Table Of ContentThe martingale representation in a progressive enlargement
of a filtration with jumps ∗
3
1
Kun Tian1, Dewen Xiong†1 and Zhongxing Ye 1,2
0
2 1. Department of Mathematics , Shanghai Jiao Tong University, Shanghai 200240,
n China
a
2. Business Information Management School, Shanghai Institute of Foreign Trade,
J
Shanghai 201620,China
7
]
R
P
.
h
Abstract
t
a
m In this paper, we assume that the filtration F is generated by a d-dimensional
[ Brownian motion W = (W1,··· ,Wd)′ as well as an integer-valued random measure
µ(du,dy). The random variable τ is the default time and L is the default loss. Let
1 G = {Gt;t ≥ 0} be the progressive enlargement of F by (τ,L), i.e, G is the smallest
9v filtration including F such that τeis a G-stopping time and L is Gτe-measurable. Un-
der the density hypothesis, we consider the G-decomposietion of a (P,F) martingale
1
1 and the representation of a G-meartingale. We characterize the conditional density
1 processbyps(s,l), θ1(u;s,l)Iu>s andθ2(u,y;s,l)Iu>s,whichallowsusto describethe
1. survivalprocessGexplicitly. ThenwegivetheexplicitG-decompositionofaFmartin-
0 gale and obtain the predictable representation theorems both for a (P,G)-martingale
3 and a (P∗,G)-martingale, which are different as shown in Callegaro, Jeanblanc and
1 Zargari(2010)[5].
:
v Key words: default time and default loss,progressiveenlargementoffiltration, con-
i ditional density, canonical decomposition, martingale representation.
X
r
a
1 Introduction
In this paper, we assume that the filtration F is generated by a d-dimensional Brownian
motion W = (W ,··· ,W )′ as well as an integer-valued random measure µ(du,dy). Let
1 d
τ ≥ 0 be the default time and L be the default loss, which are random variables. Let G =
{G ;t ≥ 0} be the progressive enlargement of F by (τ,L), i.e, G is the smallest filtration
t
iencluding F such that τ is a G-stopping time and L is Gτe-measurable. We note here that
e
theprogressive enlargement filtration inthis paperis differentfromtraditional progressive
enlargementoffiltratioenintheliterature,seee.g. inJeulin(1980)[15],Jacod(1987)[10],Jean-
blanc, Yor and Chesney(2009) [11], El Karoui, Jeanblanc and Jiao(2009)[8], Jeanblanc and
LeCam(2009)[12],JeanblancandSong(2010)[13],Callegaro, JeanblancandZargari(2010)[5]
∗Supported byNational Natural Science Foundation of China(no.11171215) and Shanghai 085 Project
forfinancialsupports. TheauthorsthankProf. MichaelKohlmannofUniversityofKonstanzforhisadvice
and revising.
†The corresponding author, his Email is [email protected].
and Jeanblanc and Song(2012)[14], among many others. In traditional progressive en-
largement, the authors didn’t consider the more practical situation that once the default
happens, the default loss is immediately generated. Hence the filtration G we consider
here includingthe default time and the default loss simultaneously is more interesting and
realistic.
It is well known that for a general enlargement of filtration, a (P,F)-martingale might
notbea(P,G)-semimartingale,thusJacod(1987)[10]andElKaroui,JeanblancandJiao(2009)[8]
proposed the concept of density hypothesis (H’), which is adapted in the following:
Assumption 1.1. i) the F-regular conditional law of (τ,L) is equivalent to the law of
(τ,L), i.e
e
P(τ ∈ ds,L ∈ dl) ∼ η(ds,dl), for every t ≥ 0;
e
ii) η(ds,dl) has no atoms.
e
From Assumption 1.1, one can see that there exits a so-called conditional density
p (s,l) to describe F-regular conditional law of (τ,L), such that
t
P((τ,L) ∈ B|F )= p (s,l)η(dse,dl), for every t ≥ 0,P-a.s.
t t
Z Z
B
One can see theat {p (s,l);t ≥ 0} is a (P,F)-martingale. In this paper, we investigate
t
{p (s,l);t ≥ 0} more deeply and find that under Assumption 2.1, it is completely deter-
t
mined by p (s,L), θ (u;s,l)′I and θ (u,y;s,l)I as
s 1 u>s 2 u>s
t
p (s,l) = E[p (s,l) F ]+p (s,l)I + p (s,l)θ (u;s,l)′I dW(u)
t s t s t≥s u− 1 u>s
Z
(cid:12) 0
t (cid:12)
+ pu−(s,l)θ2(u,y;s,l)Iu>τe{µ(du,dy)−ν(du,dy)}.
Z Z
0 E
We show in Theorem 2.9 that p (s,l), θ (u;s,l)′I and θ (u,y;s,l)I must satisfy the
s 1 u>s 2 u>s
following condition
∞ ∞ u ′
p (s,l)η(ds,dl) = 1− p (s,l)Zθ1,θ2(u−)θ (u;s,l)η(ds,dl) dW(u)
Z0 ZR s Z0 nZ0 ZR s s,l 1 o
∞ u
− p (s,l)Zθ1,θ2(u−)θ (u,y;s,l))η(ds,dl) {µ(du,dy)−ν(du,dy)},
Z0 ZEnZ0 ZR s s,l 2 o
(1.1)
where
t 1 t
Zθ1,θ2(t) = exp θ (u;s,l)′dW(u)− kθ (u;s,l)k2du
s,l (cid:26)Z 1 2 Z 1 (cid:27)
s s
t t
×exp ln(1+θ (u,y;s,l)) µ(du,dy)− θ (u,y;s,l)ν(du,dy) .
2 2
(cid:26)Z Z Z Z (cid:27)
s E(cid:8) (cid:9) s E
Then we prove that the correspondingsurvival process G has the following Doob-Meyer’s
t
decomposition
t
G = 1− p (s,l)η(ds,dl)
t s
Z0 ZR
t u ′
− p (s,l)Zθ1,θ2(u−)θ (u;s,l)η(ds,dl) dW(u)
Z0 nZ0 ZR s s,l 1 o
t u
− p (s,l)Zθ1,θ2(u−)θ (u,y;s,l))η(ds,dl) {µ(du,dy)−ν(du,dy)},
Z0 ZEnZ0 ZR s s,l 2 o
(1.2)
2
which is completely determined by p (s,l), θ (u;s,l)′I and θ (u,y;s,l)I , which
s 1 u>s 2 u>s
appears to be quite interesting.
Similar to Callegaro, Jeanblanc and Zargari(2010)[5], we show that Mt := M1(t)It<τe+
M2(t;τ,L)It≥τe is a (P,G)-martingale if only if the following two conditions are satisfied:
(i) feor η-almost every u≥ 0 and l ∈ R, {yt(u;l)pt(u;l);t ≥ u} is a (P,F)-martingale;
(ii) the process {ytGt+ 0t Ryu(u,l)pu(u,bl)η(du,dl);t ≥ 0} is a (P,F)-martingale.
R R
Then we mainly stuedy the G-debcomposition of a (P,F) martingale and the representation
of a G-martingale. This representation is very important also as it has many applications
in mathematical finance, see Duffie and Huang(1986)[6], Karatzas and Pikovsky(1996)[18],
Amendinger,BechererandSchweizer(2003)[2],Ankirchner,DereichnerandImkeller(2005)[3],
Jiao and Pham(2009)[16] and Eyraud-Loisel(2010)[7], etc. Let m be a c`adla`g (P,F)-local
martingale of the following form
t t
m = m + ξ (u)′dW(u)+ ξ (u,y){µ(du,dy)−ν(du,dy)},
t 0 1 2
Z Z Z
0 0 E
we show that
t∧τe 1 u
X = m + p (s,l)Zθ1,θ2(u−) ξ (u)′θ (u;s,l)
t t Z0 Gu−(cid:26)Z0 ZR s s,l n 1 1
+ θ (u,y;s,l))ξ (u,y)F (dy) η(ds,dl) du
2 2 u
ZE o (cid:27)
t
− ξ (u)′θ (u;τ,L)+ θ (u,y;τ,L)ξ (u,y)F (dy) du
1 1 2 2 u
Zτe (cid:26) ZE (cid:27)
e e
is a (P,G)-local martingale, from which one can see the G-decomposition of m. It is
notable that the derived G-decomposition also only depends on p (s,l), θ (u;s,l)I
s 1 u>s
and θ (u,y;s,l)I . In the end, we obtain the predictable representation theorems
2 u>s
both for a (P,G)-martingale and a (P∗,G)-martingale. Although recently Jeanblanc and
Song(2012)[14] gave the more general discussion for martingale representation property in
traditional progressively enlarged filtration, it does not apply to our filtration G which is
enlarged by the default time and the default loss. Furthermore, since we have the density
hypothesis, our method is more simple and more directly, and the results are also seen to
be different.
Thepaperis organized as follows: In Section 2, we characterize the conditional density
processandobtain amoreexplicitformof theDoob-Meyer’s decomposition ofthesurvival
process. Then we provide a necessary and sufficient condition for a G-adapted process to
be a G-martingale in Section 3. In Section 4, we will first prove Lemma 4.1 and then
explicitly describe the G-decomposition of a (P,F)-martingale. In the last section, we
obtain the martingale representation theorem for a (P∗,G)-martingale and then give the
martingale representation theorem for a (P,G)-martingale.
2 The setup and notations
We assume that F = {F ;t ≥ 0} is a filtration on (Ω,F,P) carrying an d-dimensional
t
Brownian motion W = (W ,··· ,W )′ as well as an integer-valued random measure
1 d
µ(du,dy) on R ×E, where (E,E) is a Blackwell space.
+
3
Assumption 2.1. The filtration F = {F ;t ≥ 0} is the natural filtration generated by W
t
and µ, i.e.,
F = σ W(s),µ([0,s]×A),B; 0≤ s ≤ t, A∈ B, B ∈N
t
(cid:8) (cid:9)
where N is the collection of P-null sets from F.
In the following, we let P(F) be the family of all F-predictable processes and P(F) =
P(F)⊗E. Weassumethatthecompensatorofµ(du,dy)isgivenbyν(du,dy) =F (dy)du,
uf
whereF (dy)isatransitionkernelfrom(Ω×R ,P(F))tointo(E,E)with F (dy) < ∞,
u + E u
for each u. One can see that any local (P,F)-martingale M has the formR
t t
M = M + f (u)′dW(u)+ f (u,y)(µ(du,dy)−ν(du,dy)),
t 0 1 2
Z Z Z
0 0 E
where f is an Rd-valued F-predictable process and f is a P(F)-measurable function.
1 2
f
Remark. It is easy to see that there exists an F-optional process ϕ = (ϕ ) and a sequence
t
of stopping times (τˆ ) such that for all positive P(F)-measurable function W(ω,t,y),
k
t
f
W(ω,u,y)µ(ω;du,dy) = W(τˆ ,ϕ )I .
Z Z k τˆk τˆk≤t
0 E X
(k)
Furthermore, as the compensator of µ(du,dy) is ν(du,dy) = F (dy)du, so the filtration F
u
is quasi-left continuous.
Let τ be a non-negative random variable and L be a random variable on (Ω,F), and
let the G = {G ;t ≥ 0} be the smallest progressive enlargement filtration of F such that τ
t
is a G-steopping time and L is a Gτe-measurable random variable. Let η(ds,dl) be the law
e
of (τ,L), i.e., η(ds,dl) = P(τ ∈ ds,L ∈ dl). We need the following assumptions
Asseumption 1.1. i) the Fe-regular conditional law of (τ,L) is equivalent to the law of
(τ,L), i.e.,
P(τ ∈ ds,L ∈ dl|F )∼ η(ds,dl), foreevery t ≥ 0;
t
e
ii) η(ds,dl) has no atoms.
e
Lemma 2.2. Y is a G -measurable random variable (r.v.) if and only if
t t
Yt = yt01t<τe+yt1(τ,L)1τe≤t,
where yt0 is a Ft-measurable r.v. and yt1(s,l) is aen Ft⊗B(R+×R)-measurable function.
From Pham(2010)[17], we have the following lemma
Lemma 2.3. 1) Any G-predictable process Y = (Y ) is represented as
t t≥0
Yt = Yt01t≤τe+Y1(τ,L)1t>τe,
where Y0 is a F-predictable process and where eYt1(s,l) is a P(F) ⊗ B(R+) ⊗ B(R)-
measurable function.
2) Any G-optional process Y =(Y ) is represented as
t t≥0
Yt = Yt01t<τe+Y1(τ,L)1t≥τe,
where Y0 is a F-optional process and where Yt1(s,le) is a O(F)⊗B(R+)⊗B(R)-measurable
function.
4
In this paper, we will consider the following basic questions:
1) whetheraF-martingaleisaG-martingaleandtheG-decompositionofanF-martingale;
2) the representation of a G-martingale.
From Assumption 1.1, one can see from Jacod(1987)[10] or Amendinger(1999)[1] that
there exists a strictly positive O(F) ⊗ B(R × R)-measurable function (t;ω;s,l) −→
+
p (ω;s,l), called the (P,F)-conditional density of (τ,L) with respect to η, such that for
t
every (s,l) ∈ R ×R, p(s,l) is a ca`dla`g (P,F)-martingale and for any B ∈ B(R ×R),
+ +
e
P((τ,L) ∈ B|F )= p (s,l)η(ds,dl), for every t ≥ 0,P-a.s.
t t
Z Z
B
e
One can see that p (s,l) = 1 for every s,l.
0
By the “change of probability measure” viewpoint of Song(1987)[20] and similar to
Callegaro, JeanblancandZargari(2010)[5], weintroducethefiltrationGτe,L = {Gτe,L;t ≥ 0}
t
with Gτe,L = F σ(τ,L), one can see that Gτe,L is the initial enlargement of the filtration
t t
F with τ and LWand that F ⊂ G ⊂ Gτe,L. Let
e
1
e Z = ,
t
p (τ,L)
t
similar to Grorud and Pontier(1998)[9] or Ameendinger(1999)[1], one can see that Z is a
strictly positive (P,Gτe,L)-martingale with E(Z ) = 1, for every t ≥ 0. Thus one can define
t
a locally equivalent probability measure P∗ by
dP∗
dP |Gtτe,L = Zt.
Similar to Grorud-Pontier(1998)[9] or Amendinger(1999)[1], one can show that
1. under P∗, (τ,L) is independent of F for every t ≥ 0;
t
2. P∗|F = P|Fe ;
t t
3. P∗|σ(τe,L) = P|σ(τe,L),
which implies P∗(τ ∈ ds,L ∈ dl|F ) = P(τ ∈ ds,L ∈ dl). Similar to Lemma 1.4 in
t
Callegaro, Jeanblanc and Zargari(2010)[5], we have the following lemma
e e
Lemma 2.4. 1) Let y (τ,L) be a Gτe,L-measurable r.v., then for any s ≤ t,
t t
EP∗eyt(τ,L) Gsτe,L = EP∗ yt(u,l) Fs u=τe,l=L;
(cid:0) (cid:12) (cid:1) (cid:0) (cid:12) (cid:1)(cid:12)
2) if y (τ,L) is P-integrable,ethen(cid:12) (cid:12) (cid:12)
t
1
e E y (τ,L) Gτe,L = E y (u,l)p (u,l) F .
t s p (τ,L) t t s u=τe,l=L
(cid:0) (cid:12) (cid:1) s (cid:0) (cid:12) (cid:1)(cid:12)
e (cid:12) (cid:12) (cid:12)
We have the following Corollaries e
Corollary 2.5 (Characterization of (P,Gτe,L)-martingales in terms of (P,F)-martingales).
A process y (τ,L) isa (P,Gτe,L)-martingale if and only if {y (u,l)p (u,l);t ≥ 0} is a(P,F)-
t t t
martingale, for almost every u ≥0,l ∈R.
e
5
Corollary 2.6. Let M = {M ;t ≥ 0} be a bounded (P∗,F)-martingale, then M is a
t
(P∗,Gτe,L)-martingale and hence a (P∗,G)-martingale.
Proof. From part 1) of Lemma 2.4, one can see that for any s ≤ t
EP∗ Mt Gsτe,L = EP∗ Mt Fs u=τe,l=L
(cid:0) (cid:12) (cid:1) = M ,(cid:0) (cid:12) (cid:1)(cid:12)
(cid:12) s (cid:12) (cid:12)
thus M is a (P∗,Gτe,L)-martingale. Since M is F -measurable, M is G -measurable, thus
s s s s
EP∗ Mt Gs = EP∗{EP∗ Mt Gsτe,L |Gs}
(cid:0) (cid:12) (cid:1) = EP∗{Ms|G(cid:0)s} (cid:12) (cid:1)
(cid:12) (cid:12)
= M ,
s
which completes the proof.
Let
∞
G := P(τ > t|F ) = p (s,l)η(ds,dl) and
t t t
Z Z
t R
t
G∗ := P∗e(τ > t|F ) = P∗(τ > t)= P(τ > t)= η(ds,dl),
t t
Z Z
0 R
from Callegaro, Jeanblanec andZargari(20e10)[5], we fiendthatG is a (P,F)-supermartingale
and G∗ is a deterministic continuous and decreasing function.
Theorem 2.7. Let y (τ,L) be a Gτe,L-measurable P-integrable r.v., then for s ≤ t,
t t
e E(yt(τ,L)|Gs) = ys1s<τe+ys(τ,L)1τe≤s
with e e b e
1 ∞
y = E y (u,l)p (u,l)η(du,dl) F
s t t s
Gs (cid:16)Zs ZR (cid:12) (cid:17)
1 (cid:12)
ye (u,l) = E y (u,l)p (u,l) F . (cid:12)
s t t s
p (u,l)
s (cid:8) (cid:12) (cid:9)
b (cid:12)
Proof. TheprooffollowsfromtheproofofLemma1.5ofCallegaro-Jeanblanc-Zargari(2010)([5]).
2.1 The (P,F)-density process
Sinceforanys,l,p(s,l) = {p (s,l);t ≥ 0}isastrictlypositive(P,F)-martingale,{p(s,l);s ≥
t
0,l ∈ R} is a (P,F)-martingale system. Since {p (s,l);t ≥ 0} is a strictly positive martin-
t
gale, one can see that p (s,l) can be represented in the following form
t
t 1 t
p (s,l) = E(p (s,l)|F )I +p (s,l)exp θ (u;s,l)′dW(u)− kθ (u;s,l)k2du
t s t t<s s 1 1
(cid:26)Z 2 Z (cid:27)
s s
t t
×exp ln(1+θ (u,y;s,l)) µ(du,dy)− θ (u,y;s,l)ν(du,dy) I ,
2 2 t≥s
(cid:26)Z Z Z Z (cid:27)
s E(cid:8) (cid:9) s E
(2.1)
6
whichimpliesthatp (s,l)iscompletelydeterminedbyp (s,l),θ (u;s,l)I andθ (u,y;s,l)I .
t s 1 u>s 2 u>s
Since E(p (s,l)|F ) can be written in the following form
s t
t∧s
E(p (s,l)|F ) = 1+ p (s,l)θ∗(u;s,l)′dW(u)
s t u− 1
Z
0
t∧s
+ p (s,l)θ∗(u,y;s,l){µ(du,dy)−ν(du,dy)},
u− 2
Z Z
0 E
hence p (s,l) can also be represented in the following way
t
t
p (s,l) = 1+ p (s,l){θ∗(u;s,l)I +θ (u;s,l)I }′dW(u)
t u− 1 u≤s 1 u>s
Z
0
t
+ p (s,l){θ∗(u,y;s,l)I +θ (u,y;s,l)I }{µ(du,dy)−ν(du,dy)}.
u− 2 u≤s 2 u>s
Z Z
0 E
(2.2)
First, we noted that the following equation by Fubini theorem.
∞ ∞
E(p (s,l)|F )η(ds,dl) = E p (s,l)η(ds,dl) F . (2.3)
s t s t
Zt ZR (cid:16)Zt ZR (cid:12) (cid:17)
(cid:12)
(cid:12)
We first give an examples before characterizing the conditional density process.
Example 2.8. Let θ (u;s,l)I := 0 and θ (u,y;s,l)I := 0, then
1 u>s 2 u>s
p (s,l) = E(p (s,l)|F )I +p (s,l)I .
t s t t<s s t≥s
∞
Since p (s,l)η(ds,dl) ≡ 1 for each t, one can see from (2.3) that
0 R t
R R
∞
p (s,l)η(ds,dl)
t
Z0 ZR
∞ t
= E p (s,l)η(ds,dl) F + p (s,l)η(ds,dl)
s t s
(cid:16)Zt ZR (cid:12) (cid:17) Z0 ZR
∞ (cid:12)
= E p (s,l)η(ds,dl)(cid:12)F = 1,
s t
(cid:16)Z0 ZR (cid:12) (cid:17)
(cid:12)
(cid:12)
for every t ≥ 0. Let t → ∞, one can see that p (s,l) must satisfy the following condition
s
∞
p (s,l)η(ds,dl) = 1.
s
Z0 ZR
More generally, we have the following theorem
Theorem 2.9. For any given bounded p (s,l), θ (u;s,l)I and θ (u,y;s,l)I , let
s 1 u>s 2 u>s
t 1 t
Zθ1,θ2(t) = exp θ (u;s,l)′dW(u)− kθ (u;s,l)k2du
s,l (cid:26)Z 1 2 Z 1 (cid:27)
s s
t t
×exp ln(1+θ (u,y;s,l)) µ(du,dy)− θ (u,y;s,l)ν(du,dy) .
2 2
(cid:26)Z Z Z Z (cid:27)
s E(cid:8) (cid:9) s E
If
p (s,l) = E(p (s,l)|F )I +p (s,l)Zθ1,θ2(t)I
t s t t<s s s,l t≥s
7
is the density process of a pair (τ,L) with respect to (P,F), then p (s,l), θ (u;s,l)I
s 1 u>s
and θ (u,y;s,l)I satisfies the following condition
2 u>s
e
∞ ∞ u ′
p (s,l)η(ds,dl) = 1− p (s,l)Zθ1,θ2(u−)θ (u;s,l)η(ds,dl) dW(u)
Z0 ZR s∞ u Z0 nZ0 ZR s s,l 1 o
− p (s,l)Zθ1,θ2(u−)θ (u,y;s,l))η(ds,dl) {µ(du,dy)−ν(du,dy)}.
Z0 ZEnZ0 ZR s s,l 2 o
(2.4)
Proof. One can see that
t
Zθ1,θ2(t) = 1+ Zθ1,θ2(u−)θ (u;s,l)′dW(u)
s,l Z s,l 1
s
t
+ Zθ1,θ2(u−)θ (u,y;s,l)){µ(du,dy)−ν(du,dy)},
Z Z s,l 2
s E
∞
From p (s,l)η(ds,dl) ≡ 1 for each t, one derives from (2.3) that
t
Z0 ZR
∞
1= p (s,l)η(ds,dl)
t
Z0 ZR
∞
= E p (s,l)η(ds,dl) F
s t
(cid:16)Z0 ZR (cid:12) (cid:17)
t (cid:12) t
− p (s,l)η(ds,dl)(cid:12)+ p (s,l)Zθ1,θ2(t)η(ds,dl)
Z0 ZR s Z0 ZR s s,l
∞
= E p (s,l)η(ds,dl) F
s t
(cid:16)Z0 ZR (cid:12) (cid:17)
t t (cid:12)
+ p (s,l) Zθ1(cid:12),θ2(u−)θ (u;s,l)′dW(u)
Z0 ZR s nZs s,l 1
t
+ Zθ1,θ2(u−)θ (u,y;s,l)){µ(du,dy)−ν(du,dy)} η(ds,dl)
Zs ZE s,l 2 o
∞
= E p (s,l)η(ds,dl) F
s t
(cid:16)Z0 ZR (cid:12) (cid:17)
t u (cid:12) ′
+ p (s,l)Zθ1(cid:12),θ2(u−)θ (u;s,l)η(ds,dl) dW(u)
Z0 nZ0 ZR s s,l 1 o
t u
+ p (s,l)Zθ1,θ2(u−)θ (u,y;s,l))η(ds,dl) {µ(du,dy)−ν(du,dy)},
Z0 ZEnZ0 ZR s s,l 2 o
where the last equality comes from the stochastic Funini theorem for general semimartin-
gales, see Theorem 4.1.1 of Jeanblanc, Yor and Chesney(2009) [11].
Corollary 2.10. Under the conditions of Theorem 2.9, the survival process of τ with
respect to (P,F) is given by
e
t
G = 1− p (s,l)η(ds,dl)
t s
Z0 ZR
t u ′
− p (s,l)Zθ1,θ2(u−)θ (u;s,l)η(ds,dl) dW(u)
Z0 nZ0 ZR s s,l 1 o
t u
− p (s,l)Zθ1,θ2(u−)θ (u,y;s,l))η(ds,dl) {µ(du,dy)−ν(du,dy)}.
Z0 ZEnZ0 ZR s s,l 2 o
(1.2)
8
Proof. From the definition of G , one sees that
t
∞
G = P(τ > t|F ) = p (s,l)η(ds,dl)
t t t
Zt ZR
∞
= e E[p (s,l)|F ]η(ds,dl)
s t
Zt ZR
∞
= E p (s,l)η(ds,dl) F
s t
hZt ZR (cid:12) i
t (cid:12) ∞
= − p (s,l)η(ds,dl)+(cid:12)E p (s,l)η(ds,dl) F
s s t
Z0 ZR hZ0 ZR (cid:12) i
t (cid:12)
= 1− p (s,l)η(ds,dl) (cid:12)
s
Z0 ZR
t u ′
− p (s,l)Zθ1,θ2(u−)θ (u;s,l)η(ds,dl) dW(u)
Z0 nZ0 ZR s s,l 1 o
t u
− p (s,l)Zθ1,θ2(u−)θ (u,y;s,l))η(ds,dl) {µ(du,dy)−ν(du,dy)},
Z0 ZEnZ0 ZR s s,l 2 o
which completes the proof.
Remark.We note here that (1.2) does not directly depend on θ∗(u;s,l) and θ∗(u;s,l),
1 2
that is to say, the survival process G is completely determined by p (s,l), θ (u;s,l)I
s 1 u>s
and θ (u;s,l)I .
2 u>s
Remark. From Corollary 2.14, one can this if and only if θ = θ = 0 and the survival
1 2
process G of τ with respect to (P,F) is a decreasing function, i,e, the F is immersed in G.
e
3 G-martingales’ characterization
In this section, we will characterize (P,G)-martingales in terms of (P,F)-martingales.
Similar to Proposition 2.2 of [5], we have the theorem
Theorem 3.1 (Characterization of (P,G)-martingales in terms of (P,F)-martingales).
Let y = {yt;t ≥ 0}, where yt := yt1t<τe+yt(τ,L)1t≥τe, be a G-adapted process, then y is a
(P,G)-martingale if and only if the following two conditions are satisfied:
e b e
(i) for η-almost every u ≥ 0 and l ∈ R, {y (u;l)p (u;l);t ≥ u} is a (P,F)-martingale;
t t
(ii) the process {ytGt+ 0t Ryu(u,l)pu(u,bl)η(du,dl);t ≥ 0} is a (P,F)-martingale.
R R
To give a proofe, we need thebfollowing lemma
Lemma 3.2 (Projection). Let F∗ be a filtration larger than F, i.e., F ⊂ F. If x is a
uniformly integrable (u.i.) F-martingale, then there exists an F∗-martingale x∗ such that
E(x∗|F ) = x .
t t t
Proof of Theorem 3.1. Let y is a u.i. (P,G)-martingale(otherwise consider the case of
stopped martingale as shown in [5]), since F ⊂ G ⊂ Gτe,L, one can see that there exits a
Gτe,L-martingale y∗(τ,L) such that y = E(y∗(τ,L)|G ), i.e., for η-almost every u≥ 0 and
t t t
l ∈ R, the process {y∗(u,l)p (u,l);t ≥ 0} is a u.i. (P,F)-martingale. For any t ≥ 0, since
t t
yt∗(τ,L)1t≥τe is Gt-meeasurable and 1t≥τe is Gt-meeasurable, one can see that
e yt∗(τ,L)1t≥τe = E[yt∗(τ,L)|Gt]1t≥τe = yt1t≥τe = yt(τ,L)1t≥τe,
e e b e
9
whichimpliesthaty∗(u,l)1 = y(u,l)1 ,η-a.s.,foreveryt ≥ 0,andhence{y (u;l)p (u;l);t ≥
t t≥u t≥u t t
u} is a (P,F)-martingale and (i) is proved. Furthermore, from Theorem 2.7, one can see
b b
that
1 ∞
E(yt∗(τ,L)|Gt)= It<τeGtE(cid:16)Zt ZRyt∗(u,l)pt(u,l)η(du,dl)(cid:12)Ft(cid:17)+yt(τ,L)Iτe≤t,
(cid:12)
thus e (cid:12) b e
1 ∞
ytIt<τe = ytIt<τe = E(yt∗(τ,L)|Gt)It<τe = It<τeGtE(cid:16)Zt ZRyt∗(u,l)pt(u,l)η(du,dl)(cid:12)Ft(cid:17)
e = It<τeG1tE(cid:0)Zt∞eZRyu∗(u,l)pu(u,l)η(du,dl)|Ft(cid:1), (cid:12)(cid:12)
wherethelastequalityresultsfromthe(P,F)-martingalepropertyoftheprocessy∗(u,l)p(u,l),
for η-almost every u≥ 0, l ≥ 0. We deduce
∞
y G = E y∗(u,l)p (u,l)η(du,dl)|F
t t u u t
Z Z
(cid:0) t R (cid:1)
e ∞ t
= E y∗(u,l)p (u,l)η(du,dl)|F − y∗(u,l)p (u,l)η(du,dl),
u u t u u
Z Z Z Z
(cid:0) 0 R (cid:1) 0 R
t
which implies that y G + y (u,l)p (u,l)η(du,dl), 0 ≤ t ≤ ∞ is a (P,F)-
t t u u
n Z0 ZR o
martingale (since y∗(u,l) = y (u,l)) and (ii) immediately follows.
u e u b
Conversely, assuming (i) and (ii), we verify E(y |G ) = y for s ≤ t. Indeed,
t s s
b
E(yt|Gs)= E(It<τeyt+Is<τe≤tyt(τ,L)|Gs)+E(Iτe≤syt(τ,L)|Gs)
1 1
= Is<τeGeE(It<τeyt+bIse<τe≤tyt(τ,L)|Fs)+bpe(τ,L)Iτe≤sE(yt(u,l)pt(u,l)|Fs)|u=τe,l=L.
s s
where we also use Lemmae3.1.2 and bLemema 1.5 in Bielecki, Jeanblbanc and Rutkowski [4]
e
to obtain the last equality. Next, using condition (i), it follows that
1 t 1
E(yt|Gs)= Is<τe E(ytGt + yt(u,l)pt(u,l)η(du,dy)|Fs)+Iτe≤s ys(τ,L)ps(τ,L)
Gs Zs ZR ps(τ,L)
1 e t b b e e
= Is<τe E(ytGt + yu(u,l)pu(u,l)η(du,dy)|Fs) e
Gs Z0 ZR
e1 s b
−Is<τe yu(u,l)pu(u,l)η(du,dl)+Iτe≤sys(τ,L)
Gs Z0 ZR
= Is<τe 1 ysGs+Iτe≤sybs(τ,L) b e
G
s
= y , e b e
s
where we used condition (ii) to obtain the next-to-last identity.
Similarly, we have the following corollary
Corollary 3.3. Let y = {yt;t ≥ 0}, where yt := yt1t<τe +yt(τ,L)1t≥τe, be a G-adapted
process, then y is a (P∗,G)-martingale if and only if the following two conditions are
e b e
satisfied:
(i) for η-almost every u ≥ 0 and l ∈ R, {y (u;l);t ≥ u} is a (P∗,F)-martingale;
t
(ii) the process {ytG∗t + 0t Ryu(u,l)η(du,bdl);t ≥ 0} is a (P∗,F)-martingale.
R R
e b
10
