Intensity process and compensator: A new filtration expansion approach and the Jeulin--Yor theorem

0.31 MB·English

by  Xin Guo

TheAnnalsofAppliedProbability
2008,Vol.18,No.1,120–142
DOI:10.1214/07-AAP447
(cid:13)c InstituteofMathematicalStatistics,2008
INTENSITY PROCESS AND COMPENSATOR: A NEW
FILTRATION EXPANSION APPROACH AND THE JEULIN–YOR
8
THEOREM
0
0
2 By Xin Guo and Yan Zeng

n
University of California, Berkeley and Bloomberg LP
a
J Let (Xt)t≥0 be a continuous-time, time-homogeneous strong
1 Markov process with possible jumps and let τ be its first hitting
2 time of a Borel subset of the state space. Suppose X is sampled at
random times and suppose also that X has not hit the Borel set by
]
R timet.Whatistheintensityprocessofτ basedonthisinformation?
This question from credit risk encompasses basic mathematical
P
problems concerning the existence of an intensity process and filtra-
.
h tion expansions, as well as some conceptual issues for credit risk.
at ByrevisitingandextendingthefamousJeulin–Yor[Lecture Notes in
m Math. 649 (1978) 78–97] result regarding compensators undera gen-
eral filtration expansion framework, a novel computation method-
[
ology for the intensity process of a stopping time is proposed. En

1 route, an analogous characterization result for martingales of Jacod
v andSkorohod[Lecture NotesinMath.1583(1994)21–35]underlocal
1 jumping filtration is derived.
9
1
3 1. Introduction. This paper is motivated by the following problem from
1. credit risk. Let (Xt)t≥0 be a continuous-time, time-homogeneous strong
0 Markov process and let τ be its first hitting time of a Borel subset of the
8 state space. Suppose we take samples of X at random times, and suppose
0
also that X has not hit the Borel set by time t. Does the intensity process
:
v of τ exist? And if so, how to calculate it? This question encompasses a basic
i
X mathematical problem regarding the existence of intensity process in gen-
r eral and some conceptual and computational issues in credit risk study in
a
particular.
Intensity (λ ) , compensator (A ) , stopping time τ. The notion of an
t t≥0 t t≥0
intensity process (λ ) of a stopping time is of essential interest in credit
t t≥0
Received February 2006; revised March 2007.
AMS 2000 subject classifications. Primary 60H30, 60G55; secondary 60J99.
Key words and phrases. Intensityoffirsthittingtime,filtrationexpansion,local jump-
ing filtration, Jeulin–Yor formula, Meyer’s Laplacian approximation, L´evy system.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Applied Probability,
2008,Vol. 18, No. 1, 120–142. This reprint differs from the original in pagination
and typographic detail.
1
2 X. GUO ANDY. ZENG
risk, especially in the information-based approach pioneered by Duffie and
Lando [14]. First, if τ is the default time of a firm and is a stopping time
relative to some filtration G = ( ) , then under appropriate technical
t t≥0
G
conditions (such as those in Aven [1]), λ is the instantaneous likelihood of
t
default at time t conditioned on , the information at time t. That is,
t
G
1
(1) λ =lim P(t<τ t+h ) a.s.
t t
h↓0 h ≤ |G
Thus, λ may be viewed as a first-order approximation of the default proba-
t
bility conditioned on the given information . Second, (λ ) is extremely
t t t≥0
G
useful for pricing defaultable derivatives. For example, Lando [28] showed
that when the default time is formulated as the first jump time of a Cox
process (i.e., a doubly stochastic Poisson process) with intensity (λ ) ,
t t≥0
pricing defaultable zero-coupon bond is almost identical to its default-free
counterpartexceptthatthediscountfactorr isreplacedbyr +λ .(Seealso
t t t
Duffie, Schroder and Skiadas [15] for the reduced-form pricing approach via
(λ ) , and Jeanblanc and Rutkowski [25] and B´elanger, Shreve and Wong
t t≥0
[6] for generalizations.)
Mathematically, the intensity process (λ ) of a stoppingtime τ is asso-
t t≥0
ciated withthecompensator Aof τ withrespecttoanappropriatefiltration.
Let (Ω, ,P) be a probability space, τ an arbitrary nonnegative random
F
variable, and F = ( ) any filtration. Since τ is not necessarily an F-
t t≥0
F
stopping time, (λ ) of τ is associated with the (possibly) expanded filtra-
t t≥0
tion G=( ) of F, where τ is a G-stopping time. An increasing measur-
t t≥0
G
ableprocess(A ) iscalledtheG-compensator ofτ ifA =0,1 A is
t t≥0 0 {τ≤t} t
−
a G-martingale and A is G-predictable, that is, A:R Ω [0, ] is mea-
+
× → ∞
surable with respect to the σ-field generated by G-adapted, left-continuous
processes. (This is a special case of the well-known Doob–Meyer decom-
position where a submartingale is decomposed into a martingale and an
increasing predictable process, the latter called the compensator of the sub-
martingale.) Theintensity process (λ ) of τ isthendefinedas theRadon–
t t≥0
Nikodym derivative (dA /dt) , provided that A is a.s. absolutely contin-
t t≥0
uous with respect to the Lebesgue measure. See Br´emaud [5], Chapter II,
D7, T12 and T13.
1.1. Problems and previous work.
Problem 1: Existence and characterization of (λ ) . Given a stopping
t t≥0
time τ with respect to a given filtration, its intensity (λ ) may not exist.
t t≥0
A necessary condition for τ to have an intensity is that τ be totally inacces-
sible. For example, thejumptimes of a L´evy processaretotally inaccessible,
while the first hitting times of a Brownian motion under the natural filtra-
tion are not. However, total inaccessibility is not sufficient for the existence
INTENSITY PROCESS ANDCOMPENSATOR 3
of the intensity of τ; see Davis [10] and Giesecke [17], Proposition 6.1, for
such an example. As any stopping time can be decomposed into a totally
inaccessible part and an accessible part, Zeng [36] recently showed that the
totally inaccessible part can be further uniquely decomposed into an abso-
lute continuous part (i.e., with an intensity) and a singular part. Still, the
question of the existence of the intensity for a given stopping time remains
largely open, except in some special cases.
Problem 1A: Computing the intensity (λ ) via Meyer’s Laplacian ap-
t t≥0
proximation. The existence problem of intensity has been studied in some
cases by identifying specific filtrations and stopping times and by explicit
calculations. For instance, Dellacherie ([11], Chapter V, T56) calculated the
intensity of an arbitrary positive random variable τ underthe natural filtra-
tion of 1 . C¸etin, Jarrow, Protter and Yildirim [7] studied this problem
{τ≤t}
with a filtration generated by the sign of a Brownian motion where the
stopping time τ was chosen to be the first time a Brownian motion doubles
in absolute magnitude after remaining below zero for a certain amount of
time. This result is based on Az´ema’s martingale and the excursion the-
ory of Brownian motion. Sezer [33] considered a similar but more general
problem with a filtration generated by the process (R(X )) , where X is
t t≥0
a one-dimensional diffusion process and R is a Borel function of the form
N i1 (x).Byusingstochasticintegralswithrespecttorandommea-
i=1 (xi−1,xi]
suresand amartingale representation theorem, shewas ableto calculate the
P
intensity process of any totally inaccessible stopping time under this partic-
ular filtration.
Starting from the more intuitive definition of the intensity process in
equation (1), Duffie and Lando [14] considered the first passage time τ of a
one-dimensional diffusion process under a filtration generated by 1 and
{τ≤t}
a discrete, noisy observation of the process at deterministic times. Their
approach is based on the Meyer’s Laplacian approximation for the com-
pensator of a point process and a dominated convergence type of result by
Aven [1]. Following this approach, Guo, Jarrow and Zeng [18, 19] studied
and computed (λ ) for first passage times of general Markov processes,
t t≥0
including diffusion processes with jumps, and Markov modulated processes.
The filtration is of the delayed type, and a minimal filtration expansion was
exploited for computation as in [14].
Despite its intuitive appeal, the Meyer’s Laplacian approximation ap-
proach for computing (λ ) has a serious drawback. First, it is model-
t t≥0
specific: the technical conditions for Aven’s theorem have to be verified case
by case after the first step computation. Second, if one considers the very
simple example of the first passage time of a Brownian motion under its
natural filtration, then the technical condition of Aven’s theorem will be
violated, rendering the entire approach invalid. Thus, this approach may
4 X. GUO ANDY. ZENG
give a wrong “intensity” when it actually does not exist. More importantly,
Aven’s theorem is not suitable for computing (λ ) when it is path depen-
t t≥0
dent. For example, consider a continuous process X continuously observed
exceptonaninterval [S,T]whichisrandom and nonlinearly ordered, thatis,
0<P(S <T)<1. Then (λ ) exists on (S,T), but not outside of (S,T).
t t≥0
Moreover, a direct computation using Aven’s theorem will usually involve a
nontrivial task of finding a dominating process for the approximation.
Problem 1B: Computing the compensator (A ) via the Jeulin–Yor the-
t t≥0
orem. Under a different progressive filtration expansion approach, Jeulin
and Yor [27] suggested that calculating the compensator (and its intensity)
of 1 under the enlarged filtration G can be translated into computing
{τ≤t}
the compensator of Z =E 1 under the original filtration F, since
t {τ>t} t
{ |F }
Z is an F-supermartingale. Unfortunately, the latter problem is not nec-
essarily easier than the former. And [16] considered the very special case
when Z is continuous and decreasing, which is also known as satisfying the
G-hypothesis. In this case, the F-compensator of Z is simply Z Z and
0
−
the usual “grad-log” expression 1 dZ may be analyzed through algebraic
Zt t
manipulation. However, it is not clear how this approach works for general
cases when Z′(t) may not even exist and the grad-log expression no longer
holds.
Problem 2: Consistency of filtration expansions. In the course of study-
ing(λ ) viadifferentcomputationapproachescomestheconsistencyprob-
t t≥0
lem of different filtration expansions.
For any nonnegative random variable τ and a given filtration F=( )
t t≥0
F
for which τ is not a stopping time, (λ ) of τ is associated with the ex-
t t≥0
panded filtration G=( ) of F, where τ is a G-stopping time. As afore-
t t≥0
G
mentioned,thereismorethanonewaytoexpandthefiltration F:[14,18,19]
assumed the minimal filtration expansion, and [27] used more than one fil-
tration expansion including the well-known progressive filtration expansion.
The progressive filtration expansion is different from the minimal one, and
has been recognized as inappropriate from a modeling perspective, as it re-
quires including the original filtration up to . (See Section 2.1 for more
∞
discussions.) It is thus of both mathematical and applied interests to under-
stand the implication of these different filtration expansions, and in general,
their impact on the default intensity and compensator.
1.2. Our approach and main results. In this paper Problem 2 is inves-
tigated by revisiting and extending the famous and beautiful Jeulin–Yor
theorem [27] under a general filtration expansion framework. This general-
ization resolves the consistency issue of filtration expansions, thus, unifies
the approaches and results in [14, 18, 19] and [16].
INTENSITY PROCESS ANDCOMPENSATOR 5
Based on this generalization and inspired by earlier work of Elliott, Jean-
blanc and Yor [16], a different approach to study and compute (λ ) is
t t≥0
proposed. And a class of (τ,F) is identified to obtain an explicit formula
for (λ ) of τ under the expanded filtration G. To overcome the technical
t t≥0
difficulty withoutthe G-hypothesis,the notion of local jumpingfiltrations is
introduced, and an analogous characterization result for martingales of Ja-
cod and Skorohod [24] under this filtration is established. These ensure the
finite variation property of Z =E 1 [hence, the existence of Z′(t)]
t {τ>t} t
{ |F }
on stochastic intervals. As an illustration of this method, special cases are
presentedtoreproducetheworkin[16]whenZ iscontinuousanddecreasing,
as well as the results in [14, 18, 19, 26].
Finally, for the very special case of the full filtration where filtration ex-
pansion is unnecessary, we provide explicit expressions for (λ ) in terms
t t≥0
of L´evy systems for Hunt processes.
Main results. Throughout the paper, (Ω, ,( ) ,P) is a filtered prob-
t t≥0
F F
ability space where F = ( ) satisfies the usual hypotheses and is
t t≥0
F F
any σ-field containing = . τ is a positive random variable, and
F∞ t≥0Ft
Z =E 1 . Moreover, G is any filtration expansion of F such that τ
t { {τ>t}|Ft} W
is a G-stopping time and
(2) t<τ = t<τ .
t t
G ∩{ } F ∩{ }
Theorem 1.1 (Jeulin–Yor: Extension). Given F, τ and G, the G-com-
pensator of τ is
t∧τ 1
(3) dA ,
s
Z
Z0 s−
where A is the F-compensator of Z.
Note that G here can be any filtration expansion satisfying equation (2),
instead of a progressive one used in the original statement of [27]. In fact,
the progressive and the minimal filtration expansions and the filtration ex-
pansions in [27], page 78, and [25], Proposition 5.1, all satisfy equation (2),
and hence are special cases. Therefore, this extended Jeulin–Yor theorem
immediately solves Problem 2. That is, the following:
Corollary 1.1. Compensators of τ under the progressive and the min-
imal expansions of F are identical.
Though intuitively clear, to the best of our knowledge we are not aware
of any prior mathematical result explicitly stating the consistency of com-
pensators/intensities under different filtration expansions.
6 X. GUO ANDY. ZENG
Inaddition,Theorem1.1enablesustosolve Problem1foraclassof (τ,F)
for which explicit formula of (λ ) is derived.
t t≥0
More precisely, suppose (Ω, ,( X) ,X,(P ) ,θ) is a continuous-
F Ft t≥0 x x∈E
time, time-homogeneous strong Markov process, such that F=( ) is
t t≥0
F
a sub-filtration of the completed natural filtration FX =( X) of X, and
Ft t≥0
θ is a shift operator. Assume τ is any positive random variable such that
(·)
the following assumptions hold:
Assumption (A). (The first time of X hitting a Borel subset of the
state space satisfies this condition.) τ >t τ >s and τ θ >t s ( t,
s
⇔ ◦ − ∀
s 0, t>s).
≥
Assumption (B). S and T are a pair of F-stopping times so that X
S
∈
, and S <T T < . And F jumps locally from S to T, that is,
S
F { }⊂{ ∞}
(4) S t<T = S t<T .
t S
F ∩{ ≤ } F ∩{ ≤ }
Assumption (C). (See Section 2.2 for the intuition and motivation of
this assumption.) some random variables V X =σ(X t 0) and V
∃ 1∈F∞ t| ≥ 2∈
, such that T S=g(V θ ,V ) on S<T , for some Borel function g.
S 1 S 2
F − ◦ { }
Then, we have the following.
Theorem 1.2. Given an expanded filtration G of F satisfying equa-
tion (2) and Assumptions (A), (B) and (C), let f be the following:
P (τ >t g(V ,z)>t), if P (g(V ,z)>t)>0,
f(x,z,t)= x | 1 x 1
0, if P (g(V ,z)>t)=0.
(cid:26) x 1
If f(X (ω),V (ω),t) is absolutely continuous with respect to the Lebesgue
S(ω) 2
measure on (0,T(ω) S(ω)) for a.s. ω, then the G-compensator of τ (i.e.,
−
the G-compensator N of N =1 ) satisfies
t {τ≤t}
1 dN e
(S,T] t
f′(X ,V ,t S)
(5) =1e 1 S 2 − dt
{t<τ} (S,T] −f(X ,V ,t S)
(cid:18) S 2 −
h(X ,V ,t S)P (g(V ,z) dt a)
+ S 2 − XS 1 ∈ − ,
f(X ,V ,t S) P (g(V ,z) t a)
S 2 − XS 1 ≥ − (cid:12)a=S,z=V2(cid:19)
(cid:12)
where f′ is the derivative of f with respect to t and h(x,z,t)=f(cid:12)(cid:12)(x,z,t )
− −
P (τ >t g(V ,z)=t). Here P (τ >t V) isaBorel functionof V with P (τ >
x 1 x x
| |
t V =t) taking value at t.
|
INTENSITY PROCESS ANDCOMPENSATOR 7
Finally, under the completed natural filtration of a Hunt process (i.e.,
a c`adla`g strong Markov process with totally inaccessible jump times), a
stopping time τ is totally inaccessible if and only if this process X has a
jumpatτ a.s.on τ < (see[29],pages111–116). Inthiscase,theexplicit
{ ∞}
expression of (λ ) for the associated stopping time is given by the L´evy
t t≥0
system for the Hunt process.
Theorem 1.3. Let X be a Hunt process with L´evy system (U,K). For
simplicity, we assume X has infinite lifetime. Let D be a Borel subset of
the state space, and τ =inf t>0X D . Suppose X has finitely many
t
{ | ∈ }
jumps on every bounded interval. (For a Hunt process X, this condition is
equivalent to having a strictly positive first jump time.) Then, under the
completed natural filtration ( X) of X, the compensator of τ is
Ft t≥0 Λ
t∧τΛ
dUs (1D(y)+1Dc(y)Py(τ =0))K(Xs,dy).
Z0 Z
[τ is the totally inaccessible part of τ such that
Λ
τ(ω), if ω Λ,
τΛ(ω)= , if ω∈/Λ,
(cid:26)∞ ∈
where Λ= τ = τ < ,X =X ].
τ τ−
{ ∞}∪{ ∞ 6 }
Here K(x,dy) represents intuitively the expected number per unit time
of the jumps X makes from x to dy, and the additive functional U is the
internal clock for X. If X is a continuous-time, time-homogeneous Markov
chain, K(x,dy) is its generator matrix and U =t. If X is a L´evy process,
t
K(x,dy)=ν(dy x), where ν is the L´evy measure and U =t. (See [21],
t
−
Section 2, [34] and [32], Chapter VI.)
Organization of the paper. Section 2 provides detailed proofs of the main
results, as well as discussion and comparison with the existing literature.
Section 3 illustrates some old and new explicit forms of (λ ) using our
t t≥0
approach.TheAppendixisaself-containedtreatmentofanextendedversion
of the Jeulin–Yor formula.
2. Proofs and discussions of main results.
2.1. Discussions on Theorem 1.1. The original statement of Theorem
1.1, from the insightful work of Jeulin and Yor [27], was built under a pro-
gressive filtration expansion framework.
A progressive expansion G=( ) in the classical filtration expansion
t t≥0
G
theory (see, e.g., [35]) is defined as follows. Given (Ω, ,( ) ,P) and a
t t≥0
F F
positive random variable τ, define
:= B B ,B t<τ =B t<τ ,
t ∞ t t t
G { ∈G |∃ ∈F ∩{ } ∩{ }}
8 X. GUO ANDY. ZENG
where = σ(τ).
∞ ∞
G F ∨
Note that τ t , implying intuitively that the expanded σ-
∞ t
F ∩{ ≤ }⊂G
field for t τ includes all the information from the unexpanded filtration
t
G ≥
F=( ) up to time . This seems unrealistic from a modeling perspec-
t t≥0
F ∞
tive.
An alternative is the minimal filtration expansion G′=( ′) in [11, 14,
Gt t≥0
18, 19], so that
′= σ(τ t)= σ( τ s s t).
Gt Ft∨ ∧ Ft∨ { ≤ }| ≤
Clearly, the minimal expansion is the minimal way to expand a filtration to
include τ as a stopping time. And unlike the progressive expansion, it does
not require τ t ′.
F∞∩{ ≤ }⊂Gt
Nonetheless, since ′ and G=( ) is right-continuous, we have
Gt⊂Gt Gt t≥0
t<τ = ′ t<τ ′ t<τ t<τ
Ft∩{ } Gt∩{ }⊂ Gu∩{ }!⊂ Gu∩{ }!
u>t u>t
\ \
= t<τ = t<τ .
t t
G ∩{ } F ∩{ }
Therefore, both and the right-continuous augmentation of ′ coincide
Gt Gt
with on t<τ . It is this critical observation that led us to the formula-
t
F { }
tion of the filtration expansion satisfying equation (2), and to the extended
Jeulin–Yor theorem under this general filtration expansion.
Since the original proof by Jeulin and Yor was fairly cryptic, we provide
a self-contained proof in the Appendix for completeness.
2.2. Proof of Theorem 1.2 and discussions.
Two technical lemmas. Wefirstneedtointroducethenotionoflocaljump-
ing filtration and establish an analogous characterization result for martin-
gales in [24], by adapting their proof in Section 2, pages 22–23.
Lemma 2.1. Let S and T be a pair of F-stopping times. If F jumps
locally from S to T according to equation (4), then any uniformly integrable
F-martingale M isa.s.offinitevariationon(S,T].[(S,T]:=∅on S T .]
{ ≥ }
In particular, Z is of finite variation on (S,T].
t
Anotherusefullemmacanbeprovedbymodifyingthetechniqueforjump-
ing filtrations in [20], pages 160–173, to local jumping filtrations.
Lemma 2.2. Let S and T be a pair of F-stopping times. If a filtration F
jumps locally from S to T according to equation (4), then for any F-stopping
time τ, there exists ξ , such that
S
∈F
τ =ξ .
{τ<T} {ξ<T}
INTENSITY PROCESS ANDCOMPENSATOR 9
Proof of Theorem 1.2. The key is to find the compensator of Z 1
t {S≤t<T}
byexploitingthestructureoflocaljumpingfiltration andthestrongMarkov
property.
First, for the F-optional projection Z of (1 ) , by Bayes’ formula
{τ>t} t≥0
and Assumption (B),
P(τ >t,t<T )
S
Z 1 = |F 1 .
t {S≤t<T} {S≤t<T}
P(t<T )
S
|F
HerewehavetakentheconventionthatwhenP(t<T )=0,Z :=0,since
S t
|F
in this case P(τ >t,t<T )=0. Using Assumptions (A)–(C), the strong
S
|F
Markov propertyandconditioningon X,wehaveontheevent S t<T
FS { ≤ }
P(t<T )=P (g(V ,z)>r)
|FS XS 1 |r=t−S,z=V2
and
P(τ >t,t<T )=P(τ >S,τ θ >t S,t S<g(V θ ,V ) )
S S 1 S 2 S
|F ◦ − − ◦ |F
=P(τ >S )P (τ >r,g(V ,z)>r) .
|FS XS 1 |r=t−S,z=V2
So
(6) Z 1 =f(X ,V ,t S)P(τ >S )1 .
t {S≤t<T} S 2 S {S≤t<T}
− |F
Next, by Lemma 2.1, Z is of finite variation on (S,T], since Z is an
F-supermartingale. If A is the F-compensator of Z, then we must have
∞ ∞
(7) E H 1 dZ =E H 1 dA
t {S<t≤T} t t {S<t≤T} t
−
(cid:26)Z0 (cid:27) (cid:26)Z0 (cid:27)
foranyboundedF-predictableprocessH.Bythemonotoneclasstheorem,in
ordertofindtherestriction ofAto (S,T],itsufficestoconsiderH =1 ,
t {R<t}
where R is an F-stopping time.
By Lemma 2.2, there exists a ξ , such that R =ξ . Thus,
S {R<T} {ξ<T}
∈F
equation (6) implies
∞
H 1 dZ
t {S<t≤T} t
(8) Z0
∞
= H 1 f′(X ,V ,t S)P(τ >S )dt+H ∆Z 1 .
t {S<t≤T} S 2 S T T {S<T}
− |F
Z0
Note that S <T,ξ <T . And we have by first conditioning on X
{ }∈FT FS
andthenon ,withthestrongMarkovpropertyandAssumptions(A)–(C),
S
F
and noting 1 P(t T S X)=1 P (t g(V ,z)) ,
{S<T} ≤ − |FS {S<T} XS ≤ 1 |z=V2
E H ∆Z 1
T T {S<T}
{ }
=E 1 [P(τ >T ) f(X ,V ,(T S) )P(τ >S )]
{S<T,ξ<T} T S 2 S
{ |F − − − |F }
10 X. GUO ANDY. ZENG
=P(τ >S,τ θ >g(V θ ,V ),S<T,ξ S<g(V θ ,V ))
S 1 S 2 1 S 2
◦ ◦ − ◦
E 1
− { {S<T,ξ−S<g(V1◦θS,V2)}
f(X θ ,V ,g(V θ ,V ) )P(τ >S )
0 S 2 1 S 2 S
× ◦ ◦ − |F }
∞
=E 1 P(τ >S ) h(X ,V ,u)P (g(V ,z) du)
− {S<T} |FS S 2 XS 1 ∈ |z=V2
(cid:26) Zξ−S (cid:27)
=E 1 P(τ >S )
{S<T} S
− |F
(cid:26)
∞ h(X ,V ,u)P (g(V ,z) du)
1 S 2 XS 1 ∈
{ξ−S<u≤T−S}
×Z0 PXS(g(V1,z)≥u) (cid:12)z=V2(cid:27)
(cid:12)
(cid:12)
=E P(τ >S )
S
− |F
(cid:26)
∞
H 1
t {S<t≤T}
×
Z0
h(X ,V ,t S)P (g(V ,z) dt a)
S 2 − XS 1 ∈ − .
× PXS(g(V1,z)≥t−a) (cid:12)a=S,z=V2(cid:27)
(cid:12)
One can easily check that PXS(g(V1,z)∈du) in the above eq(cid:12)uality is well de-
PXS(g(V1,z)≥u)
finedbyshowingthat u:Y =0 isa.s.negligiblefortheregularconditional
u
{ }
distribution P(T du ), where Y :=P(T t ) and can be chosen as
S t S
∈ |F ≥ |F
left-continuous.
Finally, combining the above equality with equation (8), we have
∞
E H 1 dA
t {S<t≤T} t
(cid:26)Z0 (cid:27)
∞
(9) = E H 1 dZ
t {S<t≤T} t
−
(cid:26)Z0 (cid:27)
=E P(τ >S )
S
|F
(cid:26)
∞
H 1
t {S<t≤T}
×
Z0
h(X ,V ,t S)P (g(V ,z) dt a)
S 2 − XS 1 ∈ −
× PXS(g(V1,z)≥t−a) (cid:12)a=S,z=V2(cid:27)
∞ (cid:12)
E H 1 f′(X ,V ,t S)P(τ >S (cid:12))dt .
t {S<t≤T} S 2 S
− − |F
(cid:26)Z0 (cid:27)
Thus, the F-compensator A of Z, when restricted to (S,T], satisfies
dA =P(τ >S )
t S
|F