An Explicit Model of Default Time with given Survival
Probability∗
Monique Jeanblanc
†
Shiqi Song
‡
22 juin 2010
Abstract For a given filtered probability space (Ω, F, P), where F = (Ft )t≥0 is a filtration,
an F-adapted continuous increasing process Λ and a positive P-F local martingale N such that
Zt := Nt e−Λt ≤ 1, t ≥ 0, we construct a model of default time, i.e., a probability measure
QZ and a random time τ on an extension of (Ω, F, P), such that Q[τ > t|Ft ] = Zt , t ≥ 0.
The probability QZ is linked with the well-known Cox model by an explicite density function.
There exist various properties, such as the proportionality or the invariance of conditional
laws, which characterize QZ from others. We will make a particular attention on the complete
separation property. As usual the model is equipped with an enlarged filtration G = (Gt )t≥0
with Gt = Ft ∨ σ(τ ∧ t). We will show that all P-F martingales are QZ -G semimartingale and
give an explicite semimartingale decomposition formula.
1
Introduction
1.1
The problem-A
In this paper, we consider a filtered probability space (Ω, F, P), where the elements of the
filtration F are denoted by (Ft )0≤t≤∞ with F∞ = ∨0≤t<∞ Ft . We consider N a càdlàg positive
P-F local martingale and Λ an F-adapted continuous increasing process.1 We assume that
N0 = 1, Λ0 = 0 and 0 ≤ Nt e−Λt ≤ 1 for all 0 ≤ t < ∞. Let Zt := Nt e−Λt , t ≥ 0. We note that
the process Z is a class (D) supermartingale whose Doob-Meyer decomposition,
thanks to the
R∞
−Λ
s
of Λ, is dZs = e
dNs − Zs dΛs . The random variable 0 Zs dΛs is integrable and
Rcontinuity
· −Λs
e
dN
is
a
BMO
martingale.
s
0
Let (Ω̂, Â, F̂, Q) be a second filtered probability space, where the elements of the filtration
∗
This research benefited from the support of the “Chaire Risque de Crédit”, Fédération Bancaire Française.
Equipe Analyse et Probabilités, EA2172, Université d’Évry Val d’Essonne, 91025 Évry Cedex, France and
Institut Europlace de Finance
‡
Equipe Analyse et Probabilités, EA2172, Université d’Évry Val d’Essonne, 91025 Évry Cedex, France
1
Notice that in this paper calling a a positive number means that a ≥ 0 and calling f (t), t ∈ R, an increasing
function means f (s) ≤ f (t) for s ≤ t.
†
1
F̂ are denoted by (F̂t )t≥0 . Let π be a map from Ω̂ into Ω. We say that (Ω̂, Â, F̂, Q, π) is an
extension of (Ω, F, P) if F̂t = π −1 (Ft ) for all 0 ≤ t ≤ ∞, and P = Q|F̂∞ ◦ π −1 . In this paper,
if extension holds, we shall simply identify F, P with F̂, Q|F̂∞ and equally we shall consider a
random variable Y on (Ω, F∞ ) as the random variable Y ◦π on (Ω̂, F̂∞ ). With this identification
an F-adapted process on Ω is a P-F martingale if and only if it is a Q-F̂ martingale.
In this paper we study the following problem:
Problem-A Construct an extension (Ω̂, Â, F̂, Q, π) of (Ω, F, P) and a random time τ on (Ω̂, Â)
such that Q[τ > t|Ft ] = Nt e−Λt for all 0 ≤ t < ∞.
The two processes N and Λ will be considered as the parameters of the problem-A. We solve
the problem under the hypothesis:
Hy(N, Λ): Assume that 0 ≤ Zt < 1, 0 ≤ Zt− < 1, for 0 < t < ∞, and almost surely, the
dΛt
random measure 1−Z
is bounded on any closed interval contained in (0, ∞).
t
We prove that for each pair of N and Λ satisfying Hy(N, Λ), there exists a solution to the
problem-A (in a companion paper Jeanblanc and Song [6], we shall construct other solutions).
This particular solution is built entirely in terms of the two factors N and Λ, whilst in [6] we
show that the other solutions depend some other factors. We exhibit the different properties
which characterize our solution from others. In particular, we study the relation with the
Cox construction, the conditional law invariance property, the separation condition, the (H’)
property for the enlargement of filtration problem with a decomposition formula of the same
type as in the case of honest time, whilst our solution τ is even not F∞ -measurable.
1.2
Motivation
This question arises from credit risk modelling. Usually, in that setting, the starting point
is a given increasing continuous process Λ, adapted with respect to some information filtration
F = (Ft )t≥0 . The process Λ contains the only parameters to be calibrated from the market data.
Then, one constructs a random time τ on some probability space such that (Ht −Λt∧τ , t ≥ 0) is a
G-local martingale, where Ht = 11τ ≤t , and G = (Gt )t≥0 with Gt = Ft ∨ σ(τ ∧ t). The "canonical"
construction is the Cox model, where τ := inf{t : Λt ≥ Θ} and Θ is a r.v. independent of
F∞ , with unit exponential law. However, the Cox model is a fairly simple model, because its
conditional survival probability Q[τ > t|Ft ] is equal to e−Λt , whilst it is known that, for a
general τ , for 11{τ ≤t} − Λt∧τ to be a G-local martingale, a necessary and sufficient condition
is that Q[τ > t|Ft ] has a multiplicative decomposition Nt eΛt , where N is a positive F-local
martingale. Our aim here is to take into account this second parameter N and to study the
properties of the models with conditional survival probability of the form N e−Λ .
The problem-A has been studied in Gapeev et al. [4] where, inspired by an example from
filtering, a solution has been found by solving an integral equation (see Section 6). The result
on the integral equation in [4] is important to well understand the problem-A. We shall give
below a precise description on the relation between our approach and [4]. Another related work
is Nikeghbali and Yor [11] where for a strictly positive martingale N with no positive jumps
and limt→∞ Nt = 0, the authors have established the multiplicative decomposition formula
P[g > t|Ft ] =
Nt
= Nt e− ln(St ) , 0 ≤ t < ∞,
St
where St = sup0≤s≤t Ns and g = sup{t ≥ 0 : Nt = St }. It is a particular case of the problem-A
2
with Λt = ln(St ). It is particular also because the solution (P, g) is living on the initial space Ω.
With the hypothesis Hy(N, Λ) our situation is radically different. In fact none of our solutions
can live in the initial space Ω.
1.3
Miscellaneous setting conditions
In this paper our response to the problem-A will be constructed on the product space
[0, ∞] × Ω. This space will exclusively be equipped with the product σ-field B[0, ∞] ⊗ F∞ , with
the map π : π(s, ω) = ω and the map τ : τ (s, ω) = s, with the filtration F̂ = π −1 (F) which will
immediately be identified with F. In such a setting, for ([0, ∞] × Ω, B[0, ∞] ⊗ F∞ , F̂, Q, π) to
be an extension of (Ω, F, P), or for ([0, ∞] × Ω, Q, τ ) to be a solution model of the problem-A,
the only element we have to determine is the probability Q. Therefore, in these cases, we shall
simply say that Q is an extension, or that Q is a solution. We equip systematically the product
space [0, ∞] × Ω with the progressively enlarged filtration G = (Gt )t≥0 made from F with the
random time τ , i.e. Gt = Ft ∨ σ(τ ∧ t), t ≥ 0. This fix the framework for our discussion on the
enlargement of filtration problem.
Notice that the notation τ can be used in a general situation for a general random time.
This will not cause confusion because, each time the product space is concerned, τ will be the
projection map defined above.
We notice also that it is the filtration G which is considered instead of the filtration G+ .
The reason is that the Q-G semimartingale decomposition formula we will establish for the P-F
local martingales exhibits only G-adapted càdlàg processes. This yields that in the filtration
G+ , we will have the same decomposition formula.
2
Basic Analysis
In this section, we collect diverse ideas we can have immediately on the problem-A.
2.1
Simple cases
The simplest form of the problem-A is the case where N ≡ 1 and Λ = ι, for which the
solution on the product space is the probability measure making τ an exponential random
variable independent of F∞ . More generally, if N ≡ 1 and Λ∞ = ∞, again a solution exists on
the product space given by the probability measure ν Λ :
Z t
Λ
(1)
ν [A ∩ {s < τ ≤ t}] = P[IA
e−Λv dΛv ], A ∈ F∞ , 0 < s < t < ∞
s
This construction2 coincides with the Cox construction. We will call ν Λ the Cox measure.
2.2
Föllmer’s measure
The condition of the problem-A on the conditional survival probability reminds us immediately of the Föllmer’s measure (see Meyer [9] ; or the Doléans-Dade measure in our case of a class
(D) supermartingale). We wonder if the solution of our problem is not simply the Föllmer’s
2
In this paper, for a probability measure P, we use the notation P[X] for EP [X]
3
measure. To simplify the formula, we suppose Z∞ = 0. The Föllmer’s measure in this case is
given by
Z ∞
F
Q [F ] = P[
F (s, ·)Zs dΛs ]
0
Under this measure, the Ft -conditional survival probability of τ is effectively Zt . In order to
F
be a solution
R ∞of the problem-A, Q must be an extension of (Ω, F, P). This is equivalent to the
condition: 0 Zs dΛs ≡ 1 (see Nikeghbali [10] for more discussion on such situation), because
the restriction of QF onto F∞ is calculated by
Z ∞
F
Q [A] = P[IA
Zs dΛs ], A ∈ F∞ .
0
We note that in such a case the Föllmer’s measure QF coincides with the Cox measure ν Λ .
(Although the Föllmer’s measure need not be a solution, we notice that any solution of the
problem-A coincides with the Föllmer’s measure on the F-predictable σ-field.)
The above observation is completed by another observation. If a solution (Q, τ ) exists, the
tails conditional laws of τ are linked to the supermartingale Z by the identity:
Q[τ ∈ du, t ≤ τ ≤ ∞|Ft ] = Q[11{t≤u<∞} Zu dΛu + Z∞ δ∞ (du)|Ft ]
But few information on Q[τ ∈ du, 0 ≤ τ ≤ t|Ft ] can be drawn from the supermartingale Z.
That is the main difficulty of our problem.
2.3
Time change
We can believe that if Λt = t for all 0 ≤ t < ∞, the problem could be easier. The following
result clarifies the situation. Set ι(t) = t, 0 ≤ t ≤ ∞ and
Ct = inf{s ≥ 0 : Λs > t}, 0 ≤ t < ∞
Then, (Ct , t ≥ 0) is an increasing family of F-stopping times. Since Λ is a finite increasing
process, C∞ = limt↑∞ Ct = ∞. We consider the time change induced by (Ct , t ≥ 0). We recall
that ZCt = NCt e−t∧Λ∞ , t ≥ 0, is an FC -supermartingale with 0 ≤ ZCt ≤ 1 (we use the fact
ΛCt = t ∧ Λ∞ ) and NC = (NCt , t ≥ 0) is a true FC -martingale. The following result can be
checked straightforwardly :
Theorem 2.1 Assume the hypothesis Hy(N, Λ). If (Q, τ ) is a solution of the problem-A on the
space (Ω, F, P) with parameters N and Λ, (Q, Λτ ) is a solution of the Problem-A on (Ω, FC , P)
with parameters NC and ι∧Λ∞ . Conversely if (Q, τ ) is a solution of the problem-A on (Ω, FC , P)
with parameters NC and ι, (Q, Cτ − ) is a solution of the problem-A on (Ω, F, P) with parameters
N and Λ.
2.4
Probability change
Suppose Λ∞ = ∞. Let ν Λ be Cox measure (see the formula (1)). A natural idea to solve
the problem-A is to look for solutions among the probability measures Q on the product space,
which are absolutely continuous with respect to ν Λ .
Let G be the progressively enlarged filtration from F by the random time τ . Let Q be
a probability measure on the product space, absolutely continuous with respect to ν Λ , with
4
dQ
Λ
density ξ = dν
Λ . Recall that, under the probability measure ν , the immersion property holds
(see Bielecki et al. [2]). Then, it can be proved that Q is a solution of the problem-A, if and
only if there exists a family L = (Lt , 0 ≤ t < ∞) of positive B[0, t] ⊗ Ft measurable functions,
such that
1. L satisfies the equation :
Z t
(L) : Nt e−Λt +
Lt (s, ·)e−Λs dΛs = 1, ∀t ≥ 0.
0
2. for any 0 < s < ∞, the process Lt (s), t ≥ s is a càdlàg ν Λ -G martingale
3. for 0 ≤ t < ∞, ν Λ [ξ|Gt ] = Nt I{t<τ } + Lt (τ, ·)I{τ ≤t} .
The key point here is the integral equation (L). In [4] a sufficient condition (see Section 6.2
for a detailed discussion on that condition) has been found to solve the equation (L). We do
not know how to solve the equation (L) in general.
2.5
No Solution Case
To solve the problem-A an extension of (Ω, F, P) is in general necessary. For example, no
solution can exist on Ω for the problem-A with parameter N ≡ 1 because then random variable
Λτ is independent of F∞ (see [2], p.136).
3
A solution of the problem-A
In this section, we assume Hypothesis Hy(N, Λ). We construct a solution of the problem-A
which is absolutely continuous with respect to the measure defined by
Z
P[IA
(dΛv + δ∞ (dv))], A ∈ F∞ , 0 ≤ s < t < ∞.
]s,t]
This looks like the probability change approach mentioned in the preceding section. But we
will not pass by the equation (L).
3.1
Preliminary computations
Lemma 3.1 Let 0 < θ < ∞ be fixed and consider the process defined for θ ≤ t ≤ ∞:
½ Z t
¾
Zs
Jt = Jt (θ) := (1 − Zt ) exp −
dΛs
θ 1 − Zs
Then, the process J = (Jt , θ ≤ t ≤ ∞) is a P-F uniformly integrable martingale.
Proof : Notice that Hy(N, Λ) guarantees the well definiteness of J. Recall that the process Z
is a supermartingale with the Doob-Meyer decomposition dZs = e−Λs dNs − Zs dΛs . Applying
integration by parts formula on J, for θ ≤ t < ∞,
½ Z t
¾
Zs dΛs
dJt = − exp −
e−Λt dNt
θ 1 − Zs
5
we see that J is a P-F local martingale on [θ, ∞). But clearly positive and bounded by 1, it is
a uniformly integrable martingale on [θ, ∞].
As a corollary, we obtain
Lemma 3.2 Let 0 < θ < ∞ be fixed and define
¾
½ Z t
Jt (θ)
1 − Zt
Zs dΛs
Et (θ) =
=
exp −
, θ ≤ t ≤ ∞.
1 − Zθ
1 − Zθ
θ 1 − Zs
The processes (Et (θ), θ ≤ t ≤ ∞) is a P-F uniformly integrable martingale ; and, for t ≥ θ,
E[Et (θ)|Fθ ] = 1.
Note that, in different situation, Et (θ) may be denoted in different forms such as Et (θ, ω), or
EtN,Λ (θ). We need to know the behaviour of Et (θ) when t is fixed and θ → 0.
Lemma 3.3 Let 0 < t ≤ ∞ be fixed. There is a sequence (θn ) decreasing to 0 such that
limn (1 − Zθn )Et (θn ) = 0 P-almost everywhere.
Proof. We begin with the expectation limit. Using the preceding lemma and the fact that Z
is càdlàg bounded by 1 and Z0 = 1, we can write
lim P[(1 − Zθ )Et (θ)] = lim P[(1 − Zθ )] = 0.
θ↓0
θ↓0
But everything being positive, the sequence (1 − Zθ )Et (θ) converges in L1 to zero. The lemma
follows.
Lemma 3.4 For any 0 < θ < ∞, θ ≤ t < ∞, we have
Z t
Zv Et (v)dΛv = (1 − Zt ) − (1 − Zθ )Et (θ),
θ
and, for θ ≤ t ≤ ∞,
Z
0
t
Zv Et (v)dΛv = 1 − Zt
Proof : We compute for 0 < θ < ∞, θ ≤ t < ∞,
½ Z t
¾
Z t
Z t
1 − Zt
Zs dΛs
Zv Et (v)dΛv =
Zv
exp −
dΛv
1 − Zv
θ
θ
v 1 − Zs
½ Z t
¾¯v=t
Zs dΛs ¯¯
= (1 − Zt ) exp −
¯
v 1 − Zs
v=θ
= (1 − Zt ) − (1 − Zθ )Et (θ)
Let θ goes to zero along the sequence (θn ) in the preceding lemma. We obtain
Z t
Zv Et (v)dΛv = 1 − Zt .
0
Passing to the limit, this identity holds for t = ∞.
6
Remark 3.1 Suppose that N is continuous. We have the identity
Z t −s
Z
e dNs 1 t e−2s dhN is
Et (θ) = exp{−
−
}
2 θ (1 − Zs )2
θ 1 − Zs
This explains the use of the notation of an exponential martingale. Let us write the identity:
Z t
Zu dΛu
1 − Zt = (1 − Zθ ) exp{
}Et (θ)
θ 1 − Zu
It is the multiplicative decomposition of the
submartingale (1 − Zt , t ≥ θ) where Et (θ)
R t Zpositive
u dΛu
is the martingale factor and (1 − Zθ ) exp{ θ 1−Zu } is the bounded variation factor. Notice that
we keep a θ > 0 in the decomposition, because in a neighborhood of θ = 0 the integrability of
1
1−Zθ is troublesome.
3.2
The Construction
Define

0





Zθ (ω)E∞ (θ, ω)
u(θ) = uZ (θ, ω) :=





Z∞ (ω)
θ=0
0<θ<∞
θ=∞
Notice that, for each θ, the random function u(θ) on θ ∈ [0, ∞] is non negative and it is càdlàg
in θ over (0, ∞).
Lemma 3.5 The random function u(θ) is a probability density with respect to the random
measure dΛθ + δ∞ (dθ) over [0, ∞], i.e.,
Z ∞
u(θ)dΛθ + u(∞) = 1 .
0
Proof. According to Lemma 3.4, we can write
Z ∞
Z ∞
u(v)dΛv + u(∞) =
Zv E∞ (v)dΛv + Z∞ = (1 − Z∞ ) + Z∞ = 1
0
0
Now we construct a probability QZ on the product space by:
Z
Z
0
Q [A ∩ {t < τ ≤ t }] = P[IA
u(θ)(dΛθ + δ∞ (dθ))]
(2)
]t,t0 ]
for A ∈ F∞ , 0 ≤ t < t0 ≤ ∞. Applying Lemma 3.5, we see that QZ is an extension of (Ω, F, P),
i.e. QZ [A] = P[A], for A ∈ F∞ . Compute now the conditional survival probability under QZ .
Let 0 < t < ∞ and A ∈ Ft . We have
Z ∞
Z ∞
QZ [A ∩ {τ > t}] = P[IA (
u(v)dΛv + u(∞))] = P[IA (
Zv E∞ (v)dΛv + u(∞))]
t
t
From Lemma 3.4, it follows
QZ [A ∩ {τ > t}] = P[IA ((1 − Z∞ ) − (1 − Zt )E∞ (t) + Z∞ )] = P[IA − IA (1 − Zt )E∞ (t)]
7
According to Lemma 3.2, P[E∞ (t)|Ft ] = 1 and so
QZ [A ∩ {τ > t}] = P[IA − IA (1 − Zt )] = P[IA Zt ] = QZ [IA Zt ]
i.e., QZ [τ > t|Ft ] = Zt . We can now state the following theorem.
Theorem 3.1 Suppose Hy(N, Λ). Then, the probability measure QZ defined in (2) on the
product space is a solution of the problem-A.
4
The density process with respect to Cox measure
Remark that, if N ≡ 1, it can be checked that Et (θ) ≡ 1, 0 < θ ≤ t ≤ ∞. If moreover
−Λ
Λ∞ = ∞ so that Z∞ = 0, the random function ue (θ) is simply e−Λθ and the probability
measure QZ becomes
Z t
−Λ
Qe [A ∩ {s < τ ≤ t}] = P[IA
e−Λθ dΛθ ] = ν Λ [A ∩ {s < τ ≤ t}]
s
where ν Λ is the Cox measure (see the formula (1)).
Return back to a general Z = N e−Λ . We know by the construction that QZ is absolutely
continuous with respect to ν Λ . Now we look at its density process.
Theorem 4.1 Suppose Hy(N, Λ) and Λ∞ = ∞ so that Z∞ = 0. Set
½
Ns (ω)E∞ (s, ω), 0 < s < ∞, ω ∈ Ω
L(s, ω) =
0
s = 0 or s = ∞
(in other words, L = Nτ E∞ (τ ) ν Λ -a.s.). Then, L is the probability density of QZ with respect
to ν Λ : QZ = L · ν Λ , and the G-density process Lt = ν Λ [L|Gt ], 0 ≤ t < ∞, is given by Lt =
Nt∧τ Et∨τ (τ ), 0 ≤ t ≤ ∞.
Proof. The first part of the theorem is a direct consequence of the construction. For the
assertion on G-density process (Lt , 0 ≤ t < ∞), it is the consequence of the following lemma.
Lemma 4.1 Suppose Λ∞ = ∞. Let 0 ≤ a < ∞. Let F (s, ω) be a positive B[0, ∞] ⊗ F∞
measurable ν Λ integrable function on [0, ∞] × Ω. Then, we have
ν Λ [F |Ga ](s, ω) = G(s, ω)I{s≤a} + J(ω)I{a<s}
where
G(s, ω)I{s≤a} = P[F (s, ·)|Fa ](ω)I{s≤a}
ν Λ -almost surely
R∞
J(ω) = eΛa (ω)P[ a F (s, ·)e−Λs Λ(ds)|Fa ](ω)
P-almost surely
In particular, we have
ν Λ [Nτ E∞ (τ )|Ga ] = Na∧τ Ea∨τ (τ )
8
Proof. The first part of the lemma is well-known. Applying this to F = Nτ E∞ (τ ), its G(s, ω)
part is computed by
P[Ns E∞ (s)|Fa ] = Ns Ea (s) = Na∧s Ea∨s (s), s ≤ a.
Its J(ω) part is computed by, s > a,
R∞
R∞
eΛa P[ a Nv E∞ (v)e−Λv Λ(dv)|Fa ] = eΛa P[ a Zv Λ(dv)|Fa ]
= eΛa Za (recall that Z∞ = 0)
= Na∧s Ea∨s (s)
Combining these two results we obtain the lemma.
Remark 4.1 The density process (Nt∧τ Et∨τ (τ ), 0 ≤ t < ∞) satisfies the equation (L) (see
subsection 2.4). It is just a restatement of Lemma 3.4.
Characterizations of QZ
5
The probability measure QZ possesses many properties and can be characterized in different
ways as we present now. We assume the hypothesis Hy(N, Λ) and Λ∞ = ∞.
5.1
Definition of new Properties
We introduce the following properties. Let K to be the set of all families (χt , 0 ≤ t < ∞)
such that each χt = χt (s, ω) is a function defined on [0, t] × Ω and is B[0, t] ⊗ Ft -measurable.
• Partial Separation An element χ = (χt , 0 ≤ t < ∞) in K is said to be in a partially separable
(v)
(v)
form, if, for any 0 < v < ∞, there exist two F-adapted càdlàg processes (Xt )t≥v , (Yt )t≥v
(defined only for t ≥ v) such that
(v)
χt (s, ·) = Xt Ys(v) ,
for all v ≤ s ≤ t < ∞. The processes X (v) , Y (v) will be called respectively the first and the
second components in the partially separable form.
• Complete Separation An element χ = (χt , 0 ≤ t < ∞) in K is said to be in a completely
separable form, if there exist two F-adapted càdlàg processes (Xt )t≥0 , (Yt )t≥0 such that
χt (s, ·) = Xt Ys ,
for all 0 ≤ s ≤ t < ∞. The processes (X, Y ) will be called respectively the first and the second
components in the completely separable form.
Now consider a probability measure Q on the product space.
¦ Local Density Hypothesis 3 We say that Q satisfies the local density hypothesis, if there
exists a positive element α = (αt , 0 ≤ t < ∞) in K such that, for 0 ≤ t < ∞ and B ∈ B[0, t],
Z
Q[τ ∈ B|Ft ] =
αt (s, ·)e−Λs dΛs
B
3
To be distinguished from the density hypothesis introduced in Karoui et al. [3]
9
The family α will be called a family of local density functions. To such a family α, we can
ascribe different properties (We
the implications of these properties):
R twill see later
−Λ
u
? Decreasing condition 0 αt (u, ·)e
dΛu > 0 for any 0 < t < ∞, and, for 0 ≤ u < ∞,
the map
αt (u, ·)
t → Rt
−Λs dΛ
s
0 αt (s, ·)e
is a continuous decreasing function in t ∈ [u, ∞).
? Partial separation form α is in partially separable form.
? Complete separation form α is in a completely separable form.
? Canonical form The family α takes the form αt (s, ·) = Ns Et (s) for 0 < s ≤ t < ∞.
¦ Conditional law Invariance We say that Q satisfies the conditional law invariance hypothesis, if, for any 0 < a ≤ b, for any A, B ∈ B[0, a], we have
Q[τ ∈ A|Fa ]
Q[τ ∈ A|Fb ]
=
Q[τ ∈ B|Fa ]
Q[τ ∈ B|Fb ]
whenever Q[τ ∈ B|Fb ] > 0.
¦ Proportionality Hypothesis Let Q be a probability measure on (Ω, A). We say that the
conditional proportionality hypothesis is satisfied, if Q[τ ≤ t|Ft ] > 0 for any 0 < t < ∞, and,
for any 0 < u < ∞, the map
Q[τ ≤ u|Ft ]
t→
Q[τ ≤ t|Ft ]
has a version which is a continuous decreasing function in t ∈ [u, ∞).
Remark 5.1 Chronologically it was the proportionality hypothesis which leaded us to find our
solution QZ to the problem-A.
5.2
QZ in canonical form
Theorem 5.1 The probability QZ constructed in Theorem 3.1 is the unique extension of (Ω, F, P)
on the product space which satisfies the local density hypothesis with a local density in canonical
form.
Proof. Let 0 ≤ t < ∞, A ∈ Ft , B ∈ B[0, t], by construction of QZ , we can write
Z
Z
Z
Q [A ∩ {τ ∈ B}] = Q [IA
B
Z
Ns E∞ (s)e
−Λs
Z
dΛs ] = Q [IA
B
Ns Et (s)e−Λs dΛs ]
This proves, for QZ , the local density hypothesis with canonical form. The uniqueness is obvious.
10
5.3
Computations of conditional laws
In this subsection we consider a probability measure Q on the product space which is a
solution of the problem-A. Note that
Q[τ = 0] = lim(1 − Q[τ > t]) = lim(1 − Q[Zt ]) = 0.
t↓0
t↓0
Lemma 5.1 For any 0 < t < ∞, for A ∈ B[t, ∞],
Z
Q[τ ∈ A|Ft ] = Q[ Nu e−Λu dΛu + I{∞∈A} Z∞ |Ft ]
A
Proof. For u ≥ t,
Z
Q[τ > u|Ft ] = Q[Zu − Z∞ + Z∞ |Ft ] = Q[
∞
u
Ns e−Λs dΛs + Z∞ |Ft ]
That is what should be proved.
Lemma 5.2 Suppose the proportionality hypothesis. Then, for any 0 ≤ t < ∞, for A ∈ B[0, t],
Z
Q[τ ∈ A|Ft ] =
Ns Et (s)e−Λs dΛs
A
i.e., the local density hypothesis holds in canonical form.
Proof. Let 0 < s < ∞ be fixed. Let
Q[τ ≤ s|Ft ]
Q[τ ≤ s|Ft ]
=
, s ≤ t < ∞.
Q[τ ≤ t|Ft ]
1 − Zt
rt :=
By hypothesis (rt , t ≥ s) is a continuous decreasing function. It is clearly F-adapted. Consider
the identity
Q[τ ≤ s|Ft ] = (1 − Zt )rt .
(3)
The left-hand side of this identity is an F-martingale. Therefore, the right-hand side should
have its drift null, i.e.,
rt Zt dΛt + (1 − Zt )drt = 0, s ≤ t < ∞.
Solving this equation leads to
Z
rt = exp{−
t
s
Zu dΛu
}, s ≤ t < η,
1 − Zu
where η := inf{t ≥ s : rt = 0}. But the above expression shows that necessarily η = ∞. We
substitute this expression of rt into the identity (3) to get
Z t
Zu dΛu
}
Q[τ ≤ s|Ft ] = (1 − Zt ) exp{−
s 1 − Zu
Let s ↓ 0. We see that
Z
exp{−
0
t
Zu dΛu
}=0
1 − Zu
11
Using these facts, we write finally
Z
t
Zu dΛu
Q[τ ≤ s|Ft ] = (1 − Zt ) exp{−
}
1
− Zu
s
Z s
Z t
Z s
Zv dΛv
Zu
=
(1 − Zt ) exp{−
}
dΛu =
Nu Et (u)e−Λu dΛu
1
−
Z
1
−
Z
v
u
0
u
0
That ends the proof.
Lemma 5.3 We assume that, for any 0 < v < a < ∞, on the set
{Q[v < τ ≤ a|Fa ] > 0} ∈ Fa
the map
(v)
t → rt
=
Q[v < τ ≤ a|Ft ]
Q[v < τ ≤ t|Ft ]
has a version which is a strictly positive continuous decreasing random function in t ∈ [a, ∞).
Then the local density hypothesis holds in canonical form.
Proof. The essential of the proof is the same as in the preceding lemma, but we have to take
into account the parameter v > 0. Notice
lim Q[v < τ ≤ a|Fa ] = Q[τ ≤ a|Fa ] = 1 − Za > 0
v↓0
Let 0 < v < a < ∞ and consider
(v)
rt = rt
:=
Q[v < τ ≤ a|Ft ]
.
Q[v < τ ≤ t|Ft ]
By hypothesis, on the set Av = {Q[v < τ ≤ a|Fa ] > 0} ∈ Fa , the function (rt , a ≤ t < ∞) is
strictly positive, continuous and decreasing, and clearly F-adapted. Rewrite this fact in another
form
Q[v < τ ≤ a|Ft ] = (mt − Zt )rt , t ≥ a,
(4)
where mt = Q[v < τ |Ft ] (a càdlàg version). On the set Av , the left-hand side of the identity
(4) is an F-martingale on the interval [a, ∞). Therefore the right-hand side must has its drift
null, i.e.
rt Zt dΛt + (mt − Zt )drt = 0, t ≥ a.
Let
Sn = inf{t ≥ a : mt − Zt ≤ n1 },
ηv = supn Sn ,
Because ma − Za > 0 on the set Av , one has ηv > a. Let a ≤ t < ηv . We can write
Z t
Z t
Zu dΛu
(mt − Zu ) dru
=−
−∞ < ln(rt ) =
(m
−
Z
)
r
(m
t
u
u
t − Zu )
a
a
where we use the fact that ln(ra ) = 0. In other words,
Z
rt = exp{−
a
t
Zu dΛu
}, a ≤ t < ηv .
mu − Zu
12
We have now another equation for Q[v < τ ≤ a|Fb ] on the set Av :
Z t
Zu dΛu
Q[v < τ ≤ a|Ft ] = (Q[v < τ |Ft ] − Zt ) exp{−
}, a ≤ t < ηv .
a Q[v < τ |Fu ] − Zu
Letting v ↓ 0 we get Av ↑ Ω, ηv ↑ ∞ (because of Hy(N, Λ)) and Q[v < τ ≤ a|Ft ] ↑ Q[τ ≤ a|Ft ].
We obtain
Z t
Zu dΛu
Q[τ ≤ a|Ft ] = (1 − Zt ) exp{−
}
a 1 − Zu
Let a ↓ 0. We see that
Z
exp{−
0
t
Zu dΛu
}=0
1 − Zu
We have finally
Q[τ ≤ a|Ft ]
R t u dΛu
= (1 − Zt ) exp{− a Z1−Z
}
Ra
R t ZuudΛu Zs
= R0 (1 − Zt ) exp{− s 1−Zu } 1−Zs dΛs
a
= 0 Ns Et (s)e−Λs dΛs
The lemma is proved.
5.4
Equivalences
We now prove that, if Q is a solution of the problem-A, the various properties above are
equivalent, leading to the fact that QZ is the only solution that enjoys these properties.
Theorem 5.2 Let Q be a probability measure on the product space which is a solution of the
problem-A. The following conditions are equivalent :
1. The proportionality hypothesis.
2. The local density hypothesis in canonical form, i.e. Q coincides with QZ .
3. The local density hypothesis in partially separable form with its first component strictly
positive.
4. The local density hypothesis with decreasing condition.
5. The conditional law invariance and the existence of a version of f (a, b) = Q[τ ≤ a|Fb ],
such that, outside of a negligible set, for all 0 ≤ a ≤ b < ∞, we have the limits
lim
f (a0 , b) = f (b, b), lim
f (a, b0 ) = f (a, b)
0
0
a ↑b
b ↓b
Proof. The implication 1. ⇒ 2. has been proved in Lemma 5.2.
To see the implication 2. ⇒ 3. we need only to write, for 0 < v < s < t,
R t w dΛw
1−Zt
Ns Et (s) = Ns 1−Z
exp{− s Z1−Z
}
s
R t Zw dΛww R s Zw dΛw
1−Zt
= Ns 1−Zs exp{− v 1−Zw + v 1−Zw }
=
R
w dΛw }
Ns exp{ vs Z1−Z
t
w
R1−Z
t Zw dΛw
1−Zs
exp{ v 1−Z }
w
This proves the partial separation form with
(v)
Xt
R s Zw dΛw
N
exp{
1 − Zt
s
v 1−Zw }
=
R t Zw dΛw , Ys(v) =
1 − Zs
exp{ v 1−Zw }
13
(5)
The condition 3. is proved.
Ra
v
Let us see the implication 3. ⇒ 2.. Let 0 < v < a < b < ∞. If Q[v < τ ≤ a|Fa ] > 0, we have
(v)
(v)
Ys e−Λs dΛs > 0. Since, for a ≤ t < ∞, Xt > 0 by hypothesis, we can write
(v)
rt
R a (v) −Λ
Ys e s dΛs
Q[v < τ ≤ a|Ft ]
:=
= Rvt (v)
−Λs dΛ
Q[v < τ ≤ t|Ft ]
s
v Ys e
This expression defines clearly a strictly positive continuous and decreasing function in t ∈
[a, ∞). By the Lemma 5.3, we see that the condition 2 must be satisfied.
For the implication 2. ⇒ 4., we introduce the map
½ Z t
¾
Ns
Zv
Ns Et (s)
=
exp −
dΛv
t → Rt
=
−Λv dΛ
1 − Zt
1 − Zs
s 1 − Zv
v
0 Nv Et (v)e
Ns Et (s)
on the interval [s, ∞). It is clearly continuous and decreasing. The condition 4. is proved.
Consider the implication 4. ⇒ 1.. Let 0 < s ≤ t < ∞. First of all Q[τ ≤ t|Ft ] = 1 − Zt is
strictly positive. Notice also that
Rt
Q[τ ≤ t|Ft ] = R0 αt (v, ·)e−Λv dΛv = 1 − Zt
s
Q[τ ≤ s|Ft ] = 0 αt (v, ·)e−Λv dΛv
So,
1
Q[τ ≤ s|Fs ]
=
Q[τ ≤ t|Ft ]
1 − Zt
By hypothesis,
t→
Z
s
0
αt (v, ·)e−Λv dΛv .
αt (v, ·)
1 − Zt
is a continuous decreasing function in t ∈ [v, ∞). The dominated convergence theorem yields
that
Q[τ ≤ s|Ft ]
Q[τ ≤ t|Ft ]
has a version which is a continuous decreasing function in t ∈ [s, ∞). The condition 1. is proved.
Up to now, we have proved the equivalence between the conditions 1. to 4.. To see that these
conditions are also equivalent to the condition 5., let us prove first that condition 3. implies
the conditional law invariance. We begin with the fact that, for any 0 < v < a ≤ b, for any
B ∈ B[v, a],
Z
Q[τ ∈ B|Fb ] =
R
(v) −Λu
dΛu
B Yu e
If Q[τ ∈ B|Fb ] > 0,
facts yield, for A, B ∈ B[v, a],
(v)
B
Xb Yu(v) e−Λu dΛu
(v)
> 0. Recall that, by hypothesis, Xa
R (v) −Λ
Yu e u dΛu
Q[τ ∈ A|Fa ]
Q[τ ∈ A|Fb ]
= RA (v)
=
−Λ
Q[τ ∈ B|Fa ]
Q[τ ∈ B|Fb ]
u dΛ
u
B Yu e
Such an identity being true for all v > 0, we have actually
Q[τ ∈ A|Fa ]
Q[τ ∈ A|Fb ]
=
Q[τ ∈ B|Fa ]
Q[τ ∈ B|Fb ]
14
(v)
> 0, Xb
> 0. These
for any A, B ∈ B([0, a]). The invariance hypothesis is proved. Let us now prove that condition
2. implies the property (5). It is enough to set
Z a
f (a, b) =
Nu Eb (u)e−Λu dΛu
0
and we check directly that f (a, b) is a version of Q[τ ≤ a|Fb ] which satisfies the property (5).
The implication from condition 1-4 to the condition 5. is proved.
We show finally the implication 5. ⇒ 1.. Let f (a, b), 0 ≤ a ≤ b < ∞, be the function in the
condition 5. According to the conditional law invariance hypothesis, we can write
f (a, b0 )
f (a, b)
=
, a ≤ b < b0 < ∞
f (b, b)
f (b, b0 )
whenever f (b, b0 ) > 0. The above identities hold outside of a commun exception set. Now the
property (5) together with the conditional law invariance makes valid the two limits:
lim
0
b ↑b
f (a, b0 )
f (a, b)
f (a, b)
= lim
=
0
0
0
0
f (b , b ) b ↑b f (b , b)
f (b, b)
(notice that there is no problem of zero denominator f (b0 , b), because f (b, b) = 1 − Zb > 0 so
that for b0 close to b, f (b0 , b) > 0.) and
lim
0
b ↓b
f (a, b0 )
f (a, b)
f (a, b0 )
=
lim
=
f (b0 , b0 ) b0 ↓b 1 − Zb0
f (b, b)
These two limits mean that ff (a,b)
(b,b) is a continuous function in b ∈ [a, ∞). But this function is
also decreasing as the following computation shows:
f (a, b0 )
f (a, b0 )
f (a, b)
=
≥
, a ≤ b ≤ b0 < ∞
f (b, b)
f (b, b0 )
f (b0 , b0 )
The condition 1. is proved
6
Complete Separation Condition
We assume Hy(N, Λ) and Λ∞ = ∞. We notice that the completely separable form is absent
from Theorem 5.2. This will be the main subject in this section.
6.1
Local Solution
We shall need the notion of local solution.
Local Solution Let Q be a set function on the product space. We say that Q is a local
solution to the problem-A, if there exists an increasing sequence (Tn )n≥1 of F-stopping times
such that limn→∞ Tn = ∞ and, for any n ≥ 1, Q is a probability measure on any of GTn and
P|FTn = Q ◦ π −1 |FTn , and Q[τ > Tn ∧ t|FTn ∧t ] = ZTn ∧t for all 0 ≤ t < ∞.
Notice that the above two conditions on Q are well defined because they concern only the
behaviour of Q over σ-field GTn .
15
6.2
Solution via Equation (L)
The complete separation hypothesis has been initially introduced to solve the equation (L)
(see subsection 2.4), which is a relation upon the positive elements L in K:
Z t
−Λt
(L) : Nt e
+
Lt (u, ·)e−Λu dΛu = 1, 0 ≤ t < ∞.
0
The following result is a generalization of a result proved in [4]. The proof here is based on
the same idea.
Theorem 6.1 Suppose that
Z
Γt :=
t
0
d[N, N ]u
=
(Nu− − eΛu )2
Z
t
0
e−2Λu d[N, N ]u
<∞
(1 − Zu− )2
for all 0 ≤ t < ∞. Then, the equation (L) has a positive solution L in K in completely separable
form, i.e., there are (Xt )t≥0 , (Yt )t≥0 two positive F-adapted càdlàg processes such that Lt (u, ·) =
Xt Yu . Moreover, if we introduce the random variable on the product space:
ξt = Nt I{t<τ } + Xt Yτ I{τ ≤t} , t ≥ 0.
the process ξ is a ν Λ -G local martingale. If we define a set function Q on the product space so
that, for each G-stopping time T reducing ξ to a ν Λ -G uniformly integrable martingale, Q is
the probability measure on GT defined by
¯
dQ ¯¯
= ξT ,
dν Λ ¯GT
then, Q is a local solution to the problem-A.
Proof.
Notice that N is a locally bounded P-F local martingale. The hypothesis implies that
√
Γ is a locally P-integrable process. Set
Z t
1
ϑt =
dNu , t > 0.
Λu
0 Nu− − e
The condition on Γ implies that ϑ is well defined. Let X be the Doléans-Dade exponential of
ϑ:
Z t
Xt = 1 +
Xu− dϑu , 0 ≤ t < ∞.
0
Notice that
∆u ϑ =
1
∆u N = −
Nu− − eΛu
µ
Zu − Zu−
1 − Zt−
¶
> −1.
So the Doléans-Dade exponential Xt is well defined and remains strictly positive. Set
¶
µ
Z t
1
Λu
, 0 ≤ t < ∞.
Yt = 1 +
e d
Xu
0
We note that
d(XY ) = Y− dX + X− dY + d[X,
¡ ¢Y ]
= Y− X− dϑ + eΛ X− d X1 + eΛ d[X, X1 ]
16
and
with
1
0=
dX + X− d
X−
1
X− dX
µ
1
X
¶
+ d[X,
1
]
X
= dϑ. So,
(XY )0 = 1,
N0 = 1,
d(XY )t = ((XY )t− − eΛt )dϑt
dNt = (Nt− − eΛt )dϑt
The uniqueness of the solution of the above equation (see Protter [12]) implies XY = N . This
proves that Y is positive.
Set now Lt (u) = Xt Yu , 0 ≤ u ≤ t < ∞. We check that
Z t
1
−Λt
Xt Y t e
+
Xt Yu e−Λu dΛu = Xt ·
= 1, 0 ≤ t < ∞,
Xt
0
so that L is a solution of the equation (L). Consider the local martingale property of ξ. We can
choose a sequence (Tn )n≥1 of F-stopping times such that N Tn , X Tn are uniformly integrable
for each n ≥ 1 and limn→∞ Tn = ∞. We apply Lemma 4.1 to compute ν Λ [ξTn ∧t |GTn ∧u ] for
0 ≤ u < t < ∞. We compute firstly the components G(s, ω) and J(ω):
G(s, ·)11{s≤Tn ∧u} =
=
=
=
and
J
=
=
=
=
=
=
P[NTn ∧t 11{Tn ∧t<s} + XTn ∧t Ys 11{s≤Tn ∧t} |FTn ∧t ]11{s≤Tn ∧u}
P[XTn ∧t Ys 11{s≤Tn ∧t} |FTn ∧u ]11{s≤Tn ∧u}
P[XTn ∧t |FTn ∧u ]Ys 11{s≤Tn ∧u}
XTn ∧u Ys 11{s≤Tn ∧u}
R∞
eΛTn ∧u P[ Tn ∧u (NTn ∧t 11{Tn ∧t<s} + XTn ∧t Ys 11{s≤Tn ∧t} )e−Λs dΛs |FTn ∧u ]
R∞
eΛTn ∧u P[ Tn ∧u (NTn ∧t 11{Tn ∧t<s} + Xs Ys 11{s≤Tn ∧t} )e−Λs dΛs |FTn ∧u ]
R T ∧t
eΛTn ∧u P[NTn ∧t e−ΛTn ∧t + Tnn∧u Ns e−Λs dΛs |FTn ∧u ]
eΛTn ∧u P[ZTn ∧t + ZTn ∧u − ZTn ∧t |FTn ∧u ]
eΛTn ∧u ZTn ∧u
NTn ∧u .
Combining these two terms we obtain
ν Λ [ξTn ∧t |GTn ∧u ] = NTn ∧u 11{Tn ∧u<τ } + XTn ∧u Yτ 11{τ ≤Tn ∧u} = ξTn ∧u
This shows that ξ Tn is a ν Λ -G uniformly integrable martingale, or ξ is a ν Λ -G local martingale.
Let us check that Q is a local solution of the problem-A along the sequence (Tn )n≥1 . For n ≥ 1,
we compute
ν Λ [ξTn |FTn ]
= limn→∞ ν³Λ [(NTn ∧ n)I{Tn <τ } + (XTn ∧ n)(Yτ ∧ n)I{τ ≤Tn } |F´Tn ]
RT
= limn→∞ (NTn ∧ n)e−ΛTn + (XTn ∧ n) 0 n (Ys ∧ n)e−Λs dΛs
= 1
thanks to the equation (L). This means P|FTn = Q ◦ π −1 |FTn . We compute also
Q[τ > Tn ∧ t|FTn ∧t ] = ν Λ [ξTn ∧t 11{τ >Tn ∧t} |FTn ∧t ] = ν Λ [NTn ∧t 11{τ >Tn ∧t} |FTn ∧t ] = NTn ∧t e−ΛTn ∧t .
This proves the theorem.
17
Theorem 6.2 Suppose the same hypothesis as in Theorem 6.1. The local solution Q constructed
in Theorem 6.1 is the restriction of the probability measure QZ constructed in Theorem 3.1. In
this case, the local density hypothesis holds for QZ in completely separable form with the first
component strictly positive.
Proof. We need only to repeat the proof of the implication from condition 3. =⇒ condition
1. in Theorem 5.2. For this reason we need to adapt the notions such as the local density
hypothesis to the case of a local solution of the problem-A. But this is straighforward. Now we
can conclude that Q satisfies the local density hypothesis in completely separable form, which
implies the canonical form. However, there can be only one set function with local density
hypothesis in canonical form on the product space, i.e. the probability measure QZ
6.3
Completely Separable Form
In this subsection we compare precisely the completely separable form and the canonical
form.
Theorem 6.3 Let Q be a probability measure on the product space which is a solution of the
problem-A. The following conditions are equivalent:
1. Local density hypothesis in completely separable form.
2. Local density hypothesis in canonical form and the existence of the finite and strictly
positive limit for any 0 < t < ∞:
Z t
Zu
Vt = lim(1 − Zs ) exp{
dΛu } ∈ (0, ∞)
s↓0
s 1 − Zu
In the case where one of the above conditions is satisfied, let X be the first component in the
completely separable form. We have
a
(\) lima↓0 1−Z
= X0 ∈ (0, ∞)
R a Vea−2Λu d[N,N ]u
(†) Γa = 0 (1−Zu− )2 < ∞, 0 ≤ a < ∞
−Λ
e
and X is equal to the Doléans-Dade exponential of the P-F local martingale − 1−Z
·N
−
Proof. The implication 1. ⇒ 2.. Let X, Y denote the first and the second components of the
local density function. The completely separable form implies the partially separable form.
According to Theorem 5.2, we have thus Ns Et (s) = Xt Ys for 0 < s ≤ t < ∞, and
Z t
0 < 1 − Z t = Xt
Yu e−Λu dΛu < 1, X0 Y0 = N0 = 1.
0
We obtain the equalities
1
exp{−
Ns
1 − Zs
Z
s
t
Zu
Ns Et (s)
Ys
dΛu } =
= Rt
−Λu dΛ
1 − Zu
1 − Zt
u
0 Yu e
or equivalently
Z
(1 − Zs ) exp{
s
t
Ns
Zu
dΛu } =
1 − Zu
Rt
18
0
Yu e−Λu dΛu
= Xs
Ys
Z
0
t
Yu e−Λu dΛu
so that
Z
Vt = lim(1 − Zs ) exp{
s↓0
s
t
Zu
dΛu } = X0
1 − Zu
Z
t
0
Yu e−Λu dΛu ∈ (0, ∞)
The condition 2. is proved.
The implication 2. ⇒ 1.. We write, for 0 < v < s < t,
R t w dΛw
1−Zt
exp{− s Z1−Z
}
Ns Et (s) = Ns 1−Z
s
R twZw dΛw R s
1−Zt 1−Zv
= Ns 1−Zs 1−Zv exp{− v 1−Zw + v
=
By hypothesis,
Rs
1−Zt (1−Zv ) exp{ Rv
Ns 1−Zs
(1−Zv ) exp{ vt
Z
Vt = lim(1 − Zv ) exp{
v↓0
v
t
Zw dΛw
}
1−Zw
Zw dΛw
}
1−Zw
Zw dΛw
1−Zw }
Zw dΛw
} ∈ (0, ∞)
1 − Zw
This limit leads to the identity
Ns Et (s) = Ns
1 − Zt Vs
= Xt Ys
1 − Zs Vt
Ns Vs
t
where Xt = 1−Z
Vt and Ys = 1−Zs . Notice that the so defined processes X, Y are progressively
measurable with respect to F. Let us show that Xt is integrable. In fact this is because Xt ≥ 0
and by Fatou’s lemma,
R t u dΛu
1−Zt
P[Xt ] ≤ limv↓0 P[ 1−Z
exp{− v Z1−Z
}] = 1.
v
u
For 0 < v ≤ s ≤ t, for A ∈ Fs ,
Z
Z v
Yu e−Λu dΛu Xs ] = P[IA P[τ ≤ v|Fs ]] = P[IA P[τ ≤ v|Ft ]] = P[IA Xt
P[IA
0
0
v
Yu e−Λu dΛu ]
Rv
Since 0 Yu e−Λu dΛu is strictly positive, the above computation shows that (Xt )v≤t<∞ is a
strictly positive P-F martingale. This is true for any v > 0. We conclude that the process
(Xt )0<t<∞ has a strictly positive càdlàg version. Let us abuse the notation (Xt )0<t<∞ to denote its càdlàg version. From this version we obtain de càdlàg version for (Vt )0<t<∞ that we
denote again by (Vt )0<t<∞ . Combining these two càdlàg version we get also a càdlàg version
for (Yt )0<t<∞ that we denote also by (Yt )0<t<∞ . Now repeating the argument in the first part
of the proof, we obtain
R t Zu
Rt
−Λu dΛ
(1 − Zs ) exp{ s 1−Z
dΛ
}
=
X
u
s
u
0 Yu e
u
Rt
−Λ
Xt 0 Yu e u dΛu = 1 − Zt > 0.
By hypothesis, the left-hand side of the first equality tends to Vt > 0. We prove thus X, as well
as V and Y , is also right continous at zero.
The condition (\) is true because
X0 = R t
Vt
−Λu dΛ
u
0 Yu e
> 0.
Look at the implication 1., 2. ⇒ (†).. The process X is strictly positive. The martingale
property implies that X− is also strictly posotive.
19
Let 0 < v < a < ∞. Since 1 − Zu− > 0, 0 < u < ∞, (1 − Z− ) is integrable with respect to
N on all intervals [v, a] with 0 < v < a < ∞. We have the identity
Ã
!
Z t
11(v,∞) e−Λ
Zu dΛu
1 − Zt = (1 − Zv ) exp{
}E −
·N
1 − Z−
v 1 − Zu
t
R t u dΛu
for v ≤ t < ∞. Dividing both sides by (1 − Zv ) exp{ v Z1−Z
} and taking the limit when v ↓ 0,
u
we obtain
!
Ã
11(v,∞) e−Λ
·N
Xt = lim E −
v↓0
1 − Z−
t
Since X > 0,
1
X−
is integrable with respect to X. We write for 0 < c < t,
µ
Xt = Xc E
But also
11(c,∞)
·X
X−
¶
t
³
´
11
e−Λ
Xt = limv↓0 E − (v,∞)
·
N
1−Z−
³
´t ³
´
11(v,∞) e−Λ
11
e−Λ
= limv↓0 E − 1−Z− · N E − (c,∞)
·
N
1−Z−
c
t
³
´
11(c,∞) e−Λ
= Xc E − 1−Z− · N
t
It follows that
µ
E
11(c,∞)
·X
X−
Ã
¶
=E
t
11(c,∞) e−Λ
−
·N
1 − Z−
!
t
We get then successively
Rt
dXu
c Xu− =
R t d[X,X]u
2
c Xu−
R t d[X,X]
u
2
0 Xu−
Notice that
N and
Rt
0
d[X,X]u
2
Xu−
−
Rt
=
=
e−Λu dNu
, 0<
u−
R t e1−Z
−2Λu d[N,N ]
u
,
c
(1−Zu− )2
R t e−2Λu d[N,N ]u
0
(1−Zu− )2
c
c < t.
0 < c < t.
< ∞. From this, we conclude that
e−Λ
(1−Z− )
is integrable with respect to
µ
¶
e−Λ
Xt = E −
·N
1 − Z−
t
The condition (†). is proved.
7
G-Martingales
In this last section we study the enlargement of filtration problem under the probability
measure QZ constructed in Theorem 3.1. We will show that the P-F local martingales X are
QZ -G semimartingales and give their semimartingale decomposition formula. This formula, as
the classical ones, operates on X only through the predictable bracket hN, Xi and is composed
of two parts: the part of before-τ and a part of after-τ . It is therefore an interesting example,
because until now, such a formula was known in the enlargement of filtration problem only for
honest time (see Barlow [1], Jeulin and Yor [8] and Jeulin [7]) and initial time (see Jeanblanc and
20
Le Cam [5]). But our time τ under QZ is not a honest time and in general not an initial time.
(Nevertheless Theorem 2.1 links the pair (QZ , τ ) to an initial time Λτ by a change of time.) To
end this program we will prove that the probability QZ is the only solution of the problem-A
on the product space to possess this enlargement of filtration formula. These questions are also
considered in [6] with a different approach.
Throughout this section, we assume the hypothesis Hy(N, Λ) and Λ∞ = ∞. We note then
QZ [τ = 0] = limv↓0 (1 − QZ [τ > v]) = limv↓0 (1 − QZ [Zv ]) = 0
QZ [τ = ∞] = limv↑∞ QZ [τ > v] = limv↑∞ QZ [Zv ] = 0
Moreover, QZ [τ = T ] = 0 for any F∞ -measurable random time T .
7.1
F-martingales as G-semimartingales
We begin with the equation
n R
t
1
dt Et (s) = − 1−Z
exp
− s
s
Zw dΛw
1−Zw
o
−Λt
e
e−Λt dNt = −Et− (s) 1−Z
dNt , 0 < s ≤ t < ∞
t−
consequence of Lemma 3.1. We note also Et (u) = Es (u)Et (s) for 0 < u < s < t. The decomposition of a P-F local martingale before the time τ is well-known by [8, 7]:
Lemma 7.1 Let X be a P-F local martingale. Let hN, Xi be the P-F predictable dual projection
of [N, X] (hN, Xi exists). Then,
Z
Xτ ∧· −
·
11{s≤τ }
0
e−Λs
dhN, Xis
Zs−
is a QZ -G local martingale.
As for what happens after the time τ , a more evolved computation is needed. The computation is possible because of the nice properties of Et (s).
Lemma 7.2 Let X be a P-F local martingale. Then,
Z
Xτ ∨· − Xτ +
0
·
11{τ <s}
e−Λs
dhN, Xis
1 − Zs−
is a QZ -G local martingale.
Proof. Let 0 < a < b < ∞. Let T be any F-stopping time such that everything concerned
in the computation will be integrable. Let A ∈ Fa and h(t) ∈ B[0, ∞] bounded function. We
21
compute
=
=
=
=
=
=
=
=
=
Q[IA h(τ )(XτT∨b −P
XτT∨a )]
T
T
limn↑∞ Q[IA h(τ ) ∞
k
k=0 I{ n
≤τ < k+1
} (X k+1 ∨b − X k+1 ∨a )]
n
n
n
R k+1
P
T
n
limn↑∞ ∞
Q[I
h(s)Z
E
(s)dΛ
(X
− X Tk+1 ∨a )]
s ∞
s
A k
k+1
k=0
∨b
n
n
n
R k+1
P∞
T
n
limn↑∞ k=0 Q[IA k h(s)Zs E k+1 ∨b (s)dΛs (X k+1 ∨b − X Tk+1 ∨a )]
n
n
n
n
R k+1
P∞
k+1
T
n
limn↑∞ k=0 Q[IA k h(s)Zs E k+1 (s)dΛs E k+1 ∨b ( n )(X k+1 ∨b − X Tk+1 ∨a )]
n
n
n
n
n
R k+1
R k+1
−Λ
P∞
∨b
k+1 e ξ
n
n
− limn↑∞ k=0 Q[IA k h(s)Zs E k+1 (s)dΛs k+1 Eξ− ( n ) 1−Zξ− dhN, X T iξ ]
∨a
n
n
n
R k+1
R k+1
−Λ
P∞
∨b
e ξ
n
n
− limn↑∞ k=0 Q[IA k h(s)Zs E k+1 (s)dΛs k+1 E∞ ( k+1
) 1−Z
dhN, X T iξ ]
n
ξ−
∨a
n
n
n
R k+1
R k+1
P
∨b e−Λξ
T
n
n
− limn↑∞ ∞
h(s)Zs E∞ (s)dΛs k+1
k=0 Q[IA k
1−Zξ− dhN, X iξ ]
∨a
n
n
R k+1
P
∨b e−Λξ
T
n
− limn↑∞ Q[IA h(τ ) ∞
I
k+1
k
k+1
k=0 { n ≤τ < n }
1−Zξ− dhN, X iξ ]
∨a
n
R τ ∨b e−Λξ
−Q[IA h(τ ) τ ∨a 1−Z
dhN, X T iξ ]
ξ−
This proves the lemma.
We can conclude.
Theorem 7.1 Let X be a P-F local martingale. Then,
Z ·
Z ·
e−Λu dhN, Xiu
e−Λu dhN, Xiu
X − X0 −
I{u≤τ }
+
I{τ <u}
Zu−
1 − Zu−
0
0
(6)
is a QZ -G local martingale.
7.2
Formula inversion
We now prove that QZ is the only probability measure solution of the problem-A on the
product space under which the formula (6) holds.
Theorem 7.2 Let Q be a probability measure on the product space which is a solution of
problem-A. Suppose that the formula (6) holds under Q. Then, Q = QZ .
Proof. Let 0 < a < b < ∞. Let M be any bounded P-F martingale. Let T be any F-stopping
time such that T ≥ a and everything concerned in the computation is integrable. Let f (t) =
Q[I{τ ≤a} |Ft ], a ≤ t < ∞. We compute
=
=
=
=
=
=
=
Q[f (t ∧ T )MtT ]
Q[I{τ ≤a} MtT ]
Q[I{τ ≤a} (MaT + MtT − MaT )]
R t e−Λu
Q[I{τ ≤a} (MaT − a 1−Z
dhN, M T iu )]
u−
R
t e−Λu
Q[I{τ ≤a} MaT − I{τ ≤a} a 1−Z
dhN, M T iu ]
u−
R
t
e−Λu
dhN, M T iu ]
Q[(1 − Za )MaT − a f (u−) 1−Z
u−
R t∧T
e−Λu
Q[(1 − Za )MtT − a f (u−) 1−Z
dNu MtT ]
u−
R
−Λ
u
t∧T
e
Q[MtT ((1 − Za ) − a f (u−) 1−Z
dNu )]
u−
22
This identity implies
Z
f (t) = (1 − Za ) −
t
f (u−)
a
e−Λu
dNu , a ≤ t < ∞
1 − Zu−
Comparing this equation with that of dt Et (a), we conclude
Q[I{τ ≤a} |Ft ] = f (t) = (1 − Za )Et (a)n
R t s dΛs o
= (1 − Zt ) exp − a Z1−Z
n R s
o
Ra
t s dΛs
Zv
= (1 − Zt ) 0 exp − v Z1−Z
1−Zv dΛv
s
Ra
= 0 Nv Et (v)e−Λv dΛv
= QZ [I{τ ≤a} |Ft ], a ≤ t < ∞.
n R
o
t s dΛs
Here we have used the fact that exp − 0 Z1−Z
= 0 (see the proof of Lemma 5.3). As Q
s
and QZ coincide on F∞ , they must coincide on the whole space. The theorem is proved.
Références
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307-324.
[2] Bielecki T.R., Jeanblanc M, and Rutkowski M. (2009) Credit Risk Modelling Osaka University Press.
[3] El Karoui N., Jeanblanc M. and Jiao Y. (2009) "What happens after a default: the conditional density approach". Stochastic Processes and their Applications, 120(7), 1011-1032.
[4] Gapeev, P., Jeanblanc, M., Li, L., and Rutkowski, M. : Constructing random times with
given survival processes and applications to valuation of credit derivatives. in: Contemporary Quantitative Finance, C. Chiarella and A. Novikov, eds., Springer-Verlag, Berlin
Heidelberg New York, 2010.
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times". Stochastic Processes and their Applications, 119(8), 2523-2443.
[6] Jeanblanc M. and Song S. (2010) "Random times with given survival probability and their
F-martingale decomposition formula". Working paper.
[7] Jeulin T. (1980) Semi-martingales et grossissement d’une filtration. Lecture Notes in Mathematics, 833, Springer.
[8] Jeulin T. and Yor M. (1978) "Grossissement d’une filtration et semi-martingales : formules explicites". Séminaire de Probabilités XII, 78-97. Lecture Notes in Mathematics
649. Springer-Verlag.
[9] Meyer, P.A. (1972) "La mesure de H.Föllmer en théorie des surmartingales" Séminaire de
Probabilités VI, p.118-129.
[10] Nikeghbali A. (2006) "An essay on the general theory of stochastic processes". Probability
Surveys Vol.3 345-412. http ://www.i-journals.org/.
[11] Nikeghbali, A. and Yor, M. (2007) "Doob’s Maximal Identity, Multiplicative Decompositions and Enlargements of Filtrations", in Joseph Doob : A Collection of Mathematical
Articles in his Memory, Burkholder, D. editor, Illinois Journal of Mathematics, 50, 791-814
23
[12] Protter P. (2004) Stochastic integration and differential equations. Second Edition, Springer.
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