Positive Harris recurrence and exponential ergodicity of the basic affine jump-diffusion

0.21 MB·English

by  Peng Jin

POSITIVE HARRIS RECURRENCE AND EXPONENTIAL
ERGODICITY OF THE BASIC AFFINE JUMP-DIFFUSION
5
1 PENGJIN,BARBARARU¨DIGER,ANDCHIRAZTRABELSI
0
2
Abstract. In this paper we find the transition densities of the basic affine
n
jump-diffusion(BAJD),whichisintroducedbyDuffieandGˆarleanu[D.Duffie
a
andN.Gˆarleanu,Riskandvaluationofcollateralizeddebtobligations,Finan-
J
cial Analysts Journal 57(1) (2001), pp. 41–59] as an extension of the CIR
6 model with jumps. We prove the positive Harris recurrence and exponen-
1 tialergodicityoftheBAJD.Furthermoreweprovethattheuniqueinvariant
probabilitymeasureπ ofthe BAJDisabsolutely continuous withrespect to
] theLebesguemeasureandwealsoderiveaclosedformformulaforthedensity
R
functionofπ.
P
.
h
t
a
m 1. Introduction
[
Inthispaperwestudythebasicaffinejump-diffusion(shortedasBAJD),which

2 is given as the unique strong solution X := (X ) to the following stochastic
t t≥0
v differential equation
8
3 dX =a(θ−X )dt+σ X dW +dJ , X ≥0, (1.1)
t t t t t 0
6
3 where a,θ, σ are positive constants, (Wp) is a 1-dimensional Brownian motion
t t≥0
0 and (J ) is an independent 1-dimensional pure-jump L´evy process with the
t t≥0
.
1 L´evy measure
0
cde−dydy, y ≥0,
5 ν(dy)=
1 (0, y <0,
:
v for some constants c > 0 and d > 0. We assume that all the above processes are
i
X defined on some filtered probability space (Ω,F,(F)t≥0,P).
The process X = (X ) given by (1.1) has been introduced by Duffie and
r t t≥0
a Gˆarleanu [3] to describe the dynamics of default intensity. It was also used in [4]
and [14] as a short-rate model. Due to its simple structure, it is later referred as
the basic affine jump-diffusion. The existence and uniqueness of strong solutions
to the SDE (1.1) follow from the main results of [6]. We should remark that the
BAJD process X = (X ) in (1.1) stays non-negative, given that X ≥ 0. This
t t≥0 0
fact can be shown rigorously with the help of comparison theorems for SDEs, for
more details we refer the reader to [6].
2000 Mathematics Subject Classification. primary60H10;secondary60J60.
Keywordsandphrases. stochasticdifferentialequations,CIRmodelwithjumps,basicaffine
jump-diffusion,affineprocess,Harrisrecurrence,exponential ergodicity.
1
2 P.JIN,B.RU¨DIGER,ANDC.TRABELSI
Asitsnameimplies,theBAJDbelongstotheclassofaffineprocesses. Roughly
speaking,affineprocessesareMarkovprocessesforwhichthelogarithmofthechar-
acteristic function of the process is affine with respect to the initial state. Affine
processesonthe canonicalstatespaceRm×Rn havebeenthoroughlyinvestigated
+
byDuffieetal[2],aswellasin[13]. Inparticular,itwasshownin[2](seealso[13])
that any stochastic continuous affine process on Rm×Rn is a Feller process and
+
a complete characterization of its generator has been derived. Results on affine
processes with the state space R can also be found in [4].
+
Affine processes have found vast applications in mathematical finance, because
of their complexity and computational tractability. As mentioned in [2], these
applications include the affine term structure models of interest rates, affine sto-
chastic volatility models, and many others.
Recently the long-term behavior of affine processes with the state space R
+
has been studied in [14] (see also [12]), motivated by some financial applications
in affine term structure models of interest rates. In particular they have found
some sufficient conditions such that the affine process convergesweakly to a limit
distribution. Thislimitdistributionwaslatershownin[11]astheuniqueinvariant
probability measure of the process. Under further sharper assumptions it was
evenshown in [15] that the convergenceof the law of the process to its invariance
probability measure under the total variation norm is exponentially fast, which
is called the exponential ergodicity in the literature. The method used in [15] to
show the exponential ergodicity is based on some coupling techniques.
In this paper we investigate the long-time behavior of the BAJD. More pre-
cisely, as the first main result of this paper, we show that the BAJD is positive
Harris recurrent. As a well-known fact, Harris recurrence implies the existence
of (up to the multiplication by a positive constant) unique invariant measure.
Therefore our result on the positive Harris recurrence of the BAJD provides an-
other way of proving the existence and uniqueness of invariant measures for the
BAJD.Anotherconsequenceofthe positiveHarrisrecurrenceisthelimittheorem
t
for additive functionals (see e.g. [10, Theorem 20.21]), namely (1/t) f(X )ds
0 s
converges almost surely to f(x)π(dx) as t → ∞ for any f ∈ B (R ), where
R+ b R+
π is the unique invariant probability measure for the BAJD. Some applications
R
of Harris recurrence in statistics and calibrations of some financial models can be
found in [1] and [9].
As a key step in proving the positive Harris recurrence, we will derive a closed
formula for the transition densities of the BAJD, which seems also to be a new
result. In particular this formula indicates that the law of the BAJD process
at any time t > 0 is a convolution of a mixture of Gamma-distributions with a
noncentralchi-squaredistribution. We shouldpoint outthat this facthas already
been discovered in [4] for the BAJD in some special cases, namely for the BAJD
whoseparameters(a,θ,σ,c,d)satisfyc/(a−σ2d/2)∈Z. Ingeneral,itisnoteasy
to identify the transition density functions of affine processes. However,as shown
in [5], a general density approximation procedure can be carried out for certain
affine processes and in particular for the BAJD. Different from [5] we seek in this
paper a closed formula of the transition densities of the BAJD.
POSITIVE HARRIS RECURRENCE AND EXPONENTIAL ERGODICITY OF THE BAJD 3
Finally we show the exponential ergodicity of the BAJD. We should indicate
that the BAJD does not satisfy the assumptions required in [15] in order to get
the exponentialergodicity. Our method is also a different one and is based onthe
existence of a Foster-Lyapunov.
Theremainderofthispaperisorganizedasfollows. InSection2wecollectsome
keyfactsontheBAJD.InSection3weintroducetheso-calledBesseldistributions
andsomemixturesofBessel-distributions. InSection4wederiveaclosedformula
forthetransitiondensitiesoftheBAJD.InSection5wefirststudysomecontinuity
propertiesofthetransitiondensitiesoftheBAJDandthenshowitspositiveHarris
recurrence. In Section 6 we show the exponential ergodicity of the BAJD.
2. Preliminaries
In this section we recall some key facts on the BAJD. As defined in (1.1), it is
the unique strong solution X =(X ) to the SDE
t t≥0
dX =a(θ−X )dt+σ X dW +dJ , X ≥0.
t t t t t 0
ThroughoutthispaperwedenoteE p(·)andP (·)astheexpectationandproba-
x x
bilityrespectivelygiventhe initialconditionX =x, withx≥0beingaconstant.
0
By the affine structure of the BAJD process X, the characteristic function of
X (given that X =x) is of the form
t 0
E euXt =exp φ(t,u)+xψ(t,u) , u∈U :={u∈C:ℜu≤0}, (2.1)
x
where th(cid:2)e fun(cid:3)ctions φ(cid:0)(t,u) and ψ(t,u) s(cid:1)olve the generalized Riccati equations
∂ φ(t,u)=F ψ(t,u) , φ(0,u)=0,
t
(2.2)
(∂tψ(t,u)=R(cid:0)ψ(t,u)(cid:1), ψ(0,u)=u,
with (cid:0) (cid:1)
cu
F(u)=aθu+ , u∈C\{d},
d−u
σ2u2
R(u)= −au, u∈C.
2
By solving the system (2.2) we get
ue−at
ψ(t,u)= (2.3)
1− σ2u(1−e−at)
2a
and
−2aθ log 1− σ2u(1−e−at)
σ2 2a
φ(t,u)= +(cid:0) c log d−σ22adu+(cid:1) σ22ad−1 ue−at , if ∆6=0, (2.4)
−2aθ loga−1−σ22dσ2u((cid:16)1−e−at)(cid:0)d+−ucu(1(cid:1)−e−at)(cid:17), if ∆=0,
σ2 2a a(d−u)
where ∆=a−σ2d/2. (cid:0) (cid:1)
4 P.JIN,B.RU¨DIGER,ANDC.TRABELSI
Accordingto (2.1),(2.3)and(2.4),the characteristicfunctionofX is givenby
t
1− σ2u(1−e−at) −2σa2θ · d−σ22adu+ σ22ad−1 ue−at a−cσ22d
2a d−u
(cid:0) ·exp xue(cid:1)−at (cid:16), if ∆(cid:0) 6=0,(cid:1) (cid:17)
Ex[euXt]= 1− σ2a2u(1(cid:16)−1−e−σ2aa2tu)(1−−e2σ−a2θat·)e(cid:17)xp cua(1(d−−e−u)at) + 1−σ2xuu(e1−−ate−at) ,
2a
(cid:0) (cid:1) (cid:16) if ∆=0. (cid:17) (2.5)
Obviously E exp(uX ) is continuous in t≥ 0 and thus the BAJD process X is
x t
stochastically continuous.
(cid:2) (cid:3)
Weshouldpointoutthatifweallowtheparameterctobe0,thenthestochastic
differential equation (1.1) turns into
dZ =a(θ−Z )dt+σ Z dW , Z =x≥0. (2.6)
t t t t 0
ToavoidconfusionswehaveusedZ instpeadofX here. TheuniquesolutionZ :=
t t
(Z ) to(2.6)isthewell-knownCox-Ingersoll-Ross(shortedasCIR)processand
t t≥0
it holds
Ex[euZt]= 1− σ2a2u(1−e−at) −2σa2θ ·exp 1− σ2xuu(e1−−ate−at) . (2.7)
(cid:0) (cid:1) (cid:16) 2a (cid:17)
In Section 4 we will find a distribution ν on R such that
t +
d−σ22adu+ σ22ad−1 ue−at a−cσ22d, if ∆6=0,
euyν (dy)= d−u (2.8)
ZR+ t (cid:16)exp cua(1(d−(cid:0)−e−u)at) (cid:1), (cid:17) if ∆=0.
(cid:16) (cid:17)
Then it follows from (2.5), (2.7) and (2.8) that the distribution of the BAJD is
the convolution of the distribution of the CIR process and ν . In light of this
t
observationwecanthusidentifythe transitionprobabilitiesp(t,x,y)oftheBAJD
with
p(t,x,y)= f(t,x,y−z)ν (dz), x,y ≥0, t>0, (2.9)
t
R
Z +
where f(t,x,y) denotes the transition densities of the CIR process.
Remark 2.1. For a different way of representing the distribution of X as a con-
t
volutionwe refer the readerto [4]. In factit was indicatedin [4, Remark 4.8]that
the distributionofany affine processonR canbe representedasthe convolution
+
of two distributions on R .
+
3. Mixtures of Bessel distributions
To find a distribution ν with the characteristic function of the form (2.8) and
t
study the distributional properties of the BAJD, it is inevitable to encounter the
Bessel distributions and mixtures of Bessel distributions.
We start with a slight variant of the Bessel distribution defined in [8, p.15].
Suppose that α and β are positive constants. A probability measure µ on
α,β
POSITIVE HARRIS RECURRENCE AND EXPONENTIAL ERGODICITY OF THE BAJD 5
R ,B(R ) is called a Bessel distribution with parameters α and β if
+ +
(cid:0) (cid:1)
α
µ (dx)=e−αδ (dx)+βe−α−βx ·I (2 αβx)dx, (3.1)
α,β 0 1
βx
r
p
where δ is the Dirac measure at the originand I is the modified Bessel function
0 1
of the first kind, namely
r ∞ 1r2 k
I (r)= 4 , r ∈R.
1
2 k!(k+1)!
k=0 (cid:0) (cid:1)
X
Now we consider mixtures of Besseldistributions. Let γ >0 be a constant and
define a probability measure m on R as follows:
α,β,γ +
∞ tγ−1
m (dx):= µ (dx) e−tdt.
α,β,γ αt,β
Γ(γ)
Z0
Similar to [8] we can easily calculate the characteristic function of µ and
α,β
m .
α,β,γ
Lemma 3.1. For u∈U we have:
∞
(i) euxµα,β(dx)=eβα−uu.
Z0
∞ 1 α 1 γ
(ii) euxm (dx)= + · .
α,β,γ α+1 α+1 1− α+1 ·u
Z0 (cid:16) β (cid:17)
Proof. (i) If u∈U, then
∞ ∞ α ∞ (αβx)k
euxµ (dx)=e−α+e−α βe−βx·eux αβx · dx
α,β
βx k!(k+1)!
Z0 Z0 r k=0
(cid:0)p (cid:1) X
∞ ∞ (αβx)k
=e−α+e−α αβe(u−β)x· dx
k!(k+1)!
Z0 k=0
X
∞ ∞ (αβx)k
=e−α+αβe−α e(u−β)x dx
k!(k+1)!
k=0Z0
X
∞
αβ k+1 1
=e−α+e−α ·
β−u (k+1)!
Xk=0(cid:16) (cid:17)
∞
αβ k 1
=e−α · =eβα−uu.
β−u k!
Xk=0(cid:16) (cid:17)
6 P.JIN,B.RU¨DIGER,ANDC.TRABELSI
(ii) For u∈U we get
∞ ∞ tγ−1 u −γ
euxmα,β,γ(dx) = eαt·β−uu · e−tdt= 1+α·
Γ(γ) u−β
Z0 Z0 (cid:16) (cid:17)
−β+(α+1)u −γ −β+u γ
= =
−β+u −β+(α+1)u
(cid:16) (cid:17) (cid:16) (cid:17)
1 ((α+1)u−β)+ β −β −γ
= α+1 α+1
−β+(α+1)u
(cid:16) (cid:17)
1 α 1 γ
= + · . (3.2)
α+1 α+1 1− α+1 ·u
(cid:16) β (cid:17)
(cid:3)
Lemma 3.2. (i) The measure m can be represented as follows:
α,β,γ
1 γ
m (dx)= δ +g (x)dx, x≥0, (3.3)
α,β,γ 0 α,β,γ
1+α
(cid:16) (cid:17)
where
∞ αkΓ(k+γ)
g (x):= Γ(x;k,β), x≥0, (3.4)
α,β,γ (α+1)k+γΓ(γ)k!
k=1
X
and Γ(x;k,β) denotes thedensity function of theGamma distribution with param-
eters k and β.
(ii) The function g (x) defined in (3.4) is a continuous function with variables
α,β,γ
(α,β,γ,x)∈D :=(0,∞)×(0,∞)×(0,∞)×[0,∞).
Proof. (i) We can write
∞ tγ−1
m (dx) = µ (dx) e−tdt
α,β,γ αt,β
Γ(γ)
Z0
∞ αt tγ−1
= e−αtδ (dx)+βe−αt−βx ·I (2 αtβx)dx e−tdt
0 1
βx Γ(γ)
Z0 1(cid:16) γ ∞ r ∞ (pαtβx)k (cid:17)tγ
= δ (dx)+ αβe−αt−βx · e−tdtdx
0
1+α k!(k+1)! Γ(γ)
(cid:16) (cid:17) Z0 Xk=0
1 γ ∞ ∞ e−(α+1)ttγ+k
= δ (dx)+αβe−βx. (αβx)k dt dx
0
1+α Γ(γ)k!(k+1)!
(cid:16) (cid:17) Xk=0(cid:16)Z0 (cid:17)
1 γ ∞ αk+1Γ(k+γ+1)
= δ (dx)+ Γ(x;k+1,β)dx
1+α 0 (α+1)k+γ+1Γ(γ)(k+1)!
(cid:16) (cid:17) kX=0
1 γ ∞ αkΓ(k+γ)
= δ (dx)+ Γ(x;k,β)dx.
1+α 0 (α+1)k+γΓ(γ)k!
(cid:16) (cid:17) kX=1
POSITIVE HARRIS RECURRENCE AND EXPONENTIAL ERGODICITY OF THE BAJD 7
(ii) By the definition of g (x) we have
α,β,γ
∞ αt tγ−1
g (x) = βe−αt−βx ·I (2 αtβx) e−tdt
α,β,γ 1
βx Γ(γ)
Z0 r
∞ ∞ (αtβpx)k tγ
= αβe−αt−βx e−tdt
k!(k+1)! Γ(γ)
Z0 k=0
(cid:0)X (cid:1)
∞ αβtγ ∞ (αtβx)k
= e−(α+1)t−βx dt
Γ(γ) k!(k+1)!
Z0 k=0
(cid:0)X (cid:1)
Suppose that (α ,β ,γ ,x )∈D and δ >0 is small enough such that γ −δ >0,
0 0 0 0 0
α −δ >0 and β −δ >0. Then for (α,β,γ,x)∈K with
0 0 δ
K :={(α,β,γ,x)∈D :max{|α−α |,|β−β |,|γ−γ |,|x−x |}≤δ}
δ 0 0 0 0
we get
αβtγ ∞ (αtβx)k αβtγ ∞ (αt)k(βx)k
e−(α+1)t−βx ≤ e−(α+1)t−βx
Γ(γ) k!(k+1)! Γ(γ) (k!)2
k=0 k=0
(cid:0)X (cid:1) (cid:0)X (cid:1)
αβtγ αβtγ
≤ e−(α+1)t−βx.eαteβx ≤ e−t
Γ(γ) Γ(γ)
≤c tγ0−δ1 (t)+tγ0+δe−t1 (t) (3.5)
δ [0,1] (1,∞)
for some constant c >0, since αβ is co(cid:0)ntinuous and thus bounded for (cid:1)
δ Γ(γ)
(α,β,γ,x) ∈ K . If (α ,β ,γ ,x ) → (α ,β ,γ ,x ) as n → ∞, then by domi-
δ n n n n 0 0 0 0
nated convergence we get
lim g (x )=g (x ),
n→∞ αn,βn,γn n α0,β0,γ0 0
namely g (x) is a continuous function on D. (cid:3)
α,β,γ
Remark 3.3. If we write δ = Γ(0,β), namely considering the Dirac measure δ
0 0
asadegeneratedGammadistribution, thenthe representationin(3.3)showsthat
the measure m is a mixture of Gamma distributions Γ(k,β), k ∈Z , namely
α,β,γ +
1 γ ∞ αkΓ(k+γ)
m = Γ(0,β)+ Γ(k,β).
α,β,γ 1+α (α+1)k+γΓ(γ)k!
(cid:16) (cid:17) kX=1
4. Transition density of the BAJD
Inthissectionweshallderiveaclosedformexpressionforthetransitiondensity
of the BAJD. We should mention that in [4, Chapter 7] the density functions of
the pricing semigroupassociatedto the BAJD was derivedfor some specialcases.
Essentially, the method used in [4] could be used to derive the density functions
of the BAJD in the case where c/(a−σ2d/2) ∈ Z. Here we proceed like [4] but
dealwith more generalparameters. In order to do this, we first find, by using the
resultsoftheprevioussection,aprobabilitymeasureν onR whosecharacteristic
t +
function satisfies (2.8).
WerecallthattheBAJDprocessX =(X ) isgivenby(1.1). Wedistinguish
t t≥0
between three cases according to the sign of ∆:=a−σ2d/2.
8 P.JIN,B.RU¨DIGER,ANDC.TRABELSI
4.1. Case i): ∆>0. From (2.5) we know that
Ex[euXt]= 1− σ2a2u(1−e−at) −2σa2θ ·exp 1− σ2xuu(e1−−ate−at)
(cid:0) (cid:1) (cid:16) 2a (cid:17)
d− σ2du + σ2d −1 ue−at c
· 2a 2a a−σ22d (4.1)
d−u
(cid:0) (cid:1)
(cid:16) (cid:17)
The product of the first two terms on the right-hand side of (4.1) coincides with
the characteristic function of the CIR process Z = (Z ) defined in (2.6). It is
t t≥0
well-known that the transition density function of the CIR process is given by
v q
f(t,x,y)=ρe−u−v 2Iq 2(uv)21 (4.2)
u
for t>0,x>0 and y ≥0, where (cid:16) (cid:17) (cid:0) (cid:1)
2a
ρ≡ , u≡ρxe−at,
σ2 1−e−at
(cid:16) (cid:17) 2aθ
v ≡ρy, q ≡ −1,
σ2
and I (·) is the modified Bessel function of the first kind of order q, namely
q
I (r)= r q ∞ 41r2 k , r >0.
q
2 k!Γ(q+k+1)
k=0 (cid:0) (cid:1)
(cid:0) (cid:1) X
We should remark that for x = 0 the formula of the density function f(t,x,y)
given in (4.2) is not valid any more. In this case we have
ρ
f(t,0,y)= vqe−v (4.3)
Γ(q+1)
for t>0 and y ≥0.
Thus
f(t,x,y)euydy = 1− σ2u(1−e−at) −2σa2θ ·exp xue−at .
R 2a 1− σ2u(1−e−at)
Z + (cid:16) 2a (cid:17)
(cid:0) (cid:1)
Now we want to find a probability measure ν with
t
d− σ2du + σ2d −1 ue−at c
euyνt(dy)= 2a 2a a−σ22d
ZR+ (cid:16) d(cid:0)−u (cid:1) (cid:17)
= d−uL1(t) a−cσ22d
d−u
=(cid:16)L1(t)+ 1−(cid:17)L1(t) 1−1 u a−cσ22d, (4.4)
d
(cid:16) (cid:0) (cid:1) (cid:17)
where L (t) := exp(−at)+σ2d 1−exp(−at) /(2a). If such a measure ν exists,
1 t
then the law of X can be written as the convolution of the law of Z and ν .
t t t
(cid:0) (cid:1)
Comparingthe characteristicfunctions (3.2)and (4.4),it is easy to see that we
can seek the measure ν as a mixture of Bessel distributions. More precisely, we
t
define
ν :=m (4.5)
t α1(t),β1(t),γ1
POSITIVE HARRIS RECURRENCE AND EXPONENTIAL ERGODICITY OF THE BAJD 9
with
α (t):= 1 −1
1 L1(t)
β1(t):= L1d(t) (4.6)
γ1 := a−cσ2d.
2
Then the characteristic function of νt coincides with (4.4). Since the probability
measure m is of the form (3.3), it follows now from (2.7), (4.1) and
α1(t),β1(t),γ1
(4.4) that the law of X is absolutely continuous with respect to the Lesbegue
t
measure and its density function p(t,x,y) is given by
1 γ1 y
p(t,x,y)= f(t,x,y)+ f(t,x,y−z)g (z)dz (4.7)
1+α (t) α1(t),β1(t),γ1
(cid:16) 1 (cid:17) Z0
for t>0, x≥0 and y ≥0, where the function g is defined in (3.4).
4.2. Case ii): ∆ < 0. Similar to the case (i), it suffices to find a probability
measure ν with
t
d− σ2du + σ2d −1 ue−at c
euyνt(dy) = 2a 2a a−σ22d
ZR+ (cid:16) d(cid:0)−u (cid:1) (cid:17)
= d−u a−−σc22d
d− σ2du + σ2d −1 ue−at
(cid:16) 2a 2a (cid:17)
= d−u (cid:0)a−−σc22d (cid:1)
d−L (t)u
1
(cid:16) (cid:17)
−c
= 1 +(1− 1 )· 1 a−σ22d. (4.8)
(cid:18)L1(t) L1(t) 1− L1(dt)u(cid:19)
Since ∆=a−σ2d/2<0, therefore σ2d/2a>1 and
σ2d
L (t)=e−at+ · 1−e−at >1.
1
2a
(cid:0) (cid:1)
According to the formula (3.1), we can choose
ν =m
t α2(t),β2(t),γ2
with the parameters α ,β and γ defined by
2 2 2
α (t):=L (t)−1
2 1
β2 :=d (4.9)
γ2 := a−−σc2d.
2
Similar to the case (i), the transition densities p(t,x,y) of X is given by
1 γ2 y
p(t,x,y)= f(t,x,y)+ f(t,x,y−z)g (z)dz (4.10)
1+α (t) α2(t),β2,γ2
(cid:16) 2 (cid:17) Z0
for t>0, x≥0 and y ≥0, where the function g is defined in (3.4).
10 P.JIN,B.RU¨DIGER,ANDC.TRABELSI
4.3. Case iii): ∆ = 0. In this case we need to find a probability measure ν
t
with
cu(1−e−at)
euyν (dy)=exp .
t
R a(d−u)
Z + (cid:16) (cid:17)
According to the formula (3.3) we can take ν as a Bessel distribution µ
t α3(t),β3
with the parameters α (t) and β defined by
3 3
α (t):= c(1−e−at)
3 a (4.11)
(β3 :=d.
Thus in this case the transition densities p(t,x,y) of X is given by
y α (t)
p(t,x,y)= f(t,x,y−z)β e−α3(t)−β3z 3 I (2 α (t)β z)dz
3 1 3 3
Z0 s β3z
p
+e−α3(t)f(t,x,y) (4.12)
for t>0, x≥0 and y ≥0.
Summarizing the above three cases we get the following theorem.
Theorem 4.1. Let X = (X ) be the BAJD defined in (1.1). Then the law of
t t≥0
X given that X = x ≥ 0 is absolutely continuous with respect to the Lesbegue
t 0
measure and thus posseses a density function p(t,x,y), namely
P (X ∈A)= p(t,x,y)dy, t≥0, A∈B(R )
x t +
ZA
According to the sign of ∆ = a−σ2d/2, the density p(t,x,y) is given by (4.7),
(4.10) and (4.12) respectively.
Although the density functions in (4.7), (4.10)and (4.12) are essentially differ-
ent, they do share some similarities. In the following corollary we give a unified
representationof p(t,x,y).
Corollary4.2. Irrelevantofthethesignof∆=a−σ2d/2,thetransitiondensities
p(t,x,y) of X can be expressed in a unified form as
y
p(t,x,y)=L(t)f(t,x,y)+ f(t,x,y−z)h(t,z)dz, (4.13)
Z0
where L(t) is continuous function in t>0 which satisfies 0<L(t)<1 for t>0,
the function h(t,z) is non-negative and continuous in (t,z)∈(0,∞)×[0,∞) and
satisfies h(t,z)dz =1−L(t).
R
+
R
5. Positive Harris recurrence of the BAJD
It was shownin [2] (see also [13]) that the semigroupof any stochastically con-
tinuous affine processon the canonicalstate space Rm×Rn is a Feller semigroup.
+
Define the semigroup of the BAJD by
T f(x):= p(t,x,y)f(y)dy, (5.1)
t
R
Z +