Continuous-state branching processes in Levy random environments

0.22 MB·

by  Hui He

(Version: 2016/01/19)
Continuous-state branching processes
in L´evy random environments1
6
Hui He, Zenghu Li and Wei Xu2
1
0
School of Mathematical Sciences, Beijing Normal University,
2

n Beijing 100875, People’s Republic of China
a
J E-mails: [email protected], [email protected], [email protected]

9
1
Abstract. A general continuous-state branching processes in random environment

] (CBRE-process) is defined as the strong solution of a stochastic integral equation. The
R
environment is determined by a L´evy process with no jump less than 1. We give char-
P
−
. acterizations of the quenched and annealed transition semigroups of the process in terms
h
t of a backward stochastic integral equation driven by another L´evy process determined
a
m by the environment. The process hits zero with strictly positive probability if and only
[ if its branching mechanism satisfies Grey’s condition. In that case, a characterization

of the extinction probability is given using a random differential equation with singular
1
v terminal condition. The strong Feller property of the CBRE-process is established by a
8
coupling method. We also prove a necessary and sufficient condition for the ergodicity of
0
8 the subcricital CBRE-process with immigration.
4
0 Mathematics Subject Classification (2010): Primary 60J80, 60K37; Secondary 60H20,
.
1 60G51
0
6
Key words and phrases: continuous-state branching process, random environment,
1
: L´evy process, transition semigroup, backward stochastic equation, survival probability,
v
i immigration, ergodicity.
X
r
a
1 Introduction
Galton-Watsonprocesses inrandomenvironments (GWRE-processes)wereintroduced
bySmith(1968)andSmithandWilkinson (1969)asextensions ofclassical Galton-Watson
processes (GW-processes). Thoseextensions possess manyinteresting newpropertiessuch
as phase transitions of survival probabilities. For instance, that different regimes for the
survival probability arise in the subcritical regime. For recent results on the speed of
decay of the survival probability, the reader may refer to Afanasy’ev et al. (2005, 2012),
Guivarch and Liu (2001) and Vatutin (2004) and the references therein.
1
Supported by NSFC (No.11131003,No. 11371061and No.11531001).
2
Corresponding author.
1
Let W(t) : t 0 be a Brownian motion and S(t) : t 0 a Brownian motion
{ ≥ } { ≥ }
with drift. We assume the two processes are independent. By the Yamada-Watanabe
theorem, for any constants c 0, σ 0 and b R there is a unique positive solution to
≥ ≥ ∈
the stochastic differential equation:
dX(t) = 2cX(t)dW(t) bX(t)dt+σX(t)dS(t). (1.1)
−
p
The solution X(t) : t 0 is called a continuous-state branching diffusion in ran-
{ ≥ }
dom environment (CBRE-diffusion). The environment here is determined by the process
S(t) : t 0 . It was proved in Kurtz (1978) that the CBRE-diffusion arises as the limit
{ ≥ }
of a sequence of suitably rescaled GWRE-processes; see also Helland (1981). A diffusion
approximationoftheGWRE-processwasactuallyconjecturedbyKeiding (1975). Itturns
out that the CBRE-diffusion is technically more tractable than the GWRE-process. In
the particular case of σ = 0, the CBRE-diffusion reduces to the well-known Feller branch-
ing diffusion, which belongs to an important class of positive Markov processes called
continuous-state branching processes (CB-processes); see Feller (1951), Jiˇrina (1958) and
Lamperti (1967a, 1967b).
In the work of B¨oinghoff and Hutzenthaler (2012), it was shown that the survival
probability of the CBRE-diffusion can be represented explicitly in terms of an exponential
functional of the environment process S(t) : t 0 . Based on the representation,
{ ≥ }
B¨oinghoff and Hutzenthaler (2012) gave an exact characterization for the decay rate of
the survival probability of the CBRE-diffusion in the critical and subcritical cases. The
results of B¨oinghoff and Hutzenthaler (2012) are more complete than the corresponding
results for the GWRE-processes in the sense that they calculated the accurate limiting
constants. In addition, they characterized the CBRE-process conditioned to never go
extinct and established a backbone construction for the conditioned process. See also
Hutzenthaler (2011) for some related results.
Continuous-state branching processes with immigration (CBI-processes), which gen-
eralize the CB-processes, were introduced by Kawazu and Watanabe (1971) as rescaling
limits of Galton-Watson processes with immigration (GWI-processes); see also Aliev and
Shchurenkov (1982) and Li (2006, 2011). Let b R and c 0 be given constants. Let
m(dz) be a Radon measure on (0, ) satisfying∈ ∞(z z2≥)m(dz) < . Suppose that
∞ 0 ∧ ∞
W(t) : t 0 is a Brownian motion, η(t) : t R0 is an increasing L´evy process with
η{(0) = 0 a≥nd}M˜(ds,dz,du) is a compe{nsated P≥oisso}n random measure on (0, )3 with
∞
intensity dsm(dz)du. We assume those three noses are independent of each other. By
Theorems 5.1 and 5.2 in Dawson and Li (2006), there is a unique positive strong solution
to
t t
X(t) = X(0) b X(s)ds+ 2cX(s)dW(s)
− Z Z
0 0 p
t ∞ X(s−)
+ zM˜(ds,dz,du)+η(t). (1.2)
Z Z Z
0 0 0
It was shown in Dawson and Li (2006) that the solution X(t) : t 0 is a CBI-
{ ≥ }
processes; see also Fu and Li (2010). The process η(t) : t 0 describes the inputs of
{ ≥ }
2
b ∞
the immigrants. Here and in the sequel, we understand = and = for
a (a,b] a (a,∞)
any a b R. R R R R
≤ ∈
A class of continuous-state branching processes with catastrophe was introduced by
Bansaye et al. (2013), who pointed out that the processes can be identified as continuous-
statebranchingprocessesinrandomenvironment (CBRE-processes) withtheenvironment
given by a L´evy process with bounded variation. Those authors gave a criticality classifi-
cation of their CBRE-processes according to the long time behavior of the environmental
L´evy process. They also characterized the Laplace exponent of the processes using a
backward ordinary differential equation involving the environment process. For stable
branching CBRE-processes, Bansaye et al. (2013) calculated explicitly the survival prob-
ability and characterized its decay rate in the critical and subcritical cases. In addition,
they showed some interesting applications of their results to a cell infection model. The
results of Bansaye et al. (2013) were extended in Palau and Pardo (2015a) to the case
where the environment was given by a Brownian motion with drift.
TheCBRE-processesstudiedinBansayeetal.(2013)canbegeneralizedtocontinuous-
state branching processes with immigration in random environment (CBIRE-processes).
Let L(t) : t 0 be a L´evy process with no jump less than 1 and assume it is
{ ≥ } −
independent of the three noises in (1.2). It is natural to define a CBIRE-process Y(t) :
{
t 0 by the stochastic equation
≥ }
t t
Y(t) = Y(0) b Y(s)ds+ 2cY(s)dW(s)+η(t)
− Z Z
0 0 p
t ∞ Y(s−) t
+ zM˜(ds,dz,du)+ Y(s )dL(s). (1.3)
Z Z Z Z −
0 0 0 0
Herethe impact oftheenvironment isrepresented by theL´evy process L(t) : t 0 . The
{ ≥ }
existence and uniqueness of the positive strong solution to the above equation follow from
theresult ofDawsonandLi (2012); seeSection 2 forthedetails. Inthevery recent work of
Palau and Pardo (2015b), a further generalization of the CBIRE-process was introduced
by considering a competition mechanism. The authors gave a direct construction of their
model by arguments similar to those in Dawson and Li (2012) and proved some results
on the long time behavior of the process.
The purpose of this paper is to study the basic structures of the CBRE- and CBIRE-
processes. In Section 2, we introduce two random cumulant semigroups, which are im-
portant tools in the study of those processes. The semigroups are defined in terms of a
backward stochastic equations driven by a L´evy process. The existence of them follows
from a general result in Li (2011) on Dawson-Watanabe superprocesses. In Section 3, we
construct theCBRE-process byapplying atheoreminDawsonandLi(2012)onstochastic
equationsdrivenbytime-spacenoises. Thenwegivecharacterizationsofthequenchedand
annealed transition probabilities of the CBRE-process. In Section 4, we show the CBRE-
process hits zero with strictly positive probability if and only if its branching mechanism
satisfies Grey’s condition. In that case, we give a characterization of the extinction proba-
bilities by the solution of a random differential equation with singular terminal condition.
3
The strong Feller property of the CBRE-process is established by a coupling method.
Some of the results are extended in Section 5 to CBIRE-processes. In addition, we give a
necessary and sufficient condition for the ergodicity of subcritical CBIRE-processes. Most
of the results here are obtained or presented using stochastic equations driven by L´evy
processes, which are more elegant than those in the classical discrete setting.
Acknowledgements. We thank Professors Yueyun Hu and Zhan Shi for helpful
discussions on branching processes in random environments.
2 Random cumulant semigroups
In this section, we introduce some random cumulant semigroups, which generalize
that of a classical CB-process. Those semigroups are important tools in the study of the
CBRE-process. Let I R be an interval and ζ = ζ(t) : t I a c`adla`g function. Let φ
⊂ { ∈ }
be a branching mechanism given by
∞
φ(z) = bz +cz2 + (e−uz 1+uz)m(du), z 0. (2.1)
Z − ≥
0
where b R and c 0 are constants and (z z2)m(dz) is a finite measure on (0, ).
∈ ≥ ∧ ∞
From Theorem 6.10 in Li (2011) it follows that, for any t I and λ 0, there is a unique
∈ ≥
positive solution r uζ (λ) to the integral evolution equation
7→ r,t
t
uζ (λ) = λ eζ(s)φ(e−ζ(s)uζ (λ))ds, r I ( ,t]. (2.2)
r,t −Z s,t ∈ ∩ −∞
r
Moreover, there is an inhomogeneous transition semigroup (Pζ ) on [0, ) defined
r,t t≥r∈I ∞
by
e−λyPζ (x,dy) = e−xuζr,t(λ), λ 0. (2.3)
Z r,t ≥
[0,∞)
By a simple transformation, we can define another inhomogeneous transition semigroup
(Qζ ) on [0, ) by
r,t t≥r∈I ∞
e−λyQζ (x,dy) = e−xvrζ,t(λ), λ 0, (2.4)
Z r,t ≥
[0,∞)
where
vζ (λ) = e−ζ(r)uζ (eζ(t)λ). (2.5)
r,t r,t
The uniqueness of the solution to (2.2) implies that
uζ (λ) = uζ uζ (λ), λ 0,t s r I. (2.6)
r,t r,s ◦ s,t ≥ ≥ ≥ ∈
4
There is a similar relation for (vζ ) . By Lebesgue’s theorem one can see r uζ (λ)
r,t t≥r∈I 7→ r,t
is also the unique positive continuous solution to the differential equation
d
uζ (λ) = eζ(r)φ(e−ζ(r)uζ (λ)), a.e. r I ( ,t] (2.7)
dr r,t r,t ∈ ∩ −∞
with terminal condition uζ (λ) = λ.
t,t
Proposition 2.1 Let (Pζ ) be defined by (2.3). Then for any x 0 and t r I
r,t t≥r∈I ≥ ≥ ∈
we have
yPζ (x,dy) = xe−b(t−r).
r,t
Z
[0,∞)
Proof. By differentiating both sides of (2.2) and solving the resulted integral equation
we obtain (d/dλ)uζ (0+) = e−b(t−r). Then we get the desired equality by differentiating
r,t
✷
both sides of (2.3).
Proposition 2.2 If b 0, then t uζ (λ) is decreasing on I [r, ) and r uζ (λ) is
≥ 7→ r,t ∩ ∞ 7→ r,t
increasing on I ( ,t].
∩ −∞
Proof. From (2.3) we see that λ uζ (λ) is increasing. Since b 0, we have φ(z) 0
7→ r,t ≥ ≥
for every z 0. Then (2.2) implies uζ (λ) λ. By (2.6) we see uζ (λ) uζ (λ) and
≥ r,t ≤ r,t ≤ s,t
uζ (λ) uζ (λ) for r s t I. ✷
r,t ≤ r,s ≤ ≤ ∈
When the function ζ is degenerate (ζ(t) = 0 for all t I), both (uζ ) and
∈ r,t t≥r∈I
(vζ ) reduce to the cumulant semigroup of a classical CB-process with branching
r,t t≥r∈I
mechanism φ; see, e.g., Chapter 3 of Li (2011). In the general case, we may think of
(uζ ) as an inhomogeneous cumulant semigroup determined by the time-dependent
r,t t≥r∈I
branching mechanism (s,z) eζ(s)φ(e−ζ(s)z). The idea of the proof of Theorem 6.10 in
7→
Li (2011) is to reduce the construction of an inhomogeneous cumulant semigroup to that
of a homogeneous one by some time-space processes. The transformation from (uζ )
r,t t≥r∈I
to (vζ ) is a time-dependent variation of the one used in the proof of Theorem 6.1 in
r,t t≥r∈I
Li (2011).
We next consider some randomization of the inhomogeneous cumulant semigroups
defined above. Let (Ω,F,F ,P) be a filtered probability space satisfying the usual hy-
t
potheses. Let a R and σ 0 be given constants and (1 z2)ν(dz) a finite measure
∈ ≥ ∧
on (0, ). Suppose that B(t) : t 0 is an (F )-Brownian motion and N(ds,dz) is an
t
(F )-P∞oisson random mea{sure on ≥(0, }) R with intensity dsν(dz). Let ξ(t) : t 0
t
∞ × { ≥ }
be a (F )-L´evy process with the following L´evy-Itoˆ decomposition:
t
t t
ξ(t) = ξ(0)+at+σB(t)+ zN˜(ds,dz)+ zN(ds,dz), (2.8)
Z Z Z Z
0 [−1,1] 0 [−1,1]c
5
where [ 1,1]c = R [ 1,1]. Let uξ (λ) and vξ (λ) be defined by (2.2) and (2.5) with
− \ − r,t r,t
ζ = ξ. From (2.2) we see that r vξ (λ) is the unique positive solution to
7→ r,t
t
vξ (λ) = eξ(t)−ξ(r)λ eξ(s)−ξ(r)φ(vξ (λ))ds, 0 r t. (2.9)
r,t −Z s,t ≤ ≤
r
Let L(t) : t 0 be the (F )-L´evy process with L´evy-Itoˆ decomposition:
t
{ ≥ }
t
L(t) = L(0)+βt+σB(t)+ (ez 1)N˜(ds,dz)
Z Z −
0 [−1,1]
t
+ (ez 1)N(ds,dz), (2.10)
Z Z −
0 [−1,1]c
where L(0) = ξ(0) and
σ2
β = a+ + (ez 1 z)ν(dz). (2.11)
2 Z − −
[−1,1]
Clearly, the two processes ξ(t) : t 0 and L(t) : t 0 generate the same filtration.
{ ≥ } { ≥ }
From (2.9) we see that the left-continuous process vξ (λ) : 0 s t is progressively
{ t−s,t ≤ ≤ }
measurable with respect to the filtrationgenerated by the L´evy process L (s) := L(t )
t
{ − −
L((t s) ) : 0 s t .
− − ≤ ≤ }
Theorem 2.3 For any t 0 and λ 0, the process vξ (λ) : 0 r t is the pathwise
≥ ≥ { r,t ≤ ≤ }
unique positive solution to
t t
vξ (λ) = λ φ(vξ (λ))ds+ vξ (λ)L(←d−s), 0 r t, (2.12)
r,t −Z s,t Z s,t ≤ ≤
r r
where the backward stochastic integral is defined by
t (t−r)−
vξ (λ)L(←d−s) = vξ (λ)L (ds).
s,t t−s,t t
Z Z
r 0
Proof. Let ξ (r) = ξ(t ) ξ((t r) ) and B (r) = B(t) B(t r) for 0 r t. Let
t t
− − − − − − ≤ ≤
N (ds,dz) be the Poisson random measure defined by
t
N ([0,r] B) = N([t r,t] B), 0 r t,B B(R).
t
× − × ≤ ≤ ∈
From (2.8) we have
r r
ξ (r) = ar +σB (r)+ zN˜ (ds,dz)+ zN (ds,dz).
t t t t
Z Z Z Z
0 [−1,1] 0 [−1,1]c
On the other hand, from (2.9) we have f (r) := vξ (λ) = eξt(r)F (r), where
t (t−r)−,t t
r
F (r) = λ e−ξt(s)φ(f (s))ds.
t t
−Z
0
6
By Itˆo’s formula,
r σ2 r
f (r) = λ+ eξt(s−)F (s)ξ (ds)+ eξt(s)F (s)ds
t t t t
Z 2 Z
0 0
r r
+ eξt(s−)F (s)(ez 1 z)N (ds,dz)+ eξt(s)F (ds)
t t t
Z0 ZR − − Z0
r r
= λ+ f (s )L (ds) φ(f (s))ds.
t t t
Z − −Z
0 0
It follows that
r− r
vξ (λ) = λ+ f (s )L (ds) φ(f (s))ds
r,t Z t − t −Z t
0 0
(t−r)− t−r
= λ+ vξ (λ)L (ds) φ(vξ (λ))ds
Z t−s,t t −Z t−s,t
0 0
t t
= λ+ vξ (λ)L(←d−s) φ(vξ (λ))ds.
Z s,t −Z t−s,t
r r
That proves the existence of the solution to (2.12). Conversely, assuming r vξ (λ) is
7→ r,t
a solution to (2.12), one can use similar calculations to see it also solves (2.9). Then the
✷
pathwise uniqueness for (2.12) is a consequence of that for (2.9).
3 Construction of CBRE-processes
Let (Ω,F,F ,P) be a filtered probability space satisfying the usual hypotheses. Let
t
ξ(t) : t 0 and L(t) : t 0 be (F )-L´evy processes given as in Section 2. Let
t
b{ R and≥c }0 be c{onstants a≥nd (}z z2)m(dz) a finite measure on (0, ). Suppose that
∈ ≥ ∧ ∞
W(t) : t 0 is another (F )-Brownian motion and M(ds,dz,du) is an (F )-Poisson
t t
{ ≥ }
random measure on (0, )3 with intensity dsm(dz)du. We assume both of those are
∞
independent of the L´evy process L(t) : t 0 . Given a positive F -measurable random
0
{ ≥ }
variable X(0), we consider the following stochastic integral equation:
t t
X(t) = X(0) b X(s)ds+ 2cX(s)dW(s)
− Z Z
0 0 p
t ∞ X(s−) t
+ zM˜(ds,dz,du)+ X(s )dL(s), (3.1)
Z Z Z Z −
0 0 0 0
where M˜(ds,dz,du) = M(ds,dz,du) dsm(dz)du.
−
Theorem 3.1 There is a unique positive strong solution X(t) : t 0 to (3.1).
{ ≥ }
Proof. Let E = 1,2 andU = A B , where A = 1 (0, )2 andB = 2 (0, ).
0 0 0 0 0
{ } ∪ { }× ∞ { }× ∞
Let π(du) = δ (du)+δ (du) for u E. Then W(ds,du) := dW(s)δ (du)+dB(s)δ (du)
1 2 1 2
∈
7
is a Gaussian white noise on (0, ) E with intensity dsπ(dz). Let µ (dy,dz,du) =
0
∞ ×
δ (dy)m(dz)du for (y,z,u) A and µ (dy,dz) = δ (dy)ν(dz) for (y,z) B . Then
1 0 0 2 0
∈ ∈
N (ds,dy,dz,du) := δ (dy)M(ds,dz,du) is a Poisson random measure on (0, ) A
0 1 0
∞ ×
withintensitydsδ (dy)m(dz)duandN (ds,dy,dz) := δ (dy)N(ds,dz)isaPoissonrandom
1 0 2
measure on (0, ) B with intensity dsδ (dy)ν(dz). Let b(x) = (β b)x for x [0, )
0 2
∞ × − ∈ ∞
and σ(x,u) = √2cx1 +σx1 for (x,u) [0, ) E. Let g (x,y,z,u) = z1
{u=1} {u=2} 0 {u≤x}
∈ ∞ ×
for (x,y,z,u) [0, ) A and g (x,y,z) = x(ez 1) for (x,y,z) [0, ) B . By
0 0 0
∈ ∞ × − ∈ ∞ ×
Theorem 2.5 in Dawson and Li (2012), there is a unique positive strong solution to the
stochastic equation
t t
X(t) = X(0)+ b(X(s))ds+ g (X(s ),y,z,u)N˜ (ds,dy,dz,du)
0 0
Z Z Z −
0 0 A0
t t
+ σ(X(s),u)W(ds,du)+ g (X(s ),y,u)N˜ (ds,dy,du).
0 0
Z Z Z Z −
0 E 0 B0
The above equation can be rewritten into
t t ∞ X(s−)
X(t) = X(0)+ 2cX(s)dW(s)+ zM˜(ds,dz,du)
Z Z Z Z
0 p 0 0 0
t t t
+(β b) X(s)ds+σ X(s)dB(s)+ X(s )(ez 1)N˜(ds,dz).
− Z Z Z Z − −
0 0 0 [−1,1]
Since the process
t
t (ez 1)N(ds,dz)
7→ Z Z −
0 [−1,1]c
has at most a finite number of jumps in each bounded time interval, as in the proof of
Proposition 2.2 in Fu and Li (2010), one can see that there is also a pathwise unique
positive strong solution to
t t ∞ X(s−)
˜
X(t) = X(0)+ 2cX(s)dW(s)+ zM(ds,dz,du)
Z Z Z Z
0 p 0 0 0
t t t
+(β b) X(s)ds+σ X(s)dB(s)+ X(s )(ez 1)N˜(ds,dz)
− Z Z Z Z − −
0 0 0 [−1,1]
t
+ X(s )(ez 1)N(ds,dz). (3.2)
Z Z − −
0 [−1,1]c
The above equation is just a reformulation of (3.1). Then we have the result of the
✷
theorem.
Remark 3.2 Since the stochastic equations in Section 2 of Dawson and Li (2012) are
formulated for Gaussian and Poisson noises in abstract space, they are quite flexible for
applications. For example, the proof of Theorem 3.1 given above can be modified to more
complex models with extra structures such as immigration, competition and so on.
8
We call the solution X(t) : t 0 to (3.1) a CBRE-process, which is a c`adla`g
{ ≥ }
strong Markov process. Here the random environment is provided by the L´evy process
L(t) : t 0 . By Itˆo’s formula one can see X(t) : t 0 has strong generator A defined
{ ≥ } { ≥ }
as follows: For f C2(R ),
+
∈
∞
Af(x) = (β b)xf′(x)+cxf′′(x)+x [f(x+z) f(x) zf′(x)]m(dz)
− Z − −
0
σ2
+ x2f′′(x)+ [f(xez) f(x) x(ez 1)f′(x)]ν(dz)
2 Z − − −
[−1,1]
+ [f(xez) f(x)]ν(dz). (3.3)
Z −
[−1,1]c
Proposition 3.3 Let Z(t) = X(t)e−ξ(t) for t 0. Then we have
≥
t t
Z(t) = X(0) b e−ξ(s)X(s)ds+ e−ξ(s) 2cX(s)dW(s)
− Z Z
0 0 p
t ∞ X(s−)
+ ze−ξ(s−)M˜(ds,dz,du). (3.4)
Z Z Z
0 0 0
In particular, the process Z(t) : t 0 is a positive local martingale when b = 0.
{ ≥ }
Proof. Let f(x,y) = xe−y. Then xf′(x,y) = f′(x,y) = f(x,y) and xf′′ (x,y) =
x − y − xy
f′′ (x,y) = f(x,y). Observe that the Poison random measure N(ds,dz) actually does not
yy
produce any jump of t Z(t). By (3.1) and Itˆo’s formula,
7→
t t
Z(t) = X(0) b f′(X(s),ξ(s))X(s)ds+ f′(X(s),ξ(s)) 2cX(s)dW(s)
− Z x Z x
0 0 p
t ∞ X(s−)
+ f′(X(s ),ξ(s ))zM˜(ds,dz,du)
Z Z Z x − −
0 0 0
t t
+β f′(X(s),ξ(s))X(s)ds+σ f′(X(s),ξ(s))X(s)dB((s)
Z x Z x
0 0
t
+ f′(X(s ),ξ(s ))X(s )(ez 1)N˜(ds,dz)
Z Z x − − − −
0 [−1,1]
t
+ f′(X(s ),ξ(s ))X(s )(ez 1)N(ds,dz)
Z Z x − − − −
0 [−1,1]c
t t
+a f′(X(s),ξ(s))ds+ f′(X(s ),ξ(s ))zN˜(ds,dz)
Z y Z Z y − −
0 0 [−1,1]
t t
+σ f′(X(s),ξ(s))dB((s)+ f′(X(s ),ξ(s ))zN(ds,dz)
Z y Z Z y − −
0 0 [−1,1]c
t 1
+σ2 f′′ (X(s),ξ(s))X(s)+ f′′ (X(s),ξ(s)) ds
Z0 h xy 2 yy i
t ∞ X(s−)
+ [f(X(s )+z,ξ(s)) f(X(s ),ξ(s))
Z Z Z − − −
0 0 0
9
f′(X(s ),ξ(s))z]M(ds,dz,du)
− x −
t
+ [f(X(s )ez,ξ(s )+z) f(X(s ),ξ(s ))
Z0 ZR − − − − −
f′(X(s ),ξ(s ))X(s )(ez 1) f′(X(s ),ξ(s ))z]N(ds,dz)
− x − − − − − y − −
t t
= X(0) b f′(X(s),ξ(s))X(s)ds+ f′(X(s),ξ(s)) 2cX(s)dW(s)
− Z x Z x
0 0 p
t ∞ X(s−)
+ f′(X(s ),ξ(s ))zM˜(ds,dz,du)
Z Z Z x − −
0 0 0
t t
+β f′(X(s),ξ(s))X(s)ds+σ f′(X(s),ξ(s))X(s)dB((s)
x x
Z Z
0 0
t t
+a f′(X(s),ξ(s))ds+σ f′(X(s),ξ(s))dB((s)
y y
Z Z
0 0
t 1
+σ2 f′′ (X(s),ξ(s))X(s)+ f′′ (X(s),ξ(s)) ds
Z0 h xy 2 yy i
t
ds [f′(X(s),ξ(s))X(s)(ez 1)+f′(X(s),ξ(s))z]ν(dz).
−Z Z x − y
0 [−1,1]
✷
By reorganizing the terms on the right-hand side we get the desired equality.
It is easy to see that the two L´evy processes ξ(t) : t 0 and L(t) : t 0 generate
{ ≥ } { ≥ }
the same σ-algebra. Let Pξ denote the quenched law given ξ(t) : t 0 or L(t) : t 0 .
{ ≥ } { ≥ }
Theorem 3.4 Let (Pξ ) and (Qξ ) be defined by (2.3) and (2.4), respectively,
r,t t≥r≥0 r,t t≥r≥0
with ζ = ξ. Then for any λ 0 and t r 0 we have
≥ ≥ ≥
Pξ[e−λZ(t) F ] = exp Z(r)uξ (λ) = e−λyPξ (Z(r),dy) (3.5)
| r {− r,t } Z r,t
[0,∞)
and
Pξ[e−λX(t) F ] = exp X(r)vξ (λ) = e−λyQξ (X(r),dy). (3.6)
| r {− r,t } Z r,t
[0,∞)
Proof. Fix λ 0. For t r 0, let H (r) = exp Z(r)uξ (λ) . By (2.2) and
≥ ≥ ≥ t {− r,t }
Proposition 3.3, given the environment ξ(t) : t 0 , we can use Itˆo formula to see
{ ≥ }
t
H (t) = H (r) H (s)Z(s)eξ(s)φ(e−ξ(s)uξ (λ))ds
t t −Z t s,t
r
t t
H (s )uξ (λ)dZ(s)+c H (s)uξ (λ)2e−2ξ(s)X(s)ds
− Z t − s,t Z t s,t
r r
t ∞ X(s−)
+ H (s ) exp ze−ξ(s−)uξ (λ)
Z Z Z t − {− s,t }
r 0 0 h
1+ze−ξ(s−)uξ (λ) M(ds,dz,du)
− s,t
i
10