains a strong law of large numbers
for (Yt )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 (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 ∈ R+ ) = 1 and
E(Y0 ) < ∞.
(i) Then (Yt )t∈R+ converges in law to its unique stationary distribution π given by
R∞

Z 0
Z 0

Z ∞
av + 0 (evz − 1) m(dz)
F (v)
uy
dv = exp
e π(dy) = exp
dv
σ2 2
v − bv
u
u R(v)
0
2

for u ∈ R− . Moreover, π has a finite expectation given by


Z ∞
Z ∞
1
(2.7)
∈ R+ .
z m(dz)
y π(dy) = a +
b
0
0

(ii) If, in addition, a ∈ R++ and the extra moment condition (1.4) holds, then the process (Yt )t∈R+
is exponentially ergodic, namely, there exist constants β ∈ (0, 1) and C ∈ R++ such that
k PYt | Y0 =y −πkTV 6 C(y + 1)β t ,

t ∈ R+ ,

y ∈ R+ ,

where kµkTV denotes the total-variation norm of a signed measure µ on R+ defined by
kµkTV := supA∈B(R+ ) |µ(A)|, and PYt | Y0 =y is the conditional distribution of Yt with respect
to the condition Y0 = y. Moreover, for all Borel measurable functions f : R+ → R with
R∞
0 |f (y)| π(dy) < ∞, we have
Z ∞
Z
1 T
a.s.
f (y) π(dy)
as T → ∞.
f (Ys ) ds −→
(2.8)
T 0
0
2.5 Remark. Let a ∈ R+ , b ∈ R, σ ∈ R++ , and let m be a Lévy measure on R++ satisfying
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 ∈ R+ ) = 1 and
E(Y0 ) < ∞. By (1.1), the process (Yt )t∈R+ is a semimartingale, see, e.g., Jacod and Shiryaev [23,
I.4.34]. By (1.3), we have

 Z 1
Z ∞

iθz
iθJt
e − 1 − iθzh(z) m(dz)
z m(dz) + t
E(e ) = exp itθ
0

0

for θ ∈ R and t ∈ R+ , where h(z) := z 1[−1,1] (z), z ∈ R. Using again (1.1) and the Lévy–Itô’s
representation (2.2) of J, we can write the process (Yt )t∈R+ in the form
Z tp
Z 1
Z t
z m(dz) + σ
(a − bYu ) du + t
Yt = Y0 +
Yu dWu
0

(2.9)

+

Z tZ
0

R

0

0

h(z) µ
eJ (du, dz) +

Z tZ
0

R

(z − h(z)) µJ (du, dz),

t ∈ R+ ,

where µ
eJ (ds, dz) := µJ (ds, dz) − ds m(dz). In fact, (2.9) is a so-called Grigelionis form for the
semimartingale (Yt )t∈R+ , see, e.g., Jacod and Shiryaev [23, III.2.23] or Jacod and Protter [22,
Theorem 2.1.2].
✷
8

Next we give a statistic for σ 2 using continuous time observations (Yt )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 (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 ∈ R+ ) = 1 and
E(Y0 ) < ∞. The Grigelionis representation given in (2.9) implies that the continuous martingale part
Rt√
Y cont of Y is Ytcont = σ 0 Yu dWu , t ∈ R+ , see Jacod and Shiryaev [23, III.2.28 Remarks, part
Rt
1)]. Consequently, the (predictable) quadratic variation process of Y cont is hY cont it = σ 2 0 Yu du,
t ∈ R+ . Suppose that we have P(Y0 ∈ R++ ) = 1 or a ∈ R++ . Then for all T ∈ R++ , we have
hY cont iT
=: σ
bT2 ,
σ2 = R T
Y
du
u
0

RT

since, due to Proposition 2.1, P 0 Yu du ∈ R++ = 1. We note that σ
bT2 is a statistic, i.e., there
exists a measurable function Ξ : D([0, T ], R) → R such that σ
bT2 = Ξ((Yu )u∈[0,T ] ), where D([0, T ], R)
denotes the space of real-valued càdlàg functions defined on [0, T ], since
(2.10)

⌊nT ⌋

1

1
n

P⌊nT ⌋
i=1

Y i−1
n

X
i=1

Y i − Y i−1
n

n

2

−

X

(∆Yu )2

u∈[0,T ]

!

P

−→ σ
bT2

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 (Yu )u∈[0,T ] , and one can use Theorems 4.2.2 and 4.2.8
in Dudley [12]. Next we prove (2.10). By Theorem I.4.47 a) in Jacod and Shiryaev [23],
⌊nT ⌋

X
i=1

Y i − Y i−1
n

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 [23],
X
[Y ]T = hY cont iT +
(∆Yu )2 ,
T ∈ R+ .
u∈[0,T ]

Consequently, for all T ∈ R+ , we have
⌊nT ⌋

X
i=1

Y i − Y i−1
n

n

2

−

X

P

(∆Yu )2 −→ hY cont iT

u∈[0,T ]

as n → ∞.

Moreover, for all T ∈ R+ , we have
Z T
⌊nT ⌋
1 X
P
Yu du
Y i−1 −→
n
n
0
i=1

as n → ∞,

see Proposition I.4.44 in Jacod and Shiryaev [23]. Hence (2.10) follows by the fact that convergence
in probability is closed under multiplication.
✷

9

3

Joint Laplace transform of Yt and

Rt
0

Ys ds

Rt
We study the joint Laplace transform of Yt and 0 Ys 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
[26, Theorem 4.10] in case of a regular affine process and with the one in Jiao et al. [24, 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. [24, Proposition 4.3] explicit in case
of (Yt )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 γv := b2 − 2σ 2 v.
3.1 Theorem. Let a ∈ R+ , b ∈ R, σ ∈ R++ , and let m be a Lévy measure on R++ satisfying
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1 with
some y0 ∈ R+ . Then for all u, v ∈ R− ,




 
Z t
Z t
Z ∞

zψu,v (s)
Ys ds
= exp ψu,v (t)y0 +
E exp uYt + v
aψu,v (s) +
e
− 1 m(dz) ds
0

0

0

for t ∈ R+ , where the function ψu,v : R+ → R− takes the form
(3.1)


γv t
sinh( γv2 t )

 uγv cosh(γv2t )+(−ub+2v)
2
γv cosh( 2 )+(−σ u+b) sinh( γv2 t )
ψu,v (t) =

 σu2 u
1−

2

t

if v ∈ R−− or b 6= 0 (i.e., if γv ∈ R++ ),
if v = 0 and b = 0 (i.e., if γv = 0),

3.2 Remark. (i) If v ∈ R−− , then γv > |b|, hence






γv t
γv t
γv t
+ (−σ 2 u + b) sinh
> (γv + b) sinh
∈ R++ ,
γv cosh
2
2
2

t ∈ R+ .

t ∈ R+ .

If v = 0 and b 6= 0, then γv = |b| ∈ R++ , hence









γv t
γv t
γv t
γv t
b
2
γv cosh
sinh
+ (−σ u + b) sinh
> |b| cosh
+
∈ R++ ,
t ∈ R+ .
2
2
2
|b|
2


Consequently, if γv ∈ R++ , then γv cosh γ2v t + (−σ 2 u + b) sinh γ2v t ∈ R++ , t ∈ R+ , hence the
function ψu,v in (3.1) is well-defined.
(ii) In Theorem 3.1, we have
(3.2)

Z

t

ψu,v (s) ds =
0

(

b
σ2

t−

− σ22

2
σ2

log cosh

2
log 1 − σ2u t ,

γv t 
2

+

−σ2 u+b
γv

sinh

γv t 
,
2

if v ∈ R−− or b 6= 0,
if v = 0 and b = 0,

for all t ∈ R+ , see, e.g., Lamberton and Lapeyre [32, Chapter 6, Proposition 2.5].

✷

Rt
Proof of Theorem 3.1. Introducing the process Zt := 0 Ys ds, t ∈ R+ , first, we show that
(Yt , Zt )t∈R+ is a 2-dimensional CBI process. Using the SDE (1.1) and (2.2), this process satisfies a
10

SDE of the form given in Barczy et al. [5, Section 5], namely,
"

Yt

Zt

#

=

" #
Y0
0

+

+

Z tp
0

Z

" #
a

t

0

0

+

"

−b 0
1

0

#"

Ys
Zs

#!

" # " #⊤ "
#
Z tp
1 1
dWs
2
σ Ys
ds +
fs
0 0
dW
0

" # " #⊤ "
# Z Z
t
0 0
dWs
0 · Zs
r M (ds, dr),
+
fs
1 1
dW
0
R2+ \{0}

t ∈ R+ ,

ft )t∈R are independent standard Wiener processes, and M is a Poisson
where (Wt )t∈R+ and (W
+
random measure on R+ × (R2+ \ {0}) with intensity measure ds ν(dr), where the measure ν on
R
R∞
R2+ \ {0} is given by ν(B) := 0 1B (z, 0) m(dz), B ∈ B(R2+ \ {0}), hence R2 \{0} (1 ∧ krk) ν(dr) 6
+
R∞
z
m(dz)
<
∞.
Put
0
#
" #
" #
"
"
#
1 2
σ
a
c1
−b
0
∈ R2+ ,
β :=
c :=
∈ R2+ ,
B :=
:= 2
,
µ := (µ1 , µ2 ) := (0, 0).
0
0
c2
1 0
Then B is an essentially non-negative matrix (i.e., its off-diagonal entries are non-negative), and,
due to (1.2),
Z ∞
Z
Z
z m(dz) < ∞
krk ν(dr) =
krk1{krk>1} ν(dr) 6
R2+ \{0}

R2+ \{0}

0

yielding that condition (2.7) of Barczy et al. [5] is satisfied. Thus, by Theorem 4.6 in Barczy et
al. [5], (Yt , Zt )t∈R+ is a CBI process with parameters (2, c, β, B, ν, µ). We note that the fact
that (Yt , Zt )t∈R+ is a 2-dimensional CBI process follows from Filipović et al. [17, paragraph before
Theorem 4.3] as well, where this property is stated for general affine processes without a proof. The
branching mechanism of (Yt , Zt )t∈R+ is R(u, v) = (R1 (u, v), R2 (u, v)), u, v ∈ R− , with
* " # " #+
* " # " #+
2
0
u
1
u
σ
u2 − bu + v,
R2 (u, v) = c2 v 2 + B
,
= 0,
R1 (u, v) = c1 u2 + B
,
=
2
1
v
0
v
and the immigration mechanism of (Yt , Zt )t∈R+ is
* " #+ Z
(*" # +)
!
Z ∞
u
u
F (u, v) = β,
exp
(euz − 1) m(dz),
−
,r
− 1 ν(dr) = au +
2
v
v
R+ \{0}
0
see, e.g., Theorem 2.4 in Barczy et al. [5]. 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− , are given in Proposition 2.1,
which are in accordance with Theorem 4.10 in Keller-Ressel [26]. Consequently, by Theorem 2.7 of
Duffie et al. [14] (see also Barczy et al. [5, Theorem 2.4]), we have





Z t
Z t
F (ψu,v (s), ϕu,v (s)) ds
Ys ds
= exp ψu,v (t)y0 +
(3.3)
E exp uYt + v
0

0

for t ∈ R+ , where the function (ψu,v , ϕu,v ) : R+ → R2− is the unique locally bounded solution to
the system of differential equations
(
σ2
2
′ (t) = R (ψ
t ∈ R+ ,
ψu,v
1 u,v (t), ϕu,v (t)) = 2 ψu,v (t) − bψu,v (t) + ϕu,v (t),
ϕ′u,v (t) = R2 (ψu,v (t), ϕu,v (t)) = 0,

t ∈ R+ ,

11

with initial values ψu,v (0) = u, ϕu,v (0) = v. Clearly, ϕu,v (t) = v, t ∈ R+ , hence we obtain
′
ψu,v
(t) =

σ2
ψu,v (t)2 − bψu,v (t) + v,
2

t ∈ R+ ,

ψu,v (0) = u.

The solution of this differential equation is (3.1). Indeed, in case of γv > 0, we can refer to, e.g.,
Lamberton and Lapeyre [32, Chapter 6, Proposition 2.5], and in case of γv = 0, this is a simple
2
′ (t) = σ2 ψ
separable ODE taking the form ψu,0
2 u,0 (t) , t ∈ R+ , with initial condition ψu,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 := uγv + 2v,

α(2)
u,v := uγv − 2v,

(1)
βu,v
:= λ(−σ 2 u + γv ) − α(1)
u,v ,

(2)
βu,v
:= λ(σ 2 u + γv ) − α(2)
u,v ,

√
where γv = −2σ 2 v (since now b = 0). Let (Yt )t∈R+ be the unique strong solution of the SDE
(1.1) satisfying P(Y0 = y0 ) = 1 with some y0 ∈ R+ , with b = 0 and m being a Lévy measure on
R++ satisfying (1.5). Then we check that for all u, v ∈ R− ,



Z t

(3.4)
Ys ds
= exp ψu,v (t)y0 + φu,v (t) ,
t ∈ R+ ,
E exp uYt + v
0

√
where the function ψu,v : R+ → R− is given by (3.1) with γv = −2σ 2 v (since now b = 0), and if
2 2
2 2
(2)
v ∈ R−− (i.e., if γv ∈ R++ ) and v ∈
/ {− σ 2u , − σ 2λ } (i.e., if βu,v 6= 0), then

(3.5)






γv t
γv t
σ2u
2a
sinh
−
φu,v (t) = − 2 log cosh
σ
2
γv
2
!
!!
(2)
(1)
(2)
(1)
(2)
1 αu,v
αu,v
βu,v eγv t + βu,v
αu,v
t+
− (2) log
,
+c
(2)
(1)
(1)
(2)
γv βu,v
βu,v
βu,v
βu,v + βu,v
2 2

t ∈ R+ ,

(2)

2 2

and if v ∈ {− σ 2u , − σ 2λ } and v ∈ R−− (i.e., if v ∈ R−− and βu,v = 0), then





γv t
γv t
σ2u
2a
sinh
−
φu,v (t) = − 2 log cosh
σ
2
γv
2
!
(2)
αu,v
c
t
+
(1 − e−γv t ) ,
t ∈ R+ ,
+ (1) α(1)
u,v
γ
v
βu,v

(3.6)

and if v = 0 (i.e., if γv = 0), then




2c
σ 2 λu
σ2 u
2a
(3.7)
t − 2 log 1 −
t ,
φu,v (t) = − 2 log 1 −
σ
2
σ λ
2(λ − u)
Especially, for all u ∈ R− ,
(
uYt

E(e

) = exp

uy0
1−

σ2 u
2 t

)

σ2 u
t
1−
2

− 2a2 
σ

12

σ 2 λu
t
1−
2(λ − u)

−

2c
σ2 λ

t ∈ R+ .

,

t ∈ R+ .

(1)

First, we check that the function φu,v in (3.5) is well-defined. If γv ∈ R++ , then βu,v ∈ R++
(1)
(1)
(2)
(1)
(2)
(due to αu,v ∈ R− ). If γv = 0, then αu,v = αu,v = 0, βu,v = −σ 2 λu and βu,v = σ 2 λu. Further,
(1)
(2)
(1)
(2)
(1)
(2)
βu,v eγv t + βu,v > βu,v + βu,v ∈ R+ , and βu,v + βu,v = 0 holds if and only if γv = 0 (i.e., v = 0).
(1)
Indeed, since βu,v ∈ R+ , we have
(1) γv t
(2)
(1)
(2)
βu,v
e + βu,v
> βu,v
+ βu,v
= 2γv (λ − u) ∈ R+ ,
(1)

(2)

and βu,v + βu,v = 0 holds if and only if γv = 0 (since λ − u ∈ R++ due to λ ∈ R++ and u ∈ R− ).
(1)
(1)
(2)
Consequently, if γv ∈ R++ , then βu,v ∈ R++ and βu,v + βu,v ∈ R++ , yielding that the function
φu,v in (3.5) is well-defined.
(2)

Next, we check that, provided that v ∈ R−− (i.e., γv ∈ R++ ), we have βu,v = 0 if and only if
2 2
2 2
(2)
v ∈ {− σ 2u , − σ 2λ }. Indeed, βu,v = 0 holds if and only if λ(σ 2 u + γv ) = γv 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
2 2
2 2
v ∈ {− σ 2u , − σ 2λ }, as desired.

√

−2v, we get

√
−2v ∈ {−σu, σλ}, yielding that

Finally, we check (3.4). Using Theorem 3.1 and part (ii) of Remark 3.2, it remains to calculate
R t R ∞ zψ (s)

u,v
− 1) m(dz) ds for u, v ∈ R− . For all u, v ∈ R− , we have
0
0 (e
Z ∞
Z ∞
Z ∞
zψu,v (s)
−λz
zψu,v (s)
e(ψu,v (s)−λ)z dz − c
(e
− 1)cλe
dz = cλ
(e
− 1) m(dz) =

(3.8)

=

0

0

0



cψu,v (s)
cλ
lim e(ψu,v (s)−λ)z − 1 − c = −
,
ψu,v (s) − λ z→∞
ψu,v (s) − λ

s ∈ R+ ,

where we used that ψu,v (s) − λ ∈ R−− , s ∈ R+ , following from λ ∈ R++ and ψu,v (s) ∈ R− ,
s ∈ R+ . By some algebraic transformations,
cψu,v (t)
uγv (1 + eγv t ) + 2v(eγv t − 1)
=c
λ − ψu,v (t)
−λσ 2 u(eγv t − 1) + λγv + λγv eγv t − uγv (1 + eγv t ) − 2v(eγv t − 1)
=c

(1)

(2)

(1)

(2)

αu,v eγv t + αu,v
βu,v eγv t + βu,v

,

t ∈ R+ .
(1)

As we have seen, if v ∈ R−− (i.e., if γv ∈ R++ ), then βu,v
2 2
2 2
v ∈ {− σ 2u , − σ 2λ }. Hence


  (2) 
(1) (2)
αu,v
(1)
βu,v αu,v

eγv t

c
αu,v −

(2) +
(2)
(1)
(2)

βu,v
βu,v eγv t +βu,v
 βu,v
cψu,v (t)
(1)
(2) −γv t
c
=
)
(1) (αu,v + αu,v e
λ − ψu,v (t) 
βu,v



2cu

2(λ−u)−λuσ2 t

for t ∈ R+ . By integration, we get (3.5), (3.6) and (3.7).

(2)

∈ R++ , and βu,v = 0 if and only if
2 2

2 2

if v ∈ R−− , v ∈
/ {− σ 2u , − σ 2λ },
2 2

2 2

if v ∈ {− σ 2u , − σ 2λ }, v ∈ R−− ,
if v = 0,
✷

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

Rt

13

3.4 Corollary. Let a ∈ R+ , b ∈ R, σ ∈ R++ , and let m be a Lévy measure on R++ satisfying
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1 with
some y0 ∈ R+ . Then for all u ∈ R− ,

 
Z t
Z ∞


E euYt = exp ψu,0 (t)y0 +
aψu,0 (s) +
ezψu,0 (s) − 1 m(dz) ds ,
t ∈ R+ ,
0

0

where the function ψu,0 : R+ → R− takes the form

−bt
 2 2ube
if b 6= 0,
σ u(e−bt −1)+2b
(3.9)
ψu,0 (t) =
u
 σ2 u
if b = 0,
1−

2

t ∈ R+ .

t

Proof. We have to substitute v = 0 in (3.1). The case of b = 0 is trivial. In case of b 6= 0 we
have γv = γ0 = |b|, hence

yielding (3.9).


|b|t 
−
ub
sinh
u|b| cosh |b|t
2
2
ψu,0 (t) =

,
2 u + b) sinh |b|t
|b| cosh |b|t
+
(−σ
2
2

t ∈ R+ ,
✷

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 (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1
with some y0 ∈ R+ , with b ∈ R−− and m being a Lévy measure on R++ satisfying (1.5). Then
for all u ∈ R− ,
n
o
uYt
E(e ) = exp ψu,0 (t)y0 + φu,0 (t) ,
(3.10)
t ∈ R+ ,
where the function ψu,0 : R+ → R− is given by (3.9), and if u 6=

2b
,
σ2

then





2a
(−σ 2 λ + 2b)ue−bt + (σ 2 u − 2b)λ
σ 2 u −bt
2c
(e − 1) +
log
(3.11) φu,0 (t) = − 2 log 1 +
σ
2b
−σ 2 λ + 2b
2b(u − λ)
for t ∈ R+ , and if u =
(3.12)

2b
,
σ2

then
φu,0 (t) =

2b(2ab − cσ 2 − aσ 2 λ)
t,
σ 2 (−σ 2 λ + 2b)

t ∈ R+ .

To check (3.10), using Corollary 3.4, it remains to calculate


1 m(dz) ds. Here, by (3.9),
Z

t

0



σ 2 u −bt
2
(e − 1) ,
ψu,0 (s) ds = − 2 log 1 +
σ
2b

Similarly to (3.8), for all u ∈ R− ,
Z ∞
cψu,0 (s)
(ezψu,0 (s) − 1) m(dz) = −
(3.13)
,
ψu,0 (s) − λ
0
14

Rt
0

aψu,0 (s) +

t ∈ R+ .

s ∈ R+ ,

R∞
0

ezψu,0 (s) −

By some algebraic transformations,
−

cψu,0 (s)
−2bcue−bt
=
,
ψu,0 (s) − λ
(−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 = σ2b2 (i.e., if σ 2 u − 2b = 0), then
Z t
cψu,0 (s)
−2bc
−
ds =
t,
t ∈ R+ ,
−σ 2 λ + 2b
0 ψu,0 (s) − λ
and hence



2bc
2ab
σ 2 u −bt
2bc
2
(e − 1) −
t= 2 t−
t
φu,0 (t) = − 2 log 1 +
σ
2b
−σ 2 λ + 2b
σ
−σ 2 λ + 2b
=

2b(2ab − cσ 2 − aσ 2 λ)
t,
σ 2 (−σ 2 λ + 2b)

t ∈ R+ ,

yielding (3.12). If u 6= σ2b2 (i.e., if σ 2 u − 2b 6= 0), then


Z t
cψu,0 (s)
(−σ 2 λ + 2b)ue−bt + (σ 2 u − 2b)λ
2c
ds =
log
,
−
−σ 2 λ + 2b
2b(u − λ)
0 ψu,0 (s) − λ

t ∈ R+ ,

yielding (3.11).

✷

Substituting u = 0 in (3.1), we obtain the following corollary.
3.6 Corollary. Let a ∈ R+ , b ∈ R, σ ∈ R++ , and let m be a Lévy measure on R++ satisfying
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1 with
some y0 ∈ R+ . Then for all v ∈ R− ,



 Z t
 
Z t
Z ∞

zψ0,v (s)
Ys ds
= exp ψ0,v (t)y0 +
E exp v
aψ0,v (s) +
e
− 1 m(dz) ds
0

0

0

where the function ψ0,v : R+ → R− takes the form

2v sinh( γ2v t )

if v ∈ R−− or b 6= 0,
γv t
γv t
(3.14)
ψ0,v (t) = γv cosh( 2 )+b sinh( 2 )
0
if v = 0 and b = 0,

4

t ∈ R+ .

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 Y0 = y0 ∈ R+ , and we will
consider b ∈ R as an unknown parameter.
Let Pb denote the probability measure induced by (Yt )t∈R+ on the measurable space
(D(R+ , R), D(R+ , R)) endowed with the natural filtration (Dt (R+ , R))t∈R+ , see Appendix B. Further,
for all T ∈ R++ , let Pb,T := Pb|DT (R+ ,R) be the restriction of Pb to DT (R+ , R).
The next proposition is about the form of the Radon–Nikodym derivative

d Pb,T
d Peb,T

for b, eb ∈ R. We

will consider Peb,T as a fixed reference measure, and we will derive MLE for the parameter b based
on the observations (Yt )t∈[0,T ] .
15

4.1 Proposition. Let b, eb ∈ R. Then for all T ∈ R++ , the probability measures Pb,T and Peb,T
are absolutely continuous with respect to each other, and, under P,



Z
d Pb,T e
b2 − eb2 T e
b − eb e
Ys ds,
log
(Y ) = − 2 (YT − y0 − aT − JT ) −
d Peb,T
σ
2σ 2
0

(4.1)

where Ye is the process corresponding to the parameter eb.

Proof. In what follows, we will apply Theorem III.5.34 in Jacod and Shiryaev [23] (see also Appendix
B). We will work on the canonical space (D(R+ , R), D(R+ , R)). Let (ηt )t∈R+ denote the canonical
process ηt (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 y0 ∈ R+ ), and hence, by Theorem III.2.26 in Jacod and Shiryaev
[23], under the probability measure Pb , the canonical process (ηt )t∈R+ is a semimartingale with
semimartingale characteristics (B (b) , C, ν) associated with the truncation function h, where

Z t
Z 1
(b)
z m(dz) du,
t ∈ R+ ,
Bt =
a − bηu +
0

Ct =

Z

t

0

√
(σ ηu )2 du = σ 2

0

Z

t

t ∈ R+ ,

ηu du,

0

ν(dt, dy) = K(ηt , dy) dt = dt m(dy)
with the Borel transition kernel K from R+ × R into R given by
Z
for y ∈ R+ and R ∈ B(R).
K(y, R) := 1R\{0} (z) m(dz) = m(R)
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 [23]. First note that (Ct )t∈R+ and ν(dt, dy) do not depend on the unknown parameter b,
e

and hence V (b,b) is identically one and then (B.1) and (B.2) readily hold. We also have

Pb ν({t} × R) = 0 = Pb (0 · m(R) = 0) = 1,

t ∈ R+ ,

b ∈ R,

Rt
since m(R) = m(R+ ) ∈ R+ due to (1.2). Further, (Ct )t∈R+ can be represented as Ct = 0 cu dFu ,
t ∈ R+ , where the stochastic processes (ct )t∈R+ and (Ft )t∈R+ are given by ct := σ 2 ηt , t ∈ R+ ,
and Ft = t, t ∈ R+ . Consequently, for all b, eb ∈ R,
Z t
Z t
e
(e
b)
(b)
e
cu βu(b,b) dFu
ηu du =
Bt − Bt = −(b − b)
0

0

(e
b,b)

Pb -almost surely for every t ∈ R+ , where the stochastic process (βt
(e
b,b)

βt
which yields (B.3).

=−

b − eb
,
σ2

16

t ∈ R+ ,

)t∈R+ is given by

Next we check (B.4), i.e.,
(4.2)

Pb

Z

0

We have

Z

t

t

e 2
βu(b,b) cu

e 2
βu(b,b) cu dFu

0

dFu < ∞

(b − eb)2
=
σ2



Z

t ∈ R+ .

= 1,

t

ηu du,

0

t ∈ R+ .

Since for each ω ∈ D(R+ , R), the trajectory [0, t] ∋ u 7→ ηu (ω) is càdlàg, hence bounded (see, e.g.,
Rt
Billingsley [8, (12.5)]), we have 0 ηu (ω) du < ∞, hence we obtain (4.2).

Next, we check that, under the probability measure Pb , local uniqueness holds for the martingale
problem on the canonical space corresponding to the triplet (B (b) , C, ν) with the given initial value
y0 with Pb as its unique solution. By Proposition 2.1, the SDE (1.1) has a pathwise unique strong
solution (with the given deterministic initial value y0 ∈ R+ ), and hence Theorem III.2.26 in Jacod
and Shiryaev [23] 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 (Pb ) yielding the desired local uniqueness. We
also mention that Theorem III.4.29 in Jacod and Shiryaev [23] implies that under the probability
measure Pb , all local martingales have the integral representation property relative to η.
By Theorem III.5.34 in Jacod and Shiryaev [23] (see also Appendix B), Pb,T and Peb,T are
equivalent (one can change the roles of b and eb), and under the probability measure Peb , we have
d Pb,T
(η) = exp
d Peb,T

Z

0

T

e
e
βu(b,b) d(η cont )u(b)

1
−
2

Z

T

0

e 2
βu(b,b) cu


du ,

T ∈ R++ ,

(e
b)

where ((η cont )t )t∈R+ denotes the continuous (local) martingale part of (ηt )t∈R+ under Peb . Using
part 1) of Remarks III.2.28 in Jacod and Shiryaev [23] and (2.9),
q the continuous (local) martingale
Rt
cont
cont
= σ 0 Yeu dWu , t ∈ R+ , and, by (1.1), we
part (Yet )t∈R+ of (Yet )t∈R+ takes the form Yet
have
t ∈ R+ .
dYetcont = dYet − (a − ebYet ) dt − dJt ,
Hence, under P,





 Z T
Z T
Z 
d Pb,T e
1 T
b − eb
b − eb
b − eb 2 2 e
e
e
e
log
σ Yu du
(Y ) =
− 2 (a − bYu ) du −
− 2 (dYu − dJu ) −
− 2
d Peb,T
σ
σ
2 0
σ
0
0
=−

b − eb
σ2

Z

which yields the statement.

0

T

(dYeu − dJu ) +

b − eb
σ2

Z

T

0

a du −

b2 − eb2
2σ 2

Z

T

0

Yeu du,

✷

Next, using Proposition 4.1, by considering Peb,T as a fixed reference measure, we derive an MLE
for the parameter b based on the observations (Yt )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 bbT , see
Remark 4.3. Let us denote the right hand side of (4.1) by ΛT (b, eb) replacing Ye by Y . By an MLE
bbT of the parameter b based on the observations (Yt )t∈[0,T ] , we mean
bbT := arg max ΛT (b, eb),
b∈R

17

which will turn out to be not dependent on e
b. Next, we formulate a lemma about the unique existence
b
of MLE bT of b for all T ∈ R++ .
4.2 Proposition. Let a ∈ R+ , b ∈ R, σ ∈ R++ , y0 ∈ R+ , and let m be a Lévy measure on R++
satisfying (1.2). If a ∈ R++ or y0 ∈ R++ , then for each T ∈ R++ , there exists a unique MLE bbT
of b almost surely having the form
bbT = − YT − Ry0 − aT − JT ,
T
0 Ys ds

(4.3)
provided that

RT
0

Ys ds ∈ R++ (which holds almost surely due to Proposition 2.1).


RT
Proof. Due to Proposition 2.1, P 0 Ys 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 (Yu )u∈[0,T ] (i.e., a statistic). By the Lévy–Itô’s
representation (2.2) of Jt , t ∈ R+ , we obtain
X
(4.4)
Jt =
∆Js ,
t ∈ R+ ,
s∈[0,t]

R1

R1

since −1 |z| m(dz) = 0 z m(dz) < ∞, see, e.g., Sato [44, Theorem 19.3]. Using the SDE (1.1), we
have ∆Jt = ∆Yt , t ∈ R+ , and then, by (4.4), we obtain
X
Jt =
∆Ys ,
t ∈ R+ .
s∈[0,t]

Consequently, for all t ∈ [0, T ], Jt is a measurable function of (Yu )u∈[0,T ] , yielding that the right
hand side of (4.3) is a measurable function (i.e., a statistic) of (Yu )u∈[0,T ] , as desired. By Proposition
4.1, for all b, eb ∈ R, we have
Z T
∂
1
b
ΛT (b, eb) = − 2 (YT − y0 − aT − JT ) − 2
Ys ds,
∂b
σ
σ 0
Z T
1
∂2
e
Ys ds.
Λ
(b,
b)
=
−
T
∂b2
σ2 0

Thus the MLE bbT of b based on a sample (Ys )s∈[0,T ] exists almost surely, and it takes the form
RT
(4.3) provided that 0 Ys 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.

4.3 Remark. In the literature there is another way of deriving an MLE. Sørensen [45] defined an
MLE of ψ as a solution of the equation Λ̇T (ψ) = 0, where Λ̇T (ψ) is the so-called score vector given
in formula (3.3) in Sørensen [45]. Luschgy [37], [38] called this equation as an estimating equation.
e
With the notations of the proof of Proposition 4.1, taking into account of the form of β (b,b) and the
e
fact that V (b,b) is identically one, we have

Z T
Z T
1
1
cont
Λ̇T (b) :=
− 2 dYu
(dYu − (a − bYu ) du − dJu )
=− 2
σ
σ 0
0


Z T
1
Yu du − JT
= − 2 YT − y0 − aT + b
σ
0
18

for b ∈ R and T ∈ (0, ∞). The estimating equation Λ̇T (b) = 0, b ∈ R, has a unique solution
RT
−aT −JT
R T0
− YT −y
provided that 0 Yu du is strictly positive, which holds almost surely. Recall that
0

Yu du

this unique solution coincides with b
bT , 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 (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1 with
some y0 ∈ R+ . Then


Z
Z ∞
1 t
P 1
(5.1)
Ys ds −→
z m(dz) ∈ R+
as t → ∞.
a+
t 0
b
0
Proof. Using Corollary 3.6, we have

 Z t
 


Z t
Z ∞

v
zψ v (s)
E exp
Ys ds
= exp ψ0, vt (t)y0 +
e 0, t − 1 m(dz) ds
aψ0, vt (s) +
t 0
0
0

for t ∈ R++ and v ∈ R− , where the function ψ0,v : R+ → R− is given in (3.14). We check that

 Z t

 

Z ∞
v
v
E exp
Ys ds
→ exp
z m(dz)
as t → ∞
a+
t 0
b
0

for all v ∈ R− . The case of v = 0 is trivial, so we may and do suppose that v ∈ R−− . Then we
have
ψ0, vt (t) =
since γ vt =

q
Z

−tγ v

1
2v(1 − e t )
→0
t γ v (1 + e−tγ vt ) + b(1 − e−tγ vt )
t

b2 −

2σ2 v

t

ψ

0, vt

0

→ b ∈ R++ as t → ∞. Moreover, by the dominated convergence theorem,

t

1
(s) ds =
t
=

Z

Z

−sγ v

t

0

2v(1 − e
−sγ v

γ vt (1 + e

2v(1 − e
−xtγ v

γ (1 + e
v
t

t

)
−sγ v

) + b(1 − e
−xtγ v

1
0

t

t

t

=

t

)

)

ds

dx →

v
b

as t → ∞.

−xtγ v

)

2|v|
b .

Finally, by the dominated convergence

) + b(1 − e

Indeed, the integrand in the last integral is dominated by
theorem, we obtain

Z t Z ∞

zψ0, v (s)
t
− 1 m(dz) ds
e
0

as t → ∞,

t

0

Z t Z
0

0

∞





−sγ v
t )
z
2v(1 − e
exp
−
1
m(dz)
ds
·
t γ v (1 + e−sγ vt ) + b(1 − e−sγ vt )
t






−xtγ v
t )
z
2v(1 − e
t exp
=
− 1 m(dz) dx
·
t γ v (1 + e−xtγ vt ) + b(1 − e−xtγ vt )
0
0
t

Z
Z 1 Z ∞
v ∞
v
z m(dz)
z m(dz) dx =
→
b
b 0
0
0
Z 1 Z

∞

19

as t → ∞ for all z ∈ R+ . Indeed,
At (z, x) := z ·

−xtγ v

2v(1 − e
−xtγ v

γ vt (1 + e

t

t

)
−xtγ v

) + b(1 − e

t

)

→z

v
b

where At (z, x) ∈ R− for all t ∈ R++ , z ∈ R+ and x ∈ [0, 1].

as t → ∞,
Hence t(e

At (z,x)
t

At (z,x)

− 1) → z vb as

t → ∞ for all z ∈ R+ and x ∈ [0, 1], and |t(e t − 1)| 6 |At (z, x)| 6 2z|v|
for t ∈ R++ , z ∈ R+
b
2z|v|
and x ∈ [0, 1], where b
is integrable with respect to the measure m(dz) dx on R++ × [0, 1].
By the continuity theorem, we obtain


Z
Z ∞
1 t
D 1
Ys ds −→
z m(dz) ∈ R+
as t → ∞,
a+
t 0
b
0

which implies (5.1).

✷

If a ∈ R++ or y0 ∈ R++ , then, using (4.3) and the SDE (1.1), we get
(5.2)

YT − y0 − aT − JT + b
b
bT − b = −
RT
0 Ys ds

RT
0

Ys ds

RT √
Ys dWs
= − R0 T
0 Ys ds
σ

RT
provided that 0 Ys ds ∈ R++ , which holds almost surely due to Proposition 2.1. Here note that
RT √
σ 0 Ys dWs = YTcont , T ∈ R+ , see Remark 2.6.

Despite the mistake in the formula (4.23) in Mai [40] for the MLE bbT of b, Mai [40, Theorem
4.3.1] formulated the right asymptotic behavior of b
bT 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 .

5.2 Theorem. Let a ∈ R++ , b ∈ R++ , σ ∈ R++ , and let m be a Lévy measure on R++ satisfying
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1 with
some y0 ∈ R+ . Then the MLE bbT of b is asymptotically normal, i.e.,




√
σ2b
σ2
D
b
R
R
T (bT − b) −→ N 0,
= N 0, ∞
(5.3)
as T → ∞,
∞
a + 0 z m(dz)
0 y π(dy)
where π

denotes the unique stationary distribution of (Yt )t∈R+ (see part (i) of Theorem 2.4).
P
Especially, bbT is weakly consistent, i.e., bbT −→ b as T → ∞. With a random scaling,
1
σ

Z

0

T

Ys ds

1/2

D
(bbT − b) −→ N (0, 1)

as T → ∞.

a.s.
Under the additional moment condition (1.4), bbT is strongly consistent, i.e., bbT −→ b as T → ∞.

Proof. By Proposition 4.2, there exists a unique MLE b
bT of b for all T ∈ R++ , which has the
form given in (4.3). By (i) of Theorem 2.4, (Yt )t∈R+ has a unique stationary distribution π with
1
R∞
R∞
R
P R∞
1 T
y
π(dy)
=
a
+
z
m(dz)
Y
ds
−→
∈
R
.
By
Theorem
5.1,
we
have
s
++
b
T 0
0
0
0 y π(dy)
as T → ∞. Hence, since the quadratic variation process of the square integrable martingale

20


Rt

R∞
1/2
Ys dWs t∈R+ takes the form
0 Ys ds t∈R+ , by Theorem C.2 with η :=
0 y π(dy)
and Slutsky’s lemma, we have
RT √
R∞
1/2


√1
Ys dWs D
√
N (0, 1)
y
π(dy)
σ2
0
T
0
b
R
R
= N 0, ∞
−→ −σ
T (bT − b) = −σ
R
∞
1 T
0 y π(dy)
0 y π(dy)
T 0 Ys ds
Rt√
0

as T → ∞, hence we obtain (5.3). Further, Slutsky’s lemma yields
1
σ

Z

0

T

Ys ds

1/2

1/2
Z
√
1 T
T (bbT − b)
Ys ds
T 0
1/2 

Z ∞
σ2
D 1
−→
N 0, R ∞
y π(dy)
= N (0, 1)
σ
0
0 y π(dy)

1
(bbT − b) =
σ

as T → ∞.



RT
a.s.
Under the additional moment condition (1.4), by (ii) of Theorem 2.4, we have T1 0 Ys ds −→
R∞
RT
a.s.
0 y π(dy) as T → ∞, implying also
0 Ys ds −→ ∞ as T → ∞. Using (5.2) and Theorem C.1,
a.s.
we have bbT −→ 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 (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1 with
some y0 ∈ R+ . Then




Z 1
Z
Yt 1 t
D
(6.1)
Ys ds
as t → ∞,
,
Ys ds −→ Y1 ,
t t2 0
0
where (Yt )t∈R+ is the unique strong solution of a critical (diffusion type) CIR model


Z ∞
p
z m(dz) dt + σ Yt dWt ,
dYt = a +
(6.2)
t ∈ R+ ,
0

with initial condition Y0 = 0, where (Wt )t∈R+ is a 1-dimensional standard Wiener process. MoreR1
over, the Laplace transform of (Y1 , 0 Ys ds) takes the form

− 22 (a+R ∞ z m(dz))

2u
0
σ
γv 
γv 
σ

 cosh 2 − γv sinh 2
if v ∈ R−− ,
R1

E euY1 +v 0 Ys ds = 
(6.3)
R
− 2 (a+ ∞ z m(dz))

0

 1 − σ2 u σ 2
if v = 0,
2
for all u, v ∈ R− , where γv =

√
−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 [7].
✷
21

Proof of Theorem 6.1. Using Theorem 3.1, we have




 
Z
Z t
Z ∞

zψ u , v (s)
1 t
Yt
t
2
t
Ys ds
= exp ψ ut , v2 (t)y0 +
E exp u + v 2
− 1 m(dz) ds
e
aψ ut , v2 (s) +
t
t
t
t 0
0
0

for t ∈ R+ and u, v ∈ R− , where the function ψu,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
γ 
1 uγv cosh 2v + 2v sinh

ψ ut , v2 (t) =
t
t γv cosh γ2v − σ 2 u sinh

and, for each t ∈ R+ ,
Z

t

0

γv 
2
γv 
2

→0

as t → ∞,



vs
vs
uγv cosh γ2t
+ 2v sinh γ2t
γv s 
γv s  ds
2
0 γv cosh 2t − σ u sinh 2t


Z 1
uγv cosh γv2x + 2v sinh γv2x
 dx
=
γv x 
2 u sinh γv x
−
σ
0 γv cosh
2
2
Z 1

 γ  σ2 u
 γ 
2
v
v
ψu,v (x) dx = − 2 log cosh
=
−
=: I,
sinh
σ
2
γ
2
v
0

1
ψ ut , v2 (s) ds =
t
t

(6.4)

Z

t

where we used (3.2). By the monotone convergence theorem we obtain

Z t Z ∞

zψ u , v (s)
t t2
− 1 m(dz) ds
e
0

0




γv s 
vs
+ 2v sinh γ2t
z uγv cosh 2t

 − 1 m(dz) ds
·
=
exp
vs
vs
t γv cosh γ2t
− σ 2 u sinh γ2t
0
0



Z 1 Z ∞ 
γ x
γ x
z uγv cosh v2 + 2v sinh v2


− 1 m(dz) dx
t exp
=
·
t γv cosh γv2x − σ 2 u sinh γv2x
0
0

Z 1 Z ∞
Z ∞
z
ψ
(x)
u,v
=
t(e t
z m(dz)
− 1) m(dz) dx → I
Z t Z

0

∞



0

0

as t → ∞ for all z ∈ R+ , where we used that ψu,v (t) ∈ R− for all t ∈ R+ and u, v ∈ R− , and
A
the function R++ ∋ t 7→ t(e 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 ut , v2 (t) =
t

and, for each t ∈ R+ ,
(6.5)

Z

0

t

ψ ut , v2 (s) ds =
t

1
t

Z

0

u
1
→0
t 1 − σ2 u

as t → ∞,

2

t

u
1−

σ2 us
2t

ds =

Z

0

22

1

u
1−

σ2 ux
2

dx = −



2
σ2 u
log
1
−
.
σ2
2

Moreover, by the monotone convergence theorem,

Z t Z ∞
Z t Z

zψ u , v (s)
t t2
e
− 1 m(dz) ds =
0




u
z
·
− 1 m(dz) ds
exp
t 1 − σ2 us
0
0
2t




Z 1 Z ∞ 
z
u
=
t exp
·
− 1 m(dz) dx
t 1 − σ2 ux
0
0
2

Z ∞
σ2u
2
z m(dz)
as t → ∞.
→ − 2 log 1 −
σ
2
0

0

∞



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++ , and let m be a Lévy measure on R++ satisfying
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1 with
R∞
some y0 ∈ R+ . Suppose that a ∈ R++ or a = 0, y0 ∈ R++ , 0 z m(dz) ∈ R++ . Then
R∞
a
+
D
0 z m(dz) − Y1
T (bbT − b) = T bbT −→
(6.6)
as T → ∞,
R1
0 Ys ds
where (Yt )t∈R+ is the unique strong solution of the SDE (6.2). As a consequence, the MLE bbT of
b is weakly consistent, i.e., b
bT converges to b in probability as T → ∞.
With a random scaling, we have
1
σ

Z

0

T

Ys ds

1/2

1
(bbT − b) =
σ

Z

T

Ys ds

0

1/2

D

bbT −→

R∞

z m(dz) − Y1
R1
1/2
σ 0 Ys ds

a+

0

as T → ∞.

Proof. By Proposition 4.2, there exists a unique MLE b
bT of b for all T ∈ R++ , which has the
form given in (4.3). By (4.3), we have
T bbT = −

YT
T

−

1
T2

y0
T

− a − JTT
,
RT
0 Ys ds

T ∈ R+ .

Here, by strong law of large numbers for the Lévy process (Jt )t∈R+ (see, e.g., Kyprianou [31, Exercise
7.2]),


Z ∞
JT
(6.7)
= E(J1 ) =
z m(dz) = 1.
P lim
T →∞ T
0
Hence, using Slutsky’s lemma, Theorem 6.1 and the continuous mapping theorem together with the
R1
fact that P( 0 Ys ds ∈ R++ ) = 1 (following from Proposition 2.1, since under the conditions of
R∞
Theorem 6.3, a + 0 z m(dz) ∈ R++ ), we have
R∞
Y
−
a
−
z m(dz)
1
D
T bbT −→ −
as T → ∞,
R1 0
Y
ds
s
0

as desired. Consequently, by Slutsky’s lemma, weak consistency follows by the decomposition b
bT =
1 b
T T bT , T ∈ R++ .
23

Applying Slutsky’s lemma, Theorem 6.1 and the continuous mapping theorem, we obtain
1/2
−1/2 
Z T


Z T
1
YT
1
1
y0
JT
b
Ys ds
bT = −
Ys ds
−
−a−
σ
σ T2 0
T
T
T
0
R∞

−1/2 
Z 1
Z ∞
a + 0 z m(dz) − Y1
1
D
as T → ∞,
−→ −
z m(dz) =
Ys ds
Y1 − a −
1/2
R1
σ
0
0
σ
Ys ds
0

as desired.

7

✷

Asymptotic behaviour of the MLE in the supercritical case

7.1 Theorem. Let a ∈ R+ , b ∈ R−− , σ ∈ R++ , and let m be a Lévy measure on R++ satisfying
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1 with
some y0 ∈ R+ . Then there exists a random variable V with P(V ∈ R+ ) = 1 such that
Z t
V
a.s.
a.s.
bt
bt
e Yt −→ V
and
e
Yu du −→ −
as t → ∞.
b
0
Moreover, the Laplace transform of V takes the form
uV

(7.1) E(e

) = exp



uy0
1+

σ2 u
2b



σ2u
1+
2b

− 2a2

for all u ∈ R− , and consequently,

σ

Z
exp

0

∞ Z ∞ 

exp

0



zueby
1+

σ2 u by
2b e




 
− 1 m(dz) dy

D e
e
e
V =V
+ V,

a.s. e
e
e a.s. e
e and V
e are independent random variables such that ebt Yet −→
e as
where V
V and ebt Yet −→ V
e
t → ∞, where (Yet )t∈R and (Yet )t∈R are the pathwise unique strong solutions of the supercritical
+

+

CIR models

(7.2)
and
(7.3)

dYet = (a − bYet ) dt + σ

e
e
dYet = −bYet dt + σ

q

q

ft ,
Yet dW

t ∈ R+ ,

e f
ft + dJt ,
Yet dW

t ∈ R+ ,

with

with

Ye0 = y0 ,

e
Ye0 = 0,

f
ft )t∈R and (W
ft )t∈R are independent 1-dimensional standard Wiener prorespectively, where (W
+
+
ee
e
cesses (yielding that (Yt )t∈R+ and (Y t )t∈R+ are independent). Further,
D
e=
V
Z− 1 ,
b

where (Zt )t∈R+ is the pathwise unique strong solutions of the critical CIR model
p
dZt = a dt + σ Zt dWt ,
(7.4)
t ∈ R+ ,
with
Z0 = y 0 ,
where (Wt )t∈R+ is a 1-dimensional standard Wiener process.

Moreover, P(V ∈ R++ ) = 1 if and only if a ∈ R++ or m 6= 0.
If, in addition, a ∈ R++ , then V is absolutely continuous.
24

7.2 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,
D
e
V = Xe− 1 + Xe− 1 1−

(7.5)

b

b

2b
σ2 λ

,

e
where (Xet )t∈R+ and (Xet )t∈R+ are the pathwise unique strong solutions of the critical CIR models
q
ft ,
dXet = a dt + σ Xet dW
t ∈ R+ ,
with
Xe0 = y0 ,

and

e
dXet =

c
dt + σ
λ − σ2b2

q

e f
ft ,
Xet dW

t ∈ R+ ,

with

e
Xe0 = 0,

f
ft )t∈R and (W
ft )t∈R are independent 1-dimensional standard Wiener prorespectively, where (W
+
+
ee
e
cesses (yielding that (Xt )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 Xe− 1 is absolutely conb
e
tinuous, and, if a = 0 and c ∈ R++ , then Xe− 1 1− 2b  is absolutely continuous. The independence
σ2 λ

b

of Xe− 1
b

e
and Xe− 1 1−
b

2b
σ2 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 [7]. 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
 Z t

Z ∞
Y
−b(t−s)
z m(dz) ebs
E(Yt | Fs ) = E(Yt | Ys ) = e
Ys + a +
e−bu du
0

s

for all s, t ∈ R+ with 0 6 s 6 t, where FtY := σ(Ys , s ∈ [0, t]), t ∈ R+ . The first equality
follows from the Markov property of the process (Yt )t∈R+ . The second equality is a consequence of
the time-homogeneity of the Markov process Y and the fact that
Z t

Z ∞
e−bu du,
t ∈ R+ , y0 ∈ R+ ,
z m(dz)
E(Yt | Y0 = y0 ) = e−bt y0 + a +
0

0

following from Proposition 2.2. Then
bt

E(e

Yt | FsY


Z
) = e Ys + a +
bs

0

∞


Z t
b(s+t)
z m(dz) e
e−bu du > ebs Ys
s

for all s, t ∈ R+ with 0 6 s 6 t, consequently, the process (ebt Yt )t∈R+ is a non-negative
submartingale with respect to the filtration (FtY )t∈R+ . Moreover,
Z t
 Z t


Z ∞
Z ∞
ebu du
z m(dz)
z m(dz) ebt
e−bu du = y0 + a +
E(ebt Yt ) = y0 + a +
0



6 y0 + a +

Z

0

0

0

∞

Z
z m(dz)

∞
0

ebu du = y0 −
25



1
a+
b

Z

∞
0

0


z m(dz) < ∞,

t ∈ R+ ,

hence, by the submartingale convergence theorem, there exists a non-negative random variable V
such that
a.s.

ebt Yt −→ V

(7.6)

as t → ∞.

Further, if ω ∈ Ω is such that ebt Yt (ω) → V (ω) as t → ∞, then, by the integral Toeplitz lemma
(see Küchler and Sørensen [30, Lemma B.3.2]), we have
Rt
0

Rt

Here

0

e−bu du =

(7.7)

e−bu du

e−bt −1
−b ,
bt

e

Z

1

t
0

e−bu (ebu Yu (ω)) du → V (ω)

as t → ∞.

t ∈ R+ , thus we conclude

Z

0

t

Rt
1 − ebt 0 Yu du a.s. V
Yu du =
−→ −
Rt
−b
b
e−bu du

as t → ∞.

0

Now we turn to calculate the Laplace transform of V . Since P(limt→∞ ebt Yt = V ) = 1, we also
D

have ebt Yt −→ V as t → ∞, and, by continuity theorem, limt→∞ E(exp{uebt Yt }) = euV for all
u ∈ R− . By Corollary 3.4,

 
Z t
Z ∞

zψuebt ,0 (s)
bt
E(exp{ue Yt }) = exp ψuebt ,0 (t)y0 +
aψuebt ,0 (s) +
e
− 1 m(dz) ds
0

0

for each u ∈ R− and t ∈ R+ , where we have
ψuebt ,0 (t) =

σ 2 u(1

u
2ub
→
2
bt
− e ) + 2b
1 + σ2bu

as t → ∞.

By the monotone convergence theorem we obtain
Z

t

0

ψuebt ,0 (s) ds =
→

Z

t
0

Z

2ubeb(t−s)
ds =
σ 2 u(eb(t−s) − ebt ) + 2b

∞
0

0

0

∞

zψuebt ,0 (s)

e

t
0

2ubeby
dy
σ 2 u(eby − ebt ) + 2b


2
σ2u 
2ubeby
dy
=
−
log
1
+
σ 2 ueby + 2b
σ2
2b

where we used that the function R++ ∋ t 7→
y ∈ R++ and u, b ∈ R−− . In a similar way,
Z t Z

Z

2ubeby
σ2 u(eby −ebt )+2b


Z t Z

− 1 m(dz) ds =

∞



as t → ∞,

∈ R−− is monotone decreasing for all




2zubeb(t−s)
exp 2 b(t−s)
− 1 m(dz) ds
σ u(e
− ebt ) + 2b
0
0




Z t Z ∞ 
2zubeby
=
− 1 m(dz) dy
exp 2 by
σ u(e − ebt ) + 2b
0
0




Z ∞ Z ∞ 
2zubeby
− 1 m(dz) dy
→
exp 2 by
σ ue + 2b
0
0

as t → ∞, yielding (7.1).

26

a.s. e
e a.s. e
e as t → ∞ with
Particularly, we obtain ebt Yet −→ V
and ebt Yet −→ V
e
uV

E(e

e
e
uV

E(e

) = exp

) = exp



uy0

σ2 u

1 + 2b
Z ∞ Z
0



0

∞

σ2u
1+
2b

exp



− 2a2
σ

,

zueby
1+

σ2 u by
2b e




 
− 1 m(dz) dy

D e
D
e
e The equality V
e=
for all u ∈ R− , and we conclude V = V
+ V.
Z− 1 readily follows from Corollary
b
3.4, since
(
)
(
)
− 2a2
Z t
σ
uy0
u
uy0
σ2 u
uZt
E(e ) = exp
t
+
a
ds
=
exp
1
−
2
2
2
σ u
2
1 − σ2u t
1 − σ2u t
0 1− 2 s

for t ∈ R+ , u ∈ R− , and hence
uZ−1/b

E(e

) = exp

(

uy0
1+

σ2 u
2b

)

σ2 u
1+
2b

− 2a2
σ

,

u ∈ R− .

The monotone convergence theorem yields E(euV ) ↓ E(1{V =0} ) = P(V = 0) as u → −∞. We
have






y0
2by0
uy0
∈ R++
as u → −∞
=
exp
↓
exp
exp
2
2
1
σ2
σ
1 + σ2bu
+
u
2b
and



σ2u
1+
2b

− 2a2
σ

↓

(
1
0

if a = 0,
if a ∈ R++

as u → −∞.

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




zeby
zueby
− 1 = exp 1 σ2
− 1 ∈ (−1, 0]
R− ∋ u 7→ exp
2
by
+
e
1 + σ2bu eby
u
2b
is monotone increasing, hence, again by the monotone convergence theorem, we obtain
(






Z 
Z 
0
if m = 0,
2bz
zueby
− 1 m(dz) dy ↓
exp
exp
− 1 m(dz) dy =
2u
2
σ
σ
1 + 2b eby
−∞ if m =
6 0
R2+
R2+
as u → −∞. This yields
exp

Z

0

∞ Z ∞ 
0

exp



zueby
1+

σ2 u
2b

eby




  (
1
− 1 m(dz) dy ↓
0

if m = 0,
if m 6= 0

as u → −∞.

Summarizing, by (7.1), we conclude that P(V = 0) = 0 if and only if a ∈ R++ or m 6= 0.
Consequently, P(V ∈ R++ ) = 1 if and only if a ∈ R++ or m =
6 0.

e is absolutely continuous (see,
If a ∈ R++ , then the distribution of Z− 1 (and hence that of V)
b
e.g., Alfonsi [1, Proposition 1.2.11]), and consequently V is absolutely continuous.
✷
27

7.3 Theorem. Let a ∈ R+ , b ∈ R−− , σ ∈ R++ , and let m be a Lévy measure on R++ satisfying
(1.2). Let (Yt )t∈R+ be the unique strong solution of the SDE (1.1) satisfying P(Y0 = y0 ) = 1 with
some y0 ∈ R+ . Suppose that a ∈ R++ or a = 0, m 6= 0 and y0 ∈ R++ . Then the MLE of b is
a.s.
strongly consistent, i.e., b
bT −→ b as T → ∞. Further,


V −1/2
D
−bT /2 b
as T → ∞,
e
(bT − b) −→ σZ −
b

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
1/2
Z T
1
D
(bbT − b) −→ N (0, 1)
Ys ds
σ 0

as T → ∞.

Proof. By Proposition 4.2, there exists a unique MLE b
bT 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 limT →∞ T ebT = 0, we have
bT
bT
bT
bT JT
a.s. −V
bbT = −e YT + e y0R + aT e + T e T −→
=b
T
V
bT
−
e
Y
ds
s
b
0

as T → ∞,

as desired. Further, by the second equality in (5.2),
RT √
ebT /2 0 Ys dWs
−bT /2 b
,
T ∈ R++ .
e
(bT − b) = −σ
RT
ebT 0 Ys ds
1/2
RT
a.s.
By Theorem 7.1, ebT 0 Ys ds −→ − Vb as T → ∞, and using Theorem C.2 with η := − Vb
and
V
with v := − b , we have
!

1/2


Z Tp
V
V
V
D
as T → ∞.
Z, −
−→
−
ebT /2
Ys dWs , −
b
b
b
0
Consequently,

Z
bT /2
(7.8)
e

T
0

p

bT

Ys dWs , e

Z

T

Ys ds
0



D

−→



V
−
b

D

1/2

By the continuous mapping theorem, e−bT /2 (bbT − b) −→ −σZ
first assertion.

V
Z, −
b
−V
b

!

−1/2

as T → ∞.
as T → ∞, yielding the

Applying again (7.8) and the continuous mapping theorem, we obtain

1/2
−1/2
Z T
Z Tp
Z T
1
bT /2
bT
b
Ys dWs
e
(bT − b) = − e
Ys ds
Ys ds
σ 0
0
0




V −1/2
V 1/2
D
D
−→ − −
Z = −Z = N (0, 1)
−
b
b

as desired.

as T → ∞,
✷

Appendices
28

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 η0 and η0′ be random variables independent of (Wt )t∈R+ and (Jt )t∈R+
satisfying P(η0 ∈ R+ ) = 1 and P(η0′ ∈ R+ ) = 1. Let (Yt )t∈R+ be a pathwise unique strong solution
of the SDE (1.1) such that P(Y0 = η0 ) = 1. Let (Yt′ )t∈R+ be a pathwise unique strong solution of
the SDE
q
t ∈ R+ ,
(A.1)
dYt′ = (a − bYt′ ) dt + σ Yt′ dWt ,
such that P(Y0′ = η0′ ) = 1. Then P(η0 > η0′ ) = 1 implies P(Yt > Yt′ for all t ∈ R+ ) = 1.

Proof. We follow the ideas of the proof of Theorem 3.1 of Ma [39], which is an adaptation of that
of Theorem 5.5 of Fu and Li [18]. There is a sequence φk : R → R+ , k ∈ N, of twice continuously
differentiable functions such that
(i) φk (z) ↑ z + := max(z, 0) as k → ∞;
(ii) φ′k (z) ∈ [0, 1] for all z ∈ R+ and k ∈ N;
(iii) φ′k (z) = φk (z) = 0 whenever z ∈ R− and k ∈ N;
√
√
(iv) φ′′k (x − y)( x − y)2 6 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 [39]. Let Zt := Yt′ − Yt
for all t ∈ R+ . By (1.1), (2.2) and (A.1), we have
Z tZ ∞
Z t p
Z t
p 
Ys′ − Ys dWs −
z µJ (ds, dz),
t ∈ R+ .
Zs ds + σ
Zt = Z0 − b
For each j ∈ N, put

0

0

0

0

n
o
τj := inf t ∈ R+ : max{Yt , Yt′ } > j .

By Itô’s formula (which can be used since Y and Y ′ are adapted to the augmented filtration
corresponding to (Wt )t∈R+ and (Jt )t∈R+ constructed as in Section 3 of Barczy et al. [4]), we obtain
φk (Zt∧τj ) = φk (Z0 ) +

4
X

Ij,k,ℓ(t),

ℓ=1

with
Ij,k,1(t) := −b
Ij,k,2(t) := σ

Z

Z

t∧τj

0

t∧τj
0

t ∈ R+ ,

φ′k (Zs )Zs ds,

φ′k (Zs )

p

Ys′ −

p 
Ys dWs ,

Z
p 2
p
σ 2 t∧τj ′′
φk (Zs ) Ys′ − Ys ds,
2 0
Z t∧τj Z ∞
Ij,k,4(t) :=
[φk (Zs− − z) − φk (Zs− )] µJ (ds, dz).
Ij,k,3(t) :=

0

0

29

Using formula (3.8) in Chapter II in Ikeda and Watanabe [20], the last integral can be written as
Ij,k,4(t) = Ij,k,4,1 (t) + Ij,k,4,2 (t), where
Z t∧τj Z ∞
Ij,k,4,1 (t) :=
[φk (Zs− − z) − φk (Zs− )] µ
eJ (ds, dz),
0

Ij,k,4,2 (t) :=

Z

0

t∧τj

0

Z

∞

0

[φk (Zs− − z) − φk (Zs− )] ds m(dz),

where µ
eJ (ds, dz) := µJ (ds, dz) − ds m(dz), since

Z t∧τj Z ∞
|φk (Zs− − z) − φk (Zs− )| ds m(dz)
E
0

(A.2)

6E

0

Z

t∧τj

0

Z

∞
0


Z
z ds m(dz) 6 t

∞
0

z m(dz) < ∞,

where we used that, by properties (ii) and (iii) of the function φk , we have φ′k (u) ∈ [0, 1] for all
u ∈ R, and hence, by mean value theorem,
0 6 φk (y) − φk (y − z) 6 z,

(A.3)

y ∈ R,

z ∈ R+ ,

k ∈ N.

One can check that the process (Ij,k,2 (t) + Ij,k,4,1 (t))t∈R+ is a martingale. Indeed, by properties
(ii) and (iii) of the function φk and the definition of τj ,
Z t∧τj 

Z t∧τj
p 2 
p
′
′
′
φk (Zs ) Ys − Ys
E
(Ys + Ys ) ds 6 4jt < ∞,
ds 6 2 E
0

0

hence, by Ikeda and Watanabe [20, page 55], (Ij,k,2(t))t∈R+ is a martingale. The inequality (A.2)
yields that the function R+ × R++ × Ω ∋ (s, z, ω) 7→ φk (Zs− (ω) − z) − φk (Zs− (ω))1(−∞,τj ) (s) belongs
to the class F 1p , and hence (Ij,k,4,1(t))t∈R+ is a martingale by Ikeda and Watanabe [20, page 62].
Moreover,
Ij,k,1 (t) 6 |b|

Z

t∧τj
0

Zs+ ds.

By (iv),
Ij,k,3 (t) 6 (t ∧ τj )

σ2
σ2t
6
.
k
k

By (A.3), we obtain
Ij,k,4,2(t) 6 0.
Summarizing, we have
(A.4)

φk (Zt∧τj ) 6 φk (Z0 ) + |b|

Z

t∧τj
0

Zs+ ds +

σ2 t
+ Ij,k,2(t) + Ij,k,4,1(t),
k

t ∈ R+ .

By (iii), we obtain P(φk (Z0 ) = 0) = 1. By (i), the non-negativeness of φk and monotone convergence
+
) as k → ∞ for all t ∈ R+ and j ∈ N. We have
theorem yield E(φk (Zt∧τj )) → E(Zt∧τ
j
R t∧τj +
Rt +
Zs ds 6 0 Zs∧τj ds, hence taking the expectation of (A.4) and letting k → ∞, we obtain
0
E

+
Zt∧τ
j



6 |b|

Z

t

0

30


+
E Zs∧τ
ds.
j

By Gronwall’s inequality, we conclude


+
E Zt∧τ
=0
j

′
for all t ∈ R+ and j ∈ N. Hence P(Yt∧τ
6 Yt∧τj ) = 1 for all t ∈ R+ and j ∈ N, and then
j
′
P(Yt∧τj 6 Yt∧τj for all j ∈ N) = 1 for all t ∈ R+ . Since Y and Y ′ have càdlàg trajectories, these
a.s.

trajectories are bounded almost surely on [0, T ] for all T ∈ R+ , hence τj −→ ∞ as j → ∞.
This yields P(Yt′ 6 Yt ) = 1 for all t ∈ R+ . Since the set of non-negative rational numbers Q+
is countable, we obtain P(Yt′ 6 Yt for all t ∈ Q+ ) = 1. Using again that Y and Y ′ have càdlàg
✷
trajectories almost surely, we get P(Yt′ 6 Yt for all t ∈ R+ ) = 1.

B

Likelihood-ratio process

Based on Jacod and Shiryaev [23], see also Jacod and Mémin [21], Sørensen [45] and Luschgy [38],
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. [2],
we decided to present it here as well for better readability and being self-contained.
Let D(R+ , Rd ) denote the space of Rd -valued càdlàg functions defined on R+ . Let (ηt )t∈R+
denote the canonical process ηt (ω) := ω(t), ω ∈ D(R+ , Rd ), t ∈ R+ . Put Ftη := σ(ηs , s ∈ [0, t]),
t ∈ R+ , and
!
[ η
\
η
d
d
Ft .
Ft+ε , t ∈ R+ ,
D(R+ , R ) := σ
Dt (R+ , R ) :=
t∈R+

ε∈R++

Let Ψ ⊂ Rk be an arbitrary non-empty set, and let Pψ , ψ ∈ Ψ, are probability measures on the
canonical space (D(R+ , Rd ), D(R+ , Rd )). Suppose that for each ψ ∈ Ψ, under Pψ , the canonical
process (ηt )t∈R+ is a semimartingale with semimartingale characteristics (B (ψ) , C, ν (ψ) ) associated
with a fixed Borel measurable truncation function h : Rd → Rd , see Jacod and Shiryaev [23, Definition
II.2.6 and Remark II.2.8]. Namely, Ct := h(η cont )(ψ) it , t ∈ R+ , where (h(η cont )(ψ) it )t∈R+ denotes
the (predictable) quadratic variation process (with values in Rd×d ) of the continuous martingale part
(η cont )(ψ) of η under Pψ , ν (ψ) is the compensator of the integer-valued random measure µη on
R+ × Rd associated with the jumps of η under Pψ given by
X
1{∆ηs (ω)6=0} ε(s,∆ηs (ω)) (dt, dx),
µη (ω, dt, dx) :=
ω ∈ D(R+ , Rd ),
s∈R+

where ε(t,x) denotes the Dirac measure at the point (t, x) ∈ R+ × Rd , and ∆ηt := ηt − ηt− , t ∈ R++ ,
∆η0 := 0, and B (ψ) is the predictable process (with values in Rd having finite variation over each
finite interval [0, t], t ∈ R+ ) appearing in the canonical decomposition
(ψ)

ηet = η0 + Mt

(ψ)

+ Bt

,

t ∈ R+ ,

of the special semimartingale (e
ηt )t∈R+ under Pψ given by
ηet := ηt −

X

s∈[0,t]

(ηs − h(∆ηs )),
31

t ∈ R+ ,

(ψ)

(ψ)

where (Mt )t∈R+ is a local martingale with M0 = 0. We call the attention that, by our
assumption, the process C = h(η cont )(ψ) i does not depend on ψ, although (η cont )(ψ) might depend
on ψ. In addition, assume that Pψ (ν (ψ) ({t} × Rd ) = 0) = 1 for every ψ ∈ Ψ, t ∈ R+ , and
Pψ (η0 = x0 ) = 1 with some x0 ∈ Rd for every ψ ∈ Ψ. Note that we have the semimartingale
representation
Z tZ
(ψ)
cont (ψ)
h(x) (µη − ν (ψ) )(ds, dx)
)t +
ηt = x0 + Bt + (η
Rd

0

+

Z tZ
0

Rd

(x − h(x)) µη (ds, dx),

t ∈ R+ ,

of η under Pψ , see Jacod and Shiryaev [23, Theorem II.2.34]. Moreover, for each ψ ∈ Ψ, let us
(ψ)
(ψ)
choose a nondecreasing, continuous, adapted process (Ft )t∈R+ with F0 = 0 and a predictable
(ψ)
process (ct )t∈R+ with values in the set of all symmetric positive semidefinite d × d matrices such
that
Z
t

cs(ψ) dFs(ψ)

Ct =

0

Pψ -almost sure for every t ∈ R+ . Due to the assumption Pψ (ν (ψ) ({t} × Rd ) = 0) = 1 for
(ψ)
(ψ)
every ψ ∈ Ψ, t ∈ R+ , such choices of (Ft )t∈R+ and (ct )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
e ∈ Ψ, there exist a P ⊗ B(Rd )-measurable function
D(R+ , Rd )× R+ . Assume also that for every ψ, ψ
e
e
V (ψ,ψ) : D(R+ , Rd ) × R+ × Rd → R++ and a predictable Rd -valued process β (ψ,ψ) satisfying
e

(B.2)

Rd

0

(B.3)

e

ν (ψ) (dt, dx) = V (ψ,ψ) (t, x)ν (ψ) (dt, dx),
Z t Z q
2
e
e
V (ψ,ψ)
(s, x) − 1 ν (ψ) (ds, dx) < ∞,

(B.1)

(ψ)

Bt

e
(ψ)

= Bt

+

Z

t

0

e

cs(ψ) βs(ψ,ψ) dFs(ψ) +
Z

(B.4)

t
0

Z tZ

e

0

Rd

e

e

(V (ψ,ψ) (s, x) − 1)h(x) ν (ψ) (ds, dx),

e

(βs(ψ,ψ) )⊤ cs(ψ) βs(ψ,ψ) dFs(ψ) < ∞,

Pψ -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 x0 with Pψ as its unique solution. Then for each T ∈ R+ , Pψ,T is absolutely
continuous with respect to Pψ,T
e , where Pψ,T := Pψ|DT (R+ ,Rd ) denotes the restriction of Pψ to
d
DT (R+ , R ) (similarly for Pψ,T
e ), and, under Pψ,T
e , the corresponding likelihood-ratio process takes
the form
Z T
Z
d Pψ,T
1 T (ψ,ψ)
e
e
e
e
(βs(ψ,ψ) )⊤ d(η cont )s(ψ) −
(η) =
(βs
)⊤ cs(ψ) βs(ψ,ψ) dFs(ψ)
log
d Pψ,T
2
e
0
0
(B.5)

+
+

Z

Z

T

0
T
0

Z

Z

e

e

Rd

(V (ψ,ψ) (s, x) − 1) (µη − ν (ψ) )(ds, dx)

Rd

(log(V (ψ,ψ) (s, x)) − V (ψ,ψ) (s, x) + 1) µη (ds, dx)

e

e

32

for all T ∈ R++ , see Jacod and Shiryaev [23, Theorem III.5.34]. A detailed proof of (B.5) using
Jacod and Shiryaev [23] can be found in Barczy et al. [2, Appendix A].

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, (Ft )t∈R+ , P be a filtered
probability space satisfying the usual conditions. Let (Mt )t∈R+ be a square-integrable continuous local
martingale with respect to the filtration (Ft )t∈R+ such that P(M0 = 0) = 1. Let (ξt )t∈R+ be a
progressively measurable process such that

Z t
2
ξu dhM iu < ∞ = 1,
t ∈ R+ ,
P
0

and
(C.1)

Z

t
0

a.s.

ξu2 dhM iu −→ ∞

as t → ∞,

where (hM it )t∈R+ denotes the quadratic variation process of M . Then
Rt
ξu dMu a.s.
(C.2)
−→ 0
as t → ∞.
R t0
2
0 ξu dhM iu

If (Mt )t∈R+ is a standard Wiener process, the progressive measurability of (ξt )t∈R+ can be relaxed
to measurability and adaptedness to the filtration (Ft )t∈R+ .
The next theorem is about the asymptotic behaviour of continuous multivariate local martingales,
see van Zanten [46, Theorem 4.1].

C.2 Theorem. (van Zanten [46, Theorem 4.1]) Let Ω, F, (Ft )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 (Ft )t∈R+ such that P(M 0 = 0) = 1. Suppose that
there exists a function Q : R+ → Rd×d such that Q(t) is an invertible (non-random) matrix for all
t ∈ R+ , limt→∞ kQ(t)k = 0 and
P

Q(t)hM it Q(t)⊤ −→ ηη ⊤

as t → ∞,

where η is a d× d random matrix. Then, for each Rk -valued random vector v defined on (Ω, F, P),
we have
D
(Q(t)M t , v) −→ (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 [t0 , ∞)
with some t0 ∈ R++ .
33

Acknowledgements
We would like to thank the referee for his/her comments that helped us to improve the paper.

References
[1] Alfonsi, A., (2005). On the discretization schemes for the CIR (and Bessel squared) processes.
Monte Carlo Methods and Applications 11(4) 355–384.
[2] Barczy, M., Ben Alaya, M., Kebaier, A. and Pap, G. (2016). Asymptotic behavior of maximum likelihood estimators for a jump-type Heston model. Available on ArXiv:
http://arxiv.org/abs/1509.08869
[3] Barczy, M., Döring, L., Li, Z. and Pap, G. (2013). On parameter estimation for critical
affine processes. Electronic Journal of Statistics 7 647–696.
[4] Barczy, M., Li, Z. and Pap, G. (2015). Yamada–Watanabe results for stochastic differential
equations with jumps. International Journal of Stochastic Analysis Volume 2015 Article ID
460472, 23 pages.
[5] Barczy, M., Li, Z. and Pap, G. (2015). Stochastic differential equation with jumps for multitype continuous state and continuous time branching processes with immigration. ALEA. Latin
American Journal of Probability and Mathematical Statistics. 12(1) 129–169.
[6] Ben Alaya, M. and Kebaier, A. (2012). Parameter estimation for the square root diffusions:
ergodic and nonergodic cases. Stochastic Models 28(4) 609–634.
[7] Ben Alaya, M. and Kebaier, A. (2013). Asymptotic behavior of the maximum likelihood
estimator for ergodic and nonergodic square-root diffusions. Stochastic Analysis and Applications
31(4) 552–573.
[8] Billingsley, P. (1999). Convergence of probability measures, 2nd ed. John Wiley & Sons, Inc.,
New York.
[9] Bhattacharya, R. N. (1982). On the functional central limit theorem and the law of the iterated
logarithm for Markov processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
60 185–201.
[10] Cox, J. C., Ingersoll, J. E. and Ross, S. A. (1985). A theory of the term structure of interest
rates. Econometrica 53(2) 385–407.
[11] Dawson, D. A. and Li, Z. (2006). Skew convolution semigroups and affine Markov processes.
The Annals of Probability 34(3) 1103–1142.
[12] Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth & Brooks/Cole Advanced
Books & Software, Pacific Grove, California.
[13] Duffie, D. and Gârleanu, N. (2001). Risk and valuation of collateralized debt obligations.
Financial Analysts Journal 57(1) 41–59.
34

[14] Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications
in finance. Annals of Applied Probability 13 984–1053.
[15] Feller, W. (1951). Two singular diffusion problems. Annals of Mathematics 54(1) 173–182.
[16] Filipović, D. (2001). A general characterization of one factor affine term structure models.
Finance and Stochastics 5(3) 389–412.
[17] Filipović, D., Mayerhofer, E. and Schneider, P. (2013). Density approximations for multivariate affine jump-diffusion processes. Journal of Econometrics 176(2) 93–111.
[18] Fu, Z. and Li, Z. (2010). Stochastic equations of non-negative processes with jumps. Stochastic
Processes and their Applications 120 306–330.
[19] Huang, J., Ma, C. and Zhu, C. (2011). Estimation for discretely observed continuous state
branching processes with immigration. Statistics and Probability Letters 81 1104–1111.
[20] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes,
2nd ed. North-Holland/Kodansha, Amsterdam/Tokyo.
[21] Jacod, J. and Mémin, J. (1976). Caractéristiques locales et conditions de continuité absolue
pour les semi-martingales. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 35
1–37.
[22] Jacod, J. and Protter, P. (2012). Discretization of processes, Springer, Heidelberg.
[23] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed.
Springer-Verlag, Berlin.
[24] Jiao, Y., Ma, C. and Scotti, S. (2016). Alpha-CIR model with branching processes in sovereign
interest rate modelling. Available on ArXiv: http://arxiv.org/abs/1602.05541
[25] Jin, P., Rüdiger, B. and Trabelsi, C. (2016). Exponential ergodicity of the jump-diffusion
CIR process. In: Stochastics of environmental and financial economics—Centre of Advanced
Study, Oslo, Norway, 2014–2015, pages 285–300, Springer Proceedings in Mathematics & Statistics, 138, Springer, Cham.
[26] Keller-Ressel, M. (2008). Affine Processes - Theory and Applications in Finance. PhD Thesis,
Vienna University of Technology, pages 110.
[27] Keller-Ressel, M. (2011). Moment explosions and long-term behavior of affine stochastic
volatility models. Mathematical Finance 21(1) 73–98.
[28] Keller-Ressel, M. and Mijatović, A. (2012). On the limit distributions of continuous-state
branching processes with immigration. Stochastic Processes and their Applications 122 2329–
2345.
[29] Keller-Ressel, M. and Steiner, T. (2008). Yield curve shapes and the asymptotic short rate
distribution in affine one-factor models. Finance and Stochastics 12(2) 149–172.
[30] Küchler, U. and Sørensen, M. (1997). Exponential families of stochastic processes, SpringerVerlag, New York.
35

[31] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd ed. SpringerVerlag, Berlin Heidelberg.
[32] Lamberton, D. and Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall/CRC.
[33] Li, Z. (2011). Measure-Valued Branching Markov Processes. Springer-Verlag, Heidelberg.
[34] Li,
Z.
(2012).
Continuous-state
http://arxiv.org/abs/1202.3223

branching

processes.

Available

on

ArXiv:

[35] Li, Z. and Ma, C. (2013). Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross
model. Stochastic Processes and their Applications 125(8) 3196–3233.
[36] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes II. Applications,
2nd edition. Springer-Verlag, Berlin, Heidelberg.
[37] Luschgy, H. (1992). Local asymptotic mixed normality for semimartingale experiments. Probability Theory and Related Fields 92 151–176.
[38] Luschgy, H. (1994). Asymptotic inference for semimartingale models with singular parameter
points. Journal of Statistical Planning and Inference 39 155–186.
[39] Ma, R. (2013). Stochastic equations for two-type continuous-state branching processes with
immigration. Acta Mathematica Sinica, English Series 29(2) 287–294.
[40] Mai, H. (2012). Drift estimation for jump diffusions: time-continuous and high-frequency observations. Ph.D. Dissertation. Humboldt-Universität zu Berlin.
[41] Overbeck, L. (1998). Estimation for continuous branching processes. Scandinavian Journal of
Statistics 25(1) 111–126.
[42] Overbeck, L. and Rydén, T. (1997). Estimation in the Cox-Ingersoll-Ross model. Econometric
Theory 13(3) 430–461.
[43] Pinsky, M. A. (1972). Limit theorems for continuous state branching processes with immigration. Bulletin of the American Mathematical Society 78(2) 242–244.
[44] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University
Press, Cambridge.
[45] Sørensen, M. (1991). Likelihood methods for diffusions with jumps. In: N. U. Prabhu and I.V.
Basawa, Eds., Statistical Inference in Stochastic Processes, Marcel Dekker, New York, 67–105.
[46] van Zanten, H. (2000). A multivariate central limit theorem for continuous local martingales.
Statistics & Probability Letters 50(3) 229–235.

36

