2 Special Case (ii): θ=0 and x=0
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3 A Lower Bound for the Transition Densities of JCIR
In this section we will find some conditions on the Lévy measure ν of (Jt , t ≥ 0)
such that an explicit lower bound for the transition densities of the JCIR process
given in (3) can be derived. As a first step we show that the law of X tx , t > 0, in (3)
is absolutely continuous with respect to the Lebesgue measure and thus possesses a
density function.
Lemma 1 Consider the JCIR process (X tx , t ≥ 0) (started from x ≥ 0) that is
defined in (3). Then for any t > 0 and x ≥ 0 the law of X tx is absolutely continuous with respect to the Lebesgue measure and thus possesses a density function
p(t, x, y), y ≥ 0.
Proof As shown in the previous section, it holds
x
x
E[eu X t ] = ϕ1 (t, u, x)ϕ2 (t, u) = E[euYt ]E[eu Z t ],
therefore the law of X tx , denoted by μ X tx , is the convolution of the laws of Ytx and Z t .
Since (Ytx , t ≥ 0) is the well-known CIR process and has transition density functions
f (t, x, y), t > 0, x, y ≥ 0 with respect to the Lebesgue measure, thus μ X tx is also
absolutely continuous with respect to the Lebesgue measure and possesses a density
function.
In order to get a lower bound for the transition densities of the JCIR process we
need the following lemma.
Lemma 2 Suppose that (0,1) ξ ln(1/ξ )ν(dξ ) < ∞. Then ϕ2 defined by (8) is the
characteristic function of a compound Poisson distribution. In particular, P(Z t =
0) > 0 for all t > 0, where (Z t , t ≥ 0) is defined by (14).
Proof It follows from (6), (8) and (15) that
t
E[eu Z t ] = ϕ2 (t, u) = exp
0
(0,∞)
exp
ξ ue−as
1−(σ 2 /2a)(1−e−as )u
− 1 ν(dξ )ds ,
where u ∈ U . Define
t
Δ :=
0
(0,∞)
exp
ξ ue−as
− 1 ν(dξ )ds.
1 − (σ 2 /2a)(1 − e−as )u
If we rewrite
exp
αu
ξ e−as u
= exp
,
2
−as
1 − (σ /2a)(1 − e )u
β −u
(16)
Exponential Ergodicity of the Jump-Diffusion CIR Process
⎧
⎪
⎪
⎨α :=
where
291
2aξ
> 0,
− 1)
2aeas
⎪
⎪
⎩β :=
> 0,
σ 2 (eas − 1)
σ 2 (eas
(17)
then we recognize that the right-hand side of (16) is the characteristic function of a
Bessel distribution with parameters α and β. Recall that a probability measure μα,β
on R+ , B(R+ ) is called a Bessel distribution with parameters α and β if
μα,β (d x) = e−α δ0 (d x) + βe−α−βx
α
I1 (2 αβx)d x,
βx
(18)
where δ0 is the Dirac measure at the origin and I1 is the modified Bessel function of
the first kind, namely
I1 (r ) =
r
2
∞
k=0
1 2 k
4r
k!(k + 1)!
,
r ∈ R.
For more properties of Bessel distributions we refer the readers to [8, Sect. 3] (see
also [4, p. 438] and [9, Sect. 3]). Let μˆ α,β denote the characteristic function of the
Bessel distribution μα,β with parameters α and β which are defined in (17). It follows
from [9, Lemma 3.1] that
ξ e−as u
αu
= exp
.
β −u
1 − (σ 2 /2a)(1 − e−as )u
μˆ α,β (u) = exp
Therefore
t
Δ=
(0,∞)
0
t
=
0
μˆ α,β (u) − 1 ν(dξ )ds
αu
(0,∞)
e β−u − e−α + e−α − 1 ν(dξ )ds.
Set
t
λ :=
(0,∞)
0
t
=
0
(0,∞)
1 − e−α ν(dξ )ds
1−e
−
2aξ
σ 2 (eas −1)
ν(dξ )ds.
(19)
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If λ < ∞, then
t
Δ=
αu
0
1
=λ
λ
e β−u − e−α ν(dξ )ds − λ
(0,∞)
t
0
αu
(0,∞)
e β−u − e−α ν(dξ )ds − 1 .
The fact that λ < ∞ will be shown later in this proof.
t
Next we show that the term (1/λ) 0 (0,∞) exp αu/(β−u) −exp(−α) ν(dξ )ds
can be viewed as the characteristic function of a probability measure ρ. To define ρ,
we first construct the following measures
α
I1 (2 αβx)d x, x ≥ 0,
βx
m α,β (d x) := βe−α−βx
where I1 is the modified Bessel function of the first kind. Noticing that the measure
m α,β is the absolute continuous component of the measure μα,β in (18), we easily
get
αu
mˆ α,β (u) = μˆ α,β (u) − e−α = e β−u − e−α ,
∞
where mˆ α,β (u) := 0 eux m α,β (d x) for u ∈ U . Recall that the parameters α and β
defined by (17) depend on the variables ξ and s. We can define a measure ρ on R+
as follows:
1 t
ρ :=
m α,β ν(dξ )ds.
λ 0 (0,∞)
By the definition of the constant λ in (19) we get
ρ(R+ ) =
1
λ
1
λ
= 1,
t
(0,∞)
0
t
=
0
(0,∞)
m α,β (R+ )ν(dξ )ds
(1 − e−α )ν(dξ )ds
i.e. ρ is a probability measure on R+ , and for u ∈ U
ρ(u)
ˆ
=
=
(0,∞)
1 t
λ
0
eux ρ(d x)
αu
(0,∞)
(e β−u − e−α )ν(dξ )ds.
ˆ
is the characteristic function of a
Thus Δ = λ(ρ(u)
ˆ
− 1) and E[eu Z t ] = eλ(ρ(u)−1)
compound Poisson distribution.
Exponential Ergodicity of the Jump-Diffusion CIR Process
293
Now we verify that λ < ∞. Noticing that
t
λ=
t
=
0
=
1 − e−α ν(dξ )ds
(0,∞)
0
1−e
(0,∞)
t
1−e
(0,∞) 0
we introduce the change of variables
−
2aξ
σ 2 (eas −1)
−
2aξ
σ 2 (eas −1)
ν(dξ )ds
dsν(dξ ),
2aξ
:= y and then get
σ 2 (eas − 1)
2aξ
aeas ds
σ 2 (eas − 1)2
σ 2 2aξ
+ 1 ds.
= −y 2
2ξ σ 2 y
dy = −
Therefore
λ=
(0,∞)
=
(0,∞)
=
where δ :=
(0,∞)
2aξ
σ 2 (eat −1)
ν(dξ )
∞
∞
ν(dξ )
2aξ
σ 2 (eat −1)
∞
ν(dξ )
ξ
δ
−2ξ
dy
2aξ y + σ 2 y 2
2ξ
(1 − e−y )
dy
2aξ y + σ 2 y 2
(1 − e−y )
(1 − e−y )
2ξ
dy,
2aξ y + σ 2 y 2
σ 2 (eat − 1)
. Define
2a
M(ξ ) :=
∞
ξ
δ
(1 − e−y )
2ξ
dy.
2aξ y + σ 2 y 2
Then M(ξ ) is continuous on (0, ∞). As ξ → 0 we get
M(ξ ) =
1
ξ
δ
≤ 2ξ
≤ 2ξ
(1 − e−y )
1
ξ
δ
1
ξ
δ
2ξ
dy + 2ξ
2aξ y + σ 2 y 2
y
dy + 2ξ
2aξ y + σ 2 y 2
1
dy + 2ξ
2aξ + σ 2 y
∞
1
∞
1
∞
(1 − e−y )
1
dy
2aξ y + σ 2 y 2
1
dy
2aξ y + σ 2 y 2
1
dy.
σ 2 y2
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P. Jin et al.
Since
1
2ξ
ξ
δ
2ξ
1
dy = 2 ln(2aξ + σ 2 y)
2aξ + σ 2 y
σ
1
ξ
δ
2ξ
2ξ
σ 2ξ
ln(2aξ + σ 2 ) − 2 ln(2aξ +
)
2
σ
σ
δ
1
1
≤ c1 ξ + c2 ξ ln( ) ≤ c3 ξ ln( )
ξ
ξ
=
for sufficiently small ξ , we conclude that
1
M(ξ ) ≤ c4 ξ ln( ), as ξ → 0.
ξ
If ξ → ∞, then
∞
M(ξ ) ≤
ξ
δ
≤
∞
ξ
δ
(1 − e−y )
2ξ
dy
2aξ y + σ 2 y 2
2ξ
dy ≤ 2ξ
2aξ y + σ 2 y 2
∞
2ξ
1
1
d(− ) = 2 −
y
σ
y
=
2ξ
σ2
=
2δ
2ξ δ
= 2 := c5 < ∞.
2
σ ξ
σ
Therefore,
1
λ ≤ c4
0
ξ
δ
1
ξ ln( )ν(dξ ) + c5
ξ
∞
∞
ξ
δ
1
dy
σ 2 y2
∞
ξ
δ
1ν(dξ ) < ∞.
1
With the help of the Lemma 2 we can easily prove the following proposition.
Proposition 1 Let p(t, x, y), t > 0, x, y ≥ 0 denote the transition density of the
JCIR process (X tx , t ≥ 0) defined in (3). Suppose that (0,1) ξ ln( ξ1 )ν(dξ ) < ∞.
Then for all t > 0, x, y ≥ 0 we have
p(t, x, y) ≥ P(Z t = 0) f (t, x, y),
where P(Z t = 0) > 0 for all t > 0 and f (t, x, y) are transition densities of the CIR
process (without jumps).
Proof According to Lemma 2, we have P(Z t = 0) > 0. Since
x
x
E[eu X t ] = ϕ1 (t, u, x)ϕ2 (t, u) = E[euYt ]E[eu Z t ],
Exponential Ergodicity of the Jump-Diffusion CIR Process
295
the law of X tx , denoted by μ X tx , is the convolution of the laws of Ytx and Z t . Thus
for all A ∈ B(R+ )
μ X tx (A) =
≥
R+
{0}
μYtx (A − y)μ Z t (dy)
μYtx (A − y)μ Z t (dy)
≥ μYtx (A)μ Z t ({0})
≥ P(Z t = 0)μYtx (A)
≥ P(Z t = 0)
f (t, x, y)dy,
A
where f (t, x, y) are the transition densities of the classical CIR process given in
(12). Since A ∈ B(R+ ) is arbitrary, we get
p(t, x, y) ≥ P(Z t = 0) f (t, x, y)
for all t > 0, x, y ≥ 0.
4 Exponential Ergodicity of JCIR
In this section we find some sufficient conditions such that the JCIR process is
exponentially ergodic. We have derived a lower bound for the transition densities of
the JCIR process in the previous section. Next we show that the function V (x) = x,
x ≥ 0, is a Forster-Lyapunov function for the JCIR process.
Lemma 3 Suppose that (1,∞) ξ ν(dξ ) < ∞. Then the function V (x) = x, x ≥ 0,
is a Forster-Lyapunov function for the JCIR process defined in (3), in the sense that
for all t > 0, x ≥ 0,
E[V (X tx )] ≤ e−at V (x) + M,
where 0 < M < ∞ is a constant.
Proof We know that μ X tx = μYtx ∗ μ Z t , therefore
E[X tx ] = E[Ytx ] + E[Z t ].
Since (Ytx , t ≥ 0) is the CIR process starting from x, it is known that μYtx is a
non-central Chi-squared distribution and thus E[Ytx ] < ∞. Next we show that
E[Z t ] < ∞.
Let u ∈ (−∞, 0). By using Fatou’s Lemma we get
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eu Z t − 1
u→0
u
E[eu Z t ] − 1
eu Z t − 1
= lim inf
≤ lim inf E
.
u→0
u→0
u
u
E[Z t ] = E lim
Recall that
ξ ue−as
t
E[eu Z t ] = ϕ2 (t, u) = exp
0
e 1−(σ 2 /2a)(1−e−as )u − 1 ν(dξ )ds = eΔ(u) .
(0,∞)
Then we have for all u ≤ 0
∂
∂u
ξ ue−as
exp
1 − (σ 2 /2a)(1 − e−as )u
ξ e−as
=
1 − (σ 2 /2a)(1 − e−as )u
exp
2
ξ e−as
≤
1 − (σ 2 /2a)(1 − e−as )u
and further
t
(0,∞)
0
2
−1
ξ ue−as
1 − (σ 2 /2a)(1 − e−as )u
≤ ξ e−as
ξ e−as ν(dξ )ds < ∞.
Thus Δ(u) is differentiable in u and
t
Δ (0) =
0
(0,∞)
ξ e−as ν(dξ )ds =
1 − e−at
a
(0,∞)
ξ ν(dξ ).
It follows that
ϕ2 (t, u) − ϕ2 (t, 0)
u
∂ϕ2 (t, u)
=
= eΔ(0) Δ (0)
u=0
∂u
1 − e−at
=
ξ ν(dξ ).
a
(0,∞)
E[Z t ] ≤ lim inf
u→0
Therefore under the assumption
∞. Furthermore,
E[Z t ] =
(0,∞)
∂
E[eu Z t ]
∂u
ξ ν(dξ ) < ∞ we have proved that E[Z t ] <
u=0
=
1 − e−at
a
(0,∞)
ξ ν(dξ ).
Exponential Ergodicity of the Jump-Diffusion CIR Process
297
On the other hand,
E[euYt ] = 1 − (σ 2 /2a)u(1 − e−at )
x
−2aθ/σ 2
exp
xue−at
.
1 − (σ 2 /2a)u(1 − e−at )
With a similar argument as above we get
E[Ytx ] =
x
∂
E[euYt ]
∂u
= θ (1 − e−at ) + xe−at .
u=0
Altogether we get
E[X tx ] = E[Ytx ] + E[Z t ]
= (1 − e−at ) θ +
≤θ+
1 − e−at
+ xe−at
a
1
+ xe−at ,
a
namely
E[V (X tx )] ≤ θ +
1
+ e−at V (x).
a
Remark 1 If (1,∞) ξ ν(dξ ) < ∞, then there exists a unique invariant probability
measure for the JCIR process. This fact follows from [12, Theorem 3.16] and [10,
Proposition 3.1].
Let ·
TV
denote the total-variation norm for signed measures on R+ , namely
μ
TV
=
sup
A∈B (R+ )
{|μ(A)|}.
Let P t (x, ·) := P(X tx ∈ ·) be the distribution of the JCIR process at time t started
from the initial point x ≥ 0. Now we prove the main result of this paper.
Theorem 1 Assume that
(1,∞)
ξ ν(dξ ) < ∞ and
1
ξ ln( )ν(dξ ) < ∞.
ξ
(0,1)
Let π be the unique invariant probability measure for the JCIR process. Then the
JCIR process is exponentially ergodic, namely there exist constants 0 < β < 1 and
0 < B < ∞ such that
P t (x, ·) − π
TV
≤ B x + 1 β t , t ≥ 0, x ∈ R+ .
(20)
Proof Basically, we follow the proof of [18, Theorem 6.1]. For any δ > 0 we
x , n ∈ Z , where Z denotes the set of all
consider the δ-skeleton chain ηnx := X nδ
+
+
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P. Jin et al.
non-negative integers. Then (ηnx )n∈Z+ is a Markov chain on the state space R+ with
transition kernel P δ (x, ·) and starting point η0x = x. It is easy to see that the measure
π is also an invariant probability measure for the chain (ηnx )n∈Z+ , x ≥ 0.
Let V (x) = x, x ≥ 0. It follows from the Markov property and Lemma 3 that
x
)|η0x , η1x , . . . , ηnx ] =
E[V (ηn+1
R+
V (y)P δ (ηnx , dy) ≤ e−aδ V (ηnx ) + M,
where M is a positive constant. If we set V0 := V and Vn := V (ηnx ), n ∈ N, then
E[V1 ] ≤ e−aδ V0 (x) + M
and
E[Vn+1 |η0x , η1x , . . . , ηnx ] ≤ e−aδ Vn + M, n ∈ N.
Now we proceed to show that the chain (ηnx )n∈Z+ , x ≥ 0, is λ-irreducible, strong
aperiodic, and all compact subsets of R+ are petite for the chain (ηnx )n∈Z+ .
“λ-irreducibility”: We show that the Lebesgue measure λ on R+ is an irreducibility measure for (ηnx )n∈Z+ . Let A ∈ B(R+ ) and λ(A) > 0. Then it follows from
Proposition 1 that
P[η1x ∈ A|η0x = x] = P(X δx ∈ A) ≥ P(Z δ = 0)
f (δ, x, y)dy > 0,
A
since f (δ, x, y) > 0 for any x ∈ R+ and y > 0. This shows that the chain (ηnx )n∈Z+
is irreducible with λ being an irreducibility measure.
“Strong aperiodicity”(see [16, p. 561] for a definition): To show the strong aperiodicity of (ηnx )n∈Z0 , we need to find a set B ∈ B(R+ ), a probability measure m with
m(B) = 1, and ε > 0 such that
L(x, B) > 0,
x ∈ R+ ,
(21)
and
P(η1x ∈ A) ≥ εm(A), x ∈ C,
A ∈ B(R+ ),
(22)
where L(x, B) := P(ηnx ∈ B for some n ∈ N). To this end set B := [0, 1] and
g(y) := inf x∈[0,1] f (δ, x, y), y > 0. Since for fixed y > 0 the function f (δ, x, y)
is strictly positive and continuous in x ∈ [0, 1], thus we have g(y) > 0 and 0 <
(0,1] g(y)dy ≤ 1. Define
m(A) :=
1
(0,1] g(y)dy
g(y)dy,
A∩(0,1]
A ∈ B(R+ ).
Exponential Ergodicity of the Jump-Diffusion CIR Process
299
Then for any x ∈ [0, 1] and A ∈ B(R+ ) we get
P(η1x ∈ A) = P(X δx ∈ A)
≥ P(Z δ = 0)
≥ P(Z δ = 0)
f (δ, x, y)dy
A
g(y)dy
A∩(0,1]
≥ P(Z δ = 0)m(A)
so (22) holds with ε := P(Z δ = 0)
Obviously
(0,1]
(0,1]
g(y)dy,
g(y)dy.
L(x, [0, 1]) ≥ P(η1x ∈ [0, 1]) = P(X δx ∈ [0, 1]) ≥ P(Z δ = 0)
[0,1]
f (δ, x, y)dy > 0
for all x ∈ R+ , which verifies (21).
“Compact subsets are petite”: We have shown that λ is an irreducibility measure
for (ηnx )n∈Z+ . According to [16, Theorem 3.4(ii)], to show that all compact sets are
petite, it suffices to prove the Feller property of (ηnx )n∈Z+ , x ≥ 0. But this follows
from the fact that (ηnx )n∈Z+ is a skeleton chain of the JCIR process, which is an affine
process and possess the Feller property.
According to [16, Theorem 6.3] (see also the proof of [16, Theorem 6.1]), the
probability measure π is the only invariant probability measure of the chain (ηnx )n∈Z+ ,
x ≥ 0, and there exist constants β ∈ (0, 1) and C ∈ (0, ∞) such that
P δn (x, ·) − π
TV
≤ C x + 1 β n , n ∈ Z+ , x ∈ R+ .
Then for the rest of the proof we can proceed as in [18, p. 536] and get the
inequality (20).
Acknowledgments We gratefully acknowledge the suggestions of the editors and the anonymous referee. Their valuable comments have led to a significant improvement of this manuscript.
The research was supported by the Research Program “DAAD—Transformation: Kurzmaßnahmen 2012/13 Program”. This research was also carried out with the support of CAS—Centre for
Advanced Study, at the Norwegian Academy of Science and Letter, research program SEFE.
Open Access This chapter is distributed under the terms of the Creative Commons Attribution
Noncommercial License, which permits any noncommercial use, distribution, and reproduction in
any medium, provided the original author(s) and source are credited.
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P. Jin et al.
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