An Explicit Model of Default Time with given SurvivalProbability∗

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 thatZt := Nte

−Λt ≤ 1, t ≥ 0, we construct a model of default time, i.e., a probability measureQZ 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 conditionallaws, which characterize QZ from others. We will make a particular attention on the completeseparation 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 andgive an explicite semimartingale decomposition formula.

1 Introduction

1.1 The problem-AIn 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 positiveP-F local martingale and Λ an F-adapted continuous increasing process.1 We assume thatN0 = 1,Λ0 = 0 and 0 ≤ Nte

−Λt ≤ 1 for all 0 ≤ t < ∞. Let Zt := Nte−Λt , t ≥ 0. We note that

the process Z is a class (D) supermartingale whose Doob-Meyer decomposition, thanks to thecontinuity of Λ, is dZs = e−ΛsdNs − ZsdΛs. The random variable

∫∞0 ZsdΛs is integrable and∫ ·

0 e−ΛsdNs is a BMO martingale.

Let (Ω, A, F,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, France1Notice 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.

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F are denoted by (Ft)t≥0. Let π be a map from Ω into Ω. We say that (Ω, A, F,Q, π) is anextension of (Ω,F,P) if Ft = π−1(Ft) for all 0 ≤ t ≤ ∞, and P = Q|F∞ π−1. In this paper,if extension holds, we shall simply identify F,P with F, Q|F∞ and equally we shall consider arandom variable Y on (Ω,F∞) as the random variable Y π on (Ω, F∞). With this identificationan F-adapted process on Ω is a P-F martingale if and only if it is a Q-F martingale.

In this paper we study the following problem:

Problem-A Construct an extension (Ω, A, F,Q, π) of (Ω,F,P) and a random time τ on (Ω, A)such that Q[τ > t|Ft] = Nte

−Λt for all 0 ≤ t < ∞.

The two processes N and Λ will be considered as the parameters of the problem-A. We solvethe problem under the hypothesis:

Hy(N, Λ): Assume that 0 ≤ Zt < 1, 0 ≤ Zt− < 1, for 0 < t < ∞, and almost surely, therandom measure dΛt

1−Ztis bounded on any closed interval contained in (0,∞).

We prove that for each pair of N and Λ satisfying Hy(N,Λ), there exists a solution to theproblem-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] weshow that the other solutions depend some other factors. We exhibit the different propertieswhich characterize our solution from others. In particular, we study the relation with theCox construction, the conditional law invariance property, the separation condition, the (H’)property for the enlargement of filtration problem with a decomposition formula of the sametype 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 pointis a given increasing continuous process Λ, adapted with respect to some information filtrationF = (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 aG-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 := inft : Λt ≥ Θ and Θ is a r.v. independent ofF∞, with unit exponential law. However, the Cox model is a fairly simple model, because itsconditional survival probability Q[τ > t|Ft] is equal to e−Λt , whilst it is known that, for ageneral τ , for 11τ≤t − Λt∧τ to be a G-local martingale, a necessary and sufficient conditionis that Q[τ > t|Ft] has a multiplicative decomposition Nte

Λt , where N is a positive F-localmartingale. Our aim here is to take into account this second parameter N and to study theproperties of the models with conditional survival probability of the form Ne−Λ.

The problem-A has been studied in Gapeev et al. [4] where, inspired by an example fromfiltering, a solution has been found by solving an integral equation (see Section 6). The resulton the integral equation in [4] is important to well understand the problem-A. We shall givebelow a precise description on the relation between our approach and [4]. Another related workis Nikeghbali and Yor [11] where for a strictly positive martingale N with no positive jumpsand limt→∞Nt = 0, the authors have established the multiplicative decomposition formula

P[g > t|Ft] =Nt

St= Nte

− ln(St), 0 ≤ t < ∞,

where St = sup0≤s≤t Ns and g = supt ≥ 0 : Nt = St. It is a particular case of the problem-A

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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 solutionscan 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∞, withthe map π : π(s, ω) = ω and the map τ : τ(s, ω) = s, with the filtration F = π−1(F) which willimmediately be identified with F. In such a setting, for ([0,∞] × Ω,B[0,∞] ⊗ F∞, F,Q, π) tobe 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 shallsimply say that Q is an extension, or that Q is a solution. We equip systematically the productspace [0,∞] × Ω with the progressively enlarged filtration G = (Gt)t≥0 made from F with therandom time τ , i.e. Gt = Ft ∨ σ(τ ∧ t), t ≥ 0. This fix the framework for our discussion on theenlargement 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 theprojection 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-Flocal martingales exhibits only G-adapted càdlàg processes. This yields that in the filtrationG+, 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 thesolution on the product space is the probability measure making τ an exponential randomvariable independent of F∞. More generally, if N ≡ 1 and Λ∞ = ∞, again a solution exists onthe product space given by the probability measure νΛ:

νΛ[A ∩ s < τ ≤ t] = P[IA∫ t

se−ΛvdΛv], A ∈ F∞, 0 < s < t < ∞ (1)

This construction2 coincides with the Cox construction. We will call νΛ the Cox measure.

2.2 Föllmer’s measure

The condition of the problem-A on the conditional survival probability reminds us immedia-tely 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

2In this paper, for a probability measure P, we use the notation P[X] for EP[X]

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measure. To simplify the formula, we suppose Z∞ = 0. The Föllmer’s measure in this case isgiven by

QF [F ] = P[∫ ∞

0F (s, ·)ZsdΛs]

Under this measure, the Ft-conditional survival probability of τ is effectively Zt. In order tobe a solution of the problem-A, QF must be an extension of (Ω,F,P). This is equivalent to thecondition:

∫∞0 ZsdΛs ≡ 1 (see Nikeghbali [10] for more discussion on such situation), because

the restriction of QF onto F∞ is calculated by

QF [A] = P[IA∫ ∞

0ZsdΛs], A ∈ F∞.

We note that in such a case the Föllmer’s measure QF coincides with the Cox measure νΛ.(Although the Föllmer’s measure need not be a solution, we notice that any solution of theproblem-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, thetails conditional laws of τ are linked to the supermartingale Z by the identity:

Q[τ ∈ du, t ≤ τ ≤ ∞|Ft] = Q[11t≤u<∞ZudΛu + 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 followingresult clarifies the situation. Set ι(t) = t, 0 ≤ t ≤ ∞ and

Ct = infs ≥ 0 : Λs > t, 0 ≤ t < ∞

Then, (Ct, t ≥ 0) is an increasing family of F-stopping times. Since Λ is a finite increasingprocess, C∞ = limt↑∞Ct = ∞. We consider the time change induced by (Ct, t ≥ 0). We recallthat ZCt = NCte

−t∧Λ∞ , t ≥ 0, is an FC-supermartingale with 0 ≤ ZCt ≤ 1 (we use the factΛCt = t ∧ Λ∞) and NC = (NCt , t ≥ 0) is a true FC-martingale. The following result can bechecked straightforwardly :

Theorem 2.1 Assume the hypothesis Hy(N,Λ). If (Q, τ) is a solution of the problem-A on thespace (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 parametersN and Λ.

2.4 Probability change

Suppose Λ∞ = ∞. Let νΛ be Cox measure (see the formula (1)). A natural idea to solvethe problem-A is to look for solutions among the probability measures Q on the product space,which are absolutely continuous with respect to νΛ.

Let G be the progressively enlarged filtration from F by the random time τ . Let Q bea probability measure on the product space, absolutely continuous with respect to νΛ, with

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density ξ = dQdνΛ . Recall that, under the probability measure νΛ, the immersion property holds

(see Bielecki et al. [2]). Then, it can be proved that Q is a solution of the problem-A, if andonly if there exists a family L = (Lt, 0 ≤ t < ∞) of positive B[0, t]⊗Ft measurable functions,such that

1. L satisfies the equation :

(L) : Nte−Λt +

∫ t

0Lt(s, ·)e−ΛsdΛs = 1, ∀t ≥ 0.

2. for any 0 < s < ∞, the process Lt(s), t ≥ s is a càdlàg νΛ-G martingale3. for 0 ≤ t < ∞, νΛ[ξ|Gt] = NtIt<τ + Lt(τ, ·)Iτ≤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 donot 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, nosolution can exist on Ω for the problem-A with parameter N ≡ 1 because then random variableΛτ is independent of F∞ (see [2], p.136).

3 A solution of the problem-AIn 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

P[IA∫

]s,t](dΛv + δ∞(dv))], A ∈ F∞, 0 ≤ s < t < ∞.

This looks like the probability change approach mentioned in the preceding section. But wewill 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 ≤ ∞:

Jt = Jt(θ) := (1− Zt) exp−

∫ t

θ

Zs

1− ZsdΛs

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 Zis a supermartingale with the Doob-Meyer decomposition dZs = e−ΛsdNs − ZsdΛs. Applyingintegration by parts formula on J , for θ ≤ t < ∞,

dJt = − exp−

∫ t

θ

Zs dΛs

1− Zs

e−ΛtdNt

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we see that J is a P-F local martingale on [θ,∞). But clearly positive and bounded by 1, it isa uniformly integrable martingale on [θ,∞].

As a corollary, we obtain

Lemma 3.2 Let 0 < θ < ∞ be fixed and define

Et(θ) =Jt(θ)

1− Zθ=

1− Zt

1− Zθexp

−

∫ t

θ

Zs dΛs

1− Zs

, θ ≤ t ≤ ∞.

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), orEN,Λ

t (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 thatlimn(1− Zθn)Et(θn) = 0 P-almost everywhere.

Proof. We begin with the expectation limit. Using the preceding lemma and the fact that Zis càdlàg bounded by 1 and Z0 = 1, we can write

limθ↓0P[(1− Zθ)Et(θ)] = lim

θ↓0P[(1− Zθ)] = 0.

But everything being positive, the sequence (1− Zθ)Et(θ) converges in L1 to zero. The lemmafollows.

Lemma 3.4 For any 0 < θ < ∞, θ ≤ t < ∞, we have∫ t

θZvEt(v)dΛv = (1− Zt)− (1− Zθ)Et(θ),

and, for θ ≤ t ≤ ∞, ∫ t

0ZvEt(v)dΛv = 1− Zt

Proof : We compute for 0 < θ < ∞, θ ≤ t < ∞,∫ t

θZvEt(v)dΛv =

∫ t

θZv

1− Zt

1− Zvexp

−

∫ t

v

Zs dΛs

1− Zs

dΛv

= (1− Zt) exp−

∫ t

v

Zs dΛs

1− Zs

∣∣∣∣v=t

v=θ

= (1− Zt)− (1− Zθ)Et(θ)

Let θ goes to zero along the sequence (θn) in the preceding lemma. We obtain∫ t

0ZvEt(v)dΛv = 1− Zt .

Passing to the limit, this identity holds for t = ∞.

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Remark 3.1 Suppose that N is continuous. We have the identity

Et(θ) = exp−∫ t

θ

e−sdNs

1− Zs− 1

2

∫ t

θ

e−2sd〈N〉s(1− Zs)2

This explains the use of the notation of an exponential martingale. Let us write the identity:

1− Zt = (1− Zθ) exp∫ t

θ

ZudΛu

1− ZuEt(θ)

It is the multiplicative decomposition of the positive submartingale (1−Zt, t ≥ θ) where Et(θ)is the martingale factor and (1−Zθ) exp∫ t

θZudΛu1−Zu

is the bounded variation factor. Notice thatwe keep a θ > 0 in the decomposition, because in a neighborhood of θ = 0 the integrability of

11−Zθ

is troublesome.

3.2 The Construction

Define

u(θ) = uZ(θ, ω) :=

0 θ = 0

Zθ(ω)E∞(θ, ω) 0 < θ < ∞

Z∞(ω) θ = ∞Notice that, for each θ, the random function u(θ) on θ ∈ [0,∞] is non negative and it is càdlàgin θ over (0,∞).

Lemma 3.5 The random function u(θ) is a probability density with respect to the randommeasure dΛθ + δ∞(dθ) over [0,∞], i.e.,

∫ ∞

0u(θ)dΛθ + u(∞) = 1 .

Proof. According to Lemma 3.4, we can write∫ ∞

0u(v)dΛv + u(∞) =

∫ ∞

0ZvE∞(v)dΛv + Z∞ = (1− Z∞) + Z∞ = 1

Now we construct a probability QZ on the product space by:

QZ [A ∩ t < τ ≤ t′] = P[IA∫

]t,t′]u(θ)(dΛθ + δ∞(dθ))] (2)

for A ∈ F∞, 0 ≤ t < t′ ≤ ∞. 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

QZ [A ∩ τ > t] = P[IA(∫ ∞

tu(v)dΛv + u(∞))] = P[IA(

∫ ∞

tZvE∞(v)dΛv + u(∞))]

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)]

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According to Lemma 3.2, P[E∞(t)|Ft] = 1 and so

QZ [A ∩ τ > t] = P[IA − IA(1− Zt)] = P[IAZt] = QZ [IAZt]

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 theproduct 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 moreoverΛ∞ = ∞ so that Z∞ = 0, the random function ue−Λ

(θ) is simply e−Λθ and the probabilitymeasure QZ becomes

Qe−Λ[A ∩ s < τ ≤ t] = P[IA

∫ t

se−ΛθdΛθ] = νΛ[A ∩ s < τ ≤ t]

where νΛ is the Cox measure (see the formula (1)).

Return back to a general Z = Ne−Λ. We know by the construction that QZ is absolutelycontinuous with respect to νΛ. Now we look at its density process.

Theorem 4.1 Suppose Hy(N, Λ) and Λ∞ = ∞ so that Z∞ = 0. Set

L(s, ω) =

Ns(ω)E∞(s, ω), 0 < s < ∞, ω ∈ Ω0 s = 0 or s = ∞

(in other words, L = NτE∞(τ) νΛ-a.s.). Then, L is the probability density of QZ with respectto νΛ: QZ = L · νΛ, and the G-density process Lt = νΛ[L|Gt], 0 ≤ t < ∞, is given by Lt =Nt∧τEt∨τ (τ), 0 ≤ t ≤ ∞.

Proof. The first part of the theorem is a direct consequence of the construction. For theassertion 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 νΛ integrable function on [0,∞]× Ω. Then, we have

νΛ[F |Ga](s, ω) = G(s, ω)Is≤a + J(ω)Ia<s

whereG(s, ω)Is≤a = P[F (s, ·)|Fa](ω)Is≤a νΛ-almost surely

J(ω) = eΛa(ω)P[∫∞a F (s, ·)e−ΛsΛ(ds)|Fa](ω) P-almost surely

In particular, we haveνΛ[NτE∞(τ)|Ga] = Na∧τEa∨τ (τ)

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Proof. The first part of the lemma is well-known. Applying this to F = NτE∞(τ), its G(s, ω)part is computed by

P[NsE∞(s)|Fa] = NsEa(s) = Na∧sEa∨s(s), s ≤ a.

Its J(ω) part is computed by, s > a,

eΛaP[∫∞a NvE∞(v)e−ΛvΛ(dv)|Fa] = eΛaP[

∫∞a ZvΛ(dv)|Fa]

= eΛaZa (recall that Z∞ = 0)= Na∧sEa∨s(s)

Combining these two results we obtain the lemma.

Remark 4.1 The density process (Nt∧τEt∨τ (τ), 0 ≤ t < ∞) satisfies the equation (L) (seesubsection 2.4). It is just a restatement of Lemma 3.4.

5 Characterizations of QZ

The probability measure QZ possesses many properties and can be characterized in differentways 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 separableform, if, for any 0 < v < ∞, there exist two F-adapted càdlàg processes (X(v)

t )t≥v, (Y(v)t )t≥v

(defined only for t ≥ v) such that

χt(s, ·) = X(v)t Y (v)

s ,

for all v ≤ s ≤ t < ∞. The processes X(v), Y (v) will be called respectively the first and thesecond components in the partially separable form.

• Complete Separation An element χ = (χt, 0 ≤ t < ∞) in K is said to be in a completelyseparable form, if there exist two F-adapted càdlàg processes (Xt)t≥0, (Yt)t≥0 such that

χt(s, ·) = XtYs,

for all 0 ≤ s ≤ t < ∞. The processes (X,Y ) will be called respectively the first and the secondcomponents 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 thereexists a positive element α = (αt, 0 ≤ t < ∞) in K such that, for 0 ≤ t < ∞ and B ∈ B[0, t],

Q[τ ∈ B|Ft] =∫

Bαt(s, ·)e−ΛsdΛs

3To 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 canascribe different properties (We will see later the implications of these properties):

? Decreasing condition∫ t0 αt(u, ·)e−ΛudΛu > 0 for any 0 < t < ∞, and, for 0 ≤ u < ∞,

the map

t → αt(u, ·)∫ t0 αt(s, ·)e−ΛsdΛs

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, ·) = NsEt(s) for 0 < s ≤ t < ∞.

¦ Conditional law Invariance We say that Q satisfies the conditional law invariance hypo-thesis, if, for any 0 < a ≤ b, for any A,B ∈ B[0, a], we have

Q[τ ∈ A|Fa]Q[τ ∈ B|Fa]

=Q[τ ∈ A|Fb]Q[τ ∈ B|Fb]

whenever Q[τ ∈ B|Fb] > 0.

¦ Proportionality Hypothesis Let Q be a probability measure on (Ω,A). We say that theconditional proportionality hypothesis is satisfied, if Q[τ ≤ t|Ft] > 0 for any 0 < t < ∞, and,for any 0 < u < ∞, the map

t → Q[τ ≤ u|Ft]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 oursolution 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 canonicalform.

Proof. Let 0 ≤ t < ∞, A ∈ Ft, B ∈ B[0, t], by construction of QZ , we can write

QZ [A ∩ τ ∈ B] = QZ [IA∫

BNsE∞(s)e−ΛsdΛs] = QZ [IA

∫

BNsEt(s)e−ΛsdΛs]

This proves, forQZ , the local density hypothesis with canonical form. The uniqueness is obvious.

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5.3 Computations of conditional laws

In this subsection we consider a probability measure Q on the product space which is asolution of the problem-A. Note that

Q[τ = 0] = limt↓0

(1−Q[τ > t]) = limt↓0

(1−Q[Zt]) = 0.

Lemma 5.1 For any 0 < t < ∞, for A ∈ B[t,∞],

Q[τ ∈ A|Ft] = Q[∫

ANue−ΛudΛu + I∞∈AZ∞|Ft]

Proof. For u ≥ t,

Q[τ > u|Ft] = Q[Zu − Z∞ + Z∞|Ft] = Q[∫ ∞

uNse

−ΛsdΛs + 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],

Q[τ ∈ A|Ft] =∫

ANsEt(s)e−ΛsdΛs

i.e., the local density hypothesis holds in canonical form.

Proof. Let 0 < s < ∞ be fixed. Let

rt :=Q[τ ≤ s|Ft]Q[τ ≤ t|Ft]

=Q[τ ≤ s|Ft]

1− Zt, s ≤ t < ∞.

By hypothesis (rt, t ≥ s) is a continuous decreasing function. It is clearly F-adapted. Considerthe 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 shouldhave its drift null, i.e.,

rtZtdΛt + (1− Zt)drt = 0, s ≤ t < ∞.

Solving this equation leads to

rt = exp−∫ t

s

ZudΛu

1− Zu, s ≤ t < η,

where η := inft ≥ s : rt = 0. But the above expression shows that necessarily η = ∞. Wesubstitute this expression of rt into the identity (3) to get

Q[τ ≤ s|Ft] = (1− Zt) exp−∫ t

s

ZudΛu

1− Zu

Let s ↓ 0. We see that

exp−∫ t

0

ZudΛu

1− Zu = 0

11

Using these facts, we write finally

Q[τ ≤ s|Ft] = (1− Zt) exp−∫ t

s

ZudΛu

1− Zu

=∫ s

0(1− Zt) exp−

∫ t

u

ZvdΛv

1− Zv Zu

1− ZudΛu =

∫ s

0NuEt(u)e−ΛudΛu

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

t → r(v)t =

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 takeinto account the parameter v > 0. Notice

limv↓0Q[v < τ ≤ a|Fa] = Q[τ ≤ a|Fa] = 1− Za > 0

Let 0 < v < a < ∞ and consider

rt = r(v)t :=

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 < ∞) isstrictly positive, continuous and decreasing, and clearly F-adapted. Rewrite this fact in anotherform

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 driftnull, i.e.

rtZtdΛt + (mt − Zt)drt = 0, t ≥ a.

LetSn = inft ≥ a : mt − Zt ≤ 1

n,ηv = supn Sn,

Because ma − Za > 0 on the set Av, one has ηv > a. Let a ≤ t < ηv. We can write

−∞ < ln(rt) =∫ t

a

(mt − Zu)(mt − Zu)

dru

ru= −

∫ t

a

ZudΛu

(mt − Zu)

where we use the fact that ln(ra) = 0. In other words,

rt = exp−∫ t

a

ZudΛu

mu − Zu, a ≤ t < ηv.

12

We have now another equation for Q[v < τ ≤ a|Fb] on the set Av:

Q[v < τ ≤ a|Ft] = (Q[v < τ |Ft]− Zt) exp−∫ t

a

ZudΛu

Q[v < τ |Fu]− Zu, a ≤ t < ηv.

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

Q[τ ≤ a|Ft] = (1− Zt) exp−∫ t

a

ZudΛu

1− Zu

Let a ↓ 0. We see that

exp−∫ t

0

ZudΛu

1− Zu = 0

We have finallyQ[τ ≤ a|Ft]

= (1− Zt) exp− ∫ ta

ZudΛu1−Zu

=

∫ a0 (1− Zt) exp− ∫ t

sZudΛu1−Zu

Zs1−Zs

dΛs

=∫ a0 NsEt(s)e−ΛsdΛs

The lemma is proved.

5.4 Equivalences

We now prove that, if Q is a solution of the problem-A, the various properties above areequivalent, 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 theproblem-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

lima′↑b

f(a′, b) = f(b, b), limb′↓b

f(a, b′) = f(a, b) (5)

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,

NsEt(s) = Ns1−Zt1−Zs

exp− ∫ ts

ZwdΛw1−Zw

= Ns

1−Zt1−Zs

exp− ∫ tv

ZwdΛw1−Zw

+∫ sv

ZwdΛw1−Zw

= 1−Zt

exp∫ tv

ZwdΛw1−Zw

Ns exp∫ s

vZwdΛw1−Zw

1−Zs

This proves the partial separation form with

X(v)t =

1− Zt

exp∫ tv

ZwdΛw1−Zw

, Y (v)

s =Ns exp∫ s

vZwdΛw1−Zw

1− Zs

13

The condition 3. is proved.

Let us see the implication 3. ⇒ 2.. Let 0 < v < a < b < ∞. If Q[v < τ ≤ a|Fa] > 0, we have∫ av Y

(v)s e−ΛsdΛs > 0. Since, for a ≤ t < ∞, X

(v)t > 0 by hypothesis, we can write

r(v)t :=

Q[v < τ ≤ a|Ft]Q[v < τ ≤ t|Ft]

=

∫ av Y

(v)s e−ΛsdΛs∫ t

v Y(v)s e−ΛsdΛs

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

t → NsEt(s)∫ t0 NvEt(v)e−ΛvdΛv

=NsEt(s)1− Zt

=Ns

1− Zsexp

−

∫ t

s

Zv

1− ZvdΛv

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 isstrictly positive. Notice also that

Q[τ ≤ t|Ft] =∫ t0 αt(v, ·)e−ΛvdΛv = 1− Zt

Q[τ ≤ s|Ft] =∫ s0 αt(v, ·)e−ΛvdΛv

So,Q[τ ≤ s|Fs]Q[τ ≤ t|Ft]

=1

1− Zt

∫ s

0αt(v, ·)e−ΛvdΛv.

By hypothesis,

t → αt(v, ·)1− Zt

is a continuous decreasing function in t ∈ [v,∞). The dominated convergence theorem yieldsthat

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 theseconditions are also equivalent to the condition 5., let us prove first that condition 3. impliesthe conditional law invariance. We begin with the fact that, for any 0 < v < a ≤ b, for anyB ∈ B[v, a],

Q[τ ∈ B|Fb] =∫

B

X(v)b Y (v)

u e−ΛudΛu

If Q[τ ∈ B|Fb] > 0,∫BY

(v)u e−ΛudΛu > 0. Recall that, by hypothesis, X

(v)a > 0, X

(v)b > 0. These

facts yield, for A, B ∈ B[v, a],

Q[τ ∈ A|Fa]Q[τ ∈ B|Fa]

=

∫AY

(v)u e−ΛudΛu∫

BY

(v)u e−ΛudΛu

=Q[τ ∈ A|Fb]Q[τ ∈ B|Fb]

Such an identity being true for all v > 0, we have actually

Q[τ ∈ A|Fa]Q[τ ∈ B|Fa]

=Q[τ ∈ A|Fb]Q[τ ∈ B|Fb]

14

for any A, B ∈ B([0, a]). The invariance hypothesis is proved. Let us now prove that condition2. implies the property (5). It is enough to set

f(a, b) =∫ a

0NuEb(u)e−ΛudΛu

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 thecondition 5. According to the conditional law invariance hypothesis, we can write

f(a, b)f(b, b)

=f(a, b′)f(b, b′)

, a ≤ b < b′ < ∞

whenever f(b, b′) > 0. The above identities hold outside of a commun exception set. Now theproperty (5) together with the conditional law invariance makes valid the two limits:

limb′↑b

f(a, b′)f(b′, b′)

= limb′↑b

f(a, b)f(b′, b)

=f(a, b)f(b, b)

(notice that there is no problem of zero denominator f(b′, b), because f(b, b) = 1 − Zb > 0 sothat for b′ close to b, f(b′, b) > 0.) and

limb′↓b

f(a, b′)f(b′, b′)

= limb′↓b

f(a, b′)1− Zb′

=f(a, b)f(b, b)

These two limits mean that f(a,b)f(b,b) is a continuous function in b ∈ [a,∞). But this function is

also decreasing as the following computation shows:

f(a, b)f(b, b)

=f(a, b′)f(b, b′)

≥ f(a, b′)f(b′, b′)

, a ≤ b ≤ b′ < ∞

The condition 1. is proved

6 Complete Separation Condition

We assume Hy(N,Λ) and Λ∞ = ∞. We notice that the completely separable form is absentfrom 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 localsolution to the problem-A, if there exists an increasing sequence (Tn)n≥1 of F-stopping timessuch that limn→∞ Tn = ∞ and, for any n ≥ 1, Q is a probability measure on any of GTn andP|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 thebehaviour 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:

(L) : Nte−Λt +

∫ t

0Lt(u, ·)e−ΛudΛu = 1, 0 ≤ t < ∞.

The following result is a generalization of a result proved in [4]. The proof here is based onthe same idea.

Theorem 6.1 Suppose that

Γt :=∫ t

0

d[N, N ]u(Nu− − eΛu)2

=∫ t

0

e−2Λud[N, N ]u(1− Zu−)2

< ∞

for all 0 ≤ t < ∞. Then, the equation (L) has a positive solution L in K in completely separableform, i.e., there are (Xt)t≥0, (Yt)t≥0 two positive F-adapted càdlàg processes such that Lt(u, ·) =XtYu. Moreover, if we introduce the random variable on the product space:

ξt = NtIt<τ + XtYτ Iτ≤t, t ≥ 0.

the process ξ is a νΛ-G local martingale. If we define a set function Q on the product space sothat, for each G-stopping time T reducing ξ to a νΛ-G uniformly integrable martingale, Q isthe probability measure on GT defined by

dQdνΛ

∣∣∣∣GT

= ξT ,

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

ϑt =∫ t

0

1Nu− − eΛu

dNu, t > 0.

The condition on Γ implies that ϑ is well defined. Let X be the Doléans-Dade exponential ofϑ:

Xt = 1 +∫ t

0Xu−dϑu, 0 ≤ t < ∞.

Notice that∆uϑ =

1Nu− − eΛu

∆uN = −(

Zu − Zu−1− Zt−

)> −1.

So the Doléans-Dade exponential Xt is well defined and remains strictly positive. Set

Yt = 1 +∫ t

0eΛud

(1

Xu

), 0 ≤ t < ∞.

We note thatd(XY ) = Y−dX + X−dY + d[X,Y ]

= Y−X−dϑ + eΛX−d(

1X

)+ eΛd[X, 1

X ]

16

and0 =

1X−

dX + X−d

(1X

)+ d[X,

1X

]

with 1X−dX = dϑ. So,

(XY )0 = 1, d(XY )t = ((XY )t− − eΛt)dϑt

N0 = 1, dNt = (Nt− − eΛt)dϑt

The uniqueness of the solution of the above equation (see Protter [12]) implies XY = N . Thisproves that Y is positive.

Set now Lt(u) = XtYu, 0 ≤ u ≤ t < ∞. We check that

XtYte−Λt +

∫ t

0XtYue−ΛudΛu = Xt · 1

Xt= 1, 0 ≤ t < ∞,

so that L is a solution of the equation (L). Consider the local martingale property of ξ. We canchoose a sequence (Tn)n≥1 of F-stopping times such that NTn , XTn are uniformly integrablefor each n ≥ 1 and limn→∞ Tn = ∞. We apply Lemma 4.1 to compute νΛ[ξTn∧t|GTn∧u] for0 ≤ u < t < ∞. We compute firstly the components G(s, ω) and J(ω):

G(s, ·)11s≤Tn∧u = P[NTn∧t11Tn∧t<s + XTn∧tYs11s≤Tn∧t|FTn∧t]11s≤Tn∧u= P[XTn∧tYs11s≤Tn∧t|FTn∧u]11s≤Tn∧u= P[XTn∧t|FTn∧u]Ys11s≤Tn∧u= XTn∧uYs11s≤Tn∧u

andJ = eΛTn∧uP[

∫∞Tn∧u(NTn∧t11Tn∧t<s + XTn∧tYs11s≤Tn∧t)e−ΛsdΛs|FTn∧u]

= eΛTn∧uP[∫∞Tn∧u(NTn∧t11Tn∧t<s + XsYs11s≤Tn∧t)e−ΛsdΛs|FTn∧u]

= eΛTn∧uP[NTn∧te−ΛTn∧t +

∫ Tn∧tTn∧u Nse

−ΛsdΛs|FTn∧u]= eΛTn∧uP[ZTn∧t + ZTn∧u − ZTn∧t|FTn∧u]= eΛTn∧uZTn∧u

= NTn∧u.

Combining these two terms we obtain

νΛ[ξTn∧t|GTn∧u] = NTn∧u11Tn∧u<τ + XTn∧uYτ11τ≤Tn∧u = ξTn∧u

This shows that ξTn is a νΛ-G uniformly integrable martingale, or ξ is a νΛ-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

νΛ[ξTn |FTn ]= limn→∞ νΛ[(NTn ∧ n)ITn<τ + (XTn ∧ n)(Yτ ∧ n)Iτ≤Tn|FTn ]

= limn→∞((NTn ∧ n)e−ΛTn + (XTn ∧ n)

∫ Tn

0 (Ys ∧ n)e−ΛsdΛs

)

= 1

thanks to the equation (L). This means P|FTn= Q π−1|FTn

. We compute also

Q[τ > Tn ∧ t|FTn∧t] = νΛ[ξTn∧t11τ>Tn∧t|FTn∧t] = νΛ[NTn∧t11τ>Tn∧t|FTn∧t] = NTn∧te−ΛTn∧t .

This proves the theorem.

17

Theorem 6.2 Suppose the same hypothesis as in Theorem 6.1. The local solution Q constructedin Theorem 6.1 is the restriction of the probability measure QZ constructed in Theorem 3.1. Inthis case, the local density hypothesis holds for QZ in completely separable form with the firstcomponent strictly positive.

Proof. We need only to repeat the proof of the implication from condition 3. =⇒ condition1. in Theorem 5.2. For this reason we need to adapt the notions such as the local densityhypothesis to the case of a local solution of the problem-A. But this is straighforward. Now wecan conclude that Q satisfies the local density hypothesis in completely separable form, whichimplies the canonical form. However, there can be only one set function with local densityhypothesis 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 canonicalform.

Theorem 6.3 Let Q be a probability measure on the product space which is a solution of theproblem-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 < ∞:

Vt = lims↓0

(1− Zs) exp∫ t

s

Zu

1− ZudΛu ∈ (0,∞)

In the case where one of the above conditions is satisfied, let X be the first component in thecompletely separable form. We have

(\) lima↓0 1−ZaVa

= X0 ∈ (0,∞)

(†) Γa =∫ a0

e−2Λud[N,N ]u(1−Zu−)2

< ∞, 0 ≤ a < ∞

and X is equal to the Doléans-Dade exponential of the P-F local martingale − e−Λ

1−Z− ·N

Proof. The implication 1. ⇒ 2.. Let X, Y denote the first and the second components of thelocal density function. The completely separable form implies the partially separable form.According to Theorem 5.2, we have thus NsEt(s) = XtYs for 0 < s ≤ t < ∞, and

0 < 1− Zt = Xt

∫ t

0Yue−ΛudΛu < 1, X0Y0 = N0 = 1.

We obtain the equalities

Ns1

1− Zsexp−

∫ t

s

Zu

1− ZudΛu =

NsEt(s)1− Zt

=Ys∫ t

0 Yue−ΛudΛu

or equivalently

(1− Zs) exp∫ t

s

Zu

1− ZudΛu =

Ns

∫ t0 Yue−ΛudΛu

Ys= Xs

∫ t

0Yue−ΛudΛu

18

so that

Vt = lims↓0

(1− Zs) exp∫ t

s

Zu

1− ZudΛu = X0

∫ t

0Yue−ΛudΛu ∈ (0,∞)

The condition 2. is proved.

The implication 2. ⇒ 1.. We write, for 0 < v < s < t,

NsEt(s) = Ns1−Zt1−Zs

exp− ∫ ts

ZwdΛw1−Zw

= Ns

1−Zt1−Zs

1−Zv1−Zv

exp− ∫ tv

ZwdΛw1−Zw

+∫ sv

ZwdΛw1−Zw

= Ns

1−Zt1−Zs

(1−Zv) exp∫ sv

ZwdΛw1−Zw

(1−Zv) exp∫ t

vZwdΛw1−Zw

By hypothesis,

Vt = limv↓0

(1− Zv) exp∫ t

v

ZwdΛw

1− Zw ∈ (0,∞)

This limit leads to the identity

NsEt(s) = Ns1− Zt

1− Zs

Vs

Vt= XtYs

where Xt = 1−ZtVt

and Ys = NsVs1−Zs

. Notice that the so defined processes X, Y are progressivelymeasurable with respect to F. Let us show that Xt is integrable. In fact this is because Xt ≥ 0and by Fatou’s lemma,

P[Xt] ≤ limv↓0 P[ 1−Zt1−Zv

exp− ∫ tv

ZudΛu1−Zu

] = 1.

For 0 < v ≤ s ≤ t, for A ∈ Fs,

P[IA∫ v

0Yue−ΛudΛuXs] = P[IAP[τ ≤ v|Fs]] = P[IAP[τ ≤ v|Ft]] = P[IAXt

∫ v

0Yue−ΛudΛu]

Since∫ v0 Yue−ΛudΛu 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 de-note its càdlàg version. From this version we obtain de càdlàg version for (Vt)0<t<∞ that wedenote again by (Vt)0<t<∞. Combining these two càdlàg version we get also a càdlàg versionfor (Yt)0<t<∞ that we denote also by (Yt)0<t<∞. Now repeating the argument in the first partof the proof, we obtain

(1− Zs) exp∫ ts

Zu1−Zu

dΛu = Xs

∫ t0 Yue−ΛudΛu

Xt

∫ t0 Yue−ΛudΛu = 1− Zt > 0.

By hypothesis, the left-hand side of the first equality tends to Vt > 0. We prove thus X, as wellas V and Y , is also right continous at zero.

The condition (\) is true because

X0 =Vt∫ t

0 Yue−ΛudΛu

> 0.

Look at the implication 1., 2. ⇒ (†).. The process X is strictly positive. The martingaleproperty implies that X− is also strictly posotive.

19

Let 0 < v < a < ∞. Since 1− Zu− > 0, 0 < u < ∞, (1− Z−) is integrable with respect toN on all intervals [v, a] with 0 < v < a < ∞. We have the identity

1− Zt = (1− Zv) exp∫ t

v

ZudΛu

1− ZuE

(−11(v,∞)e

−Λ

1− Z−·N

)

t

for v ≤ t < ∞. Dividing both sides by (1−Zv) exp∫ tv

ZudΛu1−Zu

and taking the limit when v ↓ 0,we obtain

Xt = limv↓0

E(−11(v,∞)e

−Λ

1− Z−·N

)

t

Since X > 0, 1X− is integrable with respect to X. We write for 0 < c < t,

Xt = XcE(

11(c,∞)

X−·X

)

t

But alsoXt = limv↓0 E

(−11(v,∞)e

−Λ

1−Z− ·N)

t

= limv↓0 E(−11(v,∞)e

−Λ

1−Z− ·N)

cE

(−11(c,∞)e

−Λ

1−Z− ·N)

t

= XcE(−11(c,∞)e

−Λ

1−Z− ·N)

t

It follows that

E(

11(c,∞)

X−·X

)

t

= E(−11(c,∞)e

−Λ

1− Z−·N

)

t

We get then successively∫ tc

dXuXu− = − ∫ t

ce−ΛudNu1−Zu− , 0 < c < t.∫ t

cd[X,X]u

X2u−

=∫ tc

e−2Λud[N,N ]u(1−Zu−)2

, 0 < c < t.∫ t0

d[X,X]uX2

u−=

∫ t0

e−2Λud[N,N ]u(1−Zu−)2

Notice that∫ t0

d[X,X]uX2

u−< ∞. From this, we conclude that e−Λ

(1−Z−) is integrable with respect toN and

Xt = E(− e−Λ

1− Z−·N

)

t

The condition (†). is proved.

7 G-Martingales

In this last section we study the enlargement of filtration problem under the probabilitymeasure QZ constructed in Theorem 3.1. We will show that the P-F local martingales X areQZ-G semimartingales and give their semimartingale decomposition formula. This formula, asthe classical ones, operates on X only through the predictable bracket 〈N,X〉 and is composedof 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 forhonest 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.) Toend this program we will prove that the probability QZ is the only solution of the problem-Aon the product space to possess this enlargement of filtration formula. These questions are alsoconsidered 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]) = 0QZ [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

dtEt(s) = − 11−Zs

exp− ∫ t

sZw dΛw1−Zw

e−ΛtdNt = −Et−(s) e−Λt

1−Zt−dNt, 0 < s ≤ t < ∞

consequence of Lemma 3.1. We note also Et(u) = Es(u)Et(s) for 0 < u < s < t. The decompo-sition 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 〈N, X〉 be the P-F predictable dual projectionof [N, X] (〈N, X〉 exists). Then,

Xτ∧· −∫ ·

011s≤τ

e−Λs

Zs−d〈N, X〉s

is a QZ-G local martingale.

As for what happens after the time τ , a more evolved computation is needed. The compu-tation is possible because of the nice properties of Et(s).

Lemma 7.2 Let X be a P-F local martingale. Then,

Xτ∨· −Xτ +∫ ·

011τ<s

e−Λs

1− Zs−d〈N, X〉s

is a QZ-G local martingale.

Proof. Let 0 < a < b < ∞. Let T be any F-stopping time such that everything concernedin the computation will be integrable. Let A ∈ Fa and h(t) ∈ B[0,∞] bounded function. We

21

compute

Q[IAh(τ)(XTτ∨b −XT

τ∨a)]= limn↑∞Q[IAh(τ)

∑∞k=0 I k

n≤τ< k+1

n(X

Tk+1

n∨b−XT

k+1n∨a

)]

= limn↑∞∑∞

k=0Q[IA∫ k+1

nkn

h(s)ZsE∞(s)dΛs(XTk+1

n∨b−XT

k+1n∨a

)]

= limn↑∞∑∞

k=0Q[IA∫ k+1

nkn

h(s)ZsE k+1n∨b(s)dΛs(XT

k+1n∨b−XT

k+1n∨a

)]

= limn↑∞∑∞

k=0Q[IA∫ k+1

nkn

h(s)ZsE k+1n

(s)dΛs E k+1n∨b(

k+1n )(XT

k+1n∨b−XT

k+1n∨a

)]

= − limn↑∞∑∞

k=0Q[IA∫ k+1

nkn

h(s)ZsE k+1n

(s)dΛs

∫ k+1n∨b

k+1n∨aEξ−(k+1

n ) e−Λξ

1−Zξ−d〈N,XT 〉ξ]

= − limn↑∞∑∞

k=0Q[IA∫ k+1

nkn

h(s)ZsE k+1n

(s)dΛs

∫ k+1n∨b

k+1n∨aE∞(k+1

n ) e−Λξ

1−Zξ−d〈N,XT 〉ξ]

= − limn↑∞∑∞

k=0Q[IA∫ k+1

nkn

h(s)ZsE∞(s)dΛs

∫ k+1n∨b

k+1n∨a

e−Λξ

1−Zξ−d〈N,XT 〉ξ]

= − limn↑∞Q[IAh(τ)∑∞

k=0 I kn≤τ< k+1

n

∫ k+1n∨b

k+1n∨a

e−Λξ

1−Zξ−d〈N,XT 〉ξ]

= −Q[IAh(τ)∫ τ∨bτ∨a

e−Λξ

1−Zξ−d〈N, XT 〉ξ]

This proves the lemma.

We can conclude.

Theorem 7.1 Let X be a P-F local martingale. Then,

X −X0 −∫ ·

0Iu≤τ

e−Λu d〈N, X〉uZu−

+∫ ·

0Iτ<u

e−Λu d〈N, X〉u1− Zu−

(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 theproduct space under which the formula (6) holds.

Theorem 7.2 Let Q be a probability measure on the product space which is a solution ofproblem-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-stoppingtime such that T ≥ a and everything concerned in the computation is integrable. Let f(t) =Q[Iτ≤a|Ft], a ≤ t < ∞. We compute

Q[f(t ∧ T )MTt ]

= Q[Iτ≤aMTt ]

= Q[Iτ≤a(MTa + MT

t −MTa )]

= Q[Iτ≤a(MTa −

∫ ta

e−Λu

1−Zu−d〈N, MT 〉u)]

= Q[Iτ≤aMTa − Iτ≤a

∫ ta

e−Λu

1−Zu−d〈N, MT 〉u]

= Q[(1− Za)MTa −

∫ ta f(u−) e−Λu

1−Zu−d〈N, MT 〉u]

= Q[(1− Za)MTt −

∫ t∧Ta f(u−) e−Λu

1−Zu−dNuMTt ]

= Q[MTt ((1− Za)−

∫ t∧Ta f(u−) e−Λu

1−Zu−dNu)]

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This identity implies

f(t) = (1− Za)−∫ t

af(u−)

e−Λu

1− Zu−dNu, a ≤ t < ∞

Comparing this equation with that of dtEt(a), we conclude

Q[Iτ≤a|Ft] = f(t) = (1− Za)Et(a)

= (1− Zt) exp− ∫ t

aZs dΛs1−Zs

= (1− Zt)∫ a0 exp

− ∫ t

vZs dΛs1−Zs

Zv

1−ZvdΛv

=∫ a0 NvEt(v)e−ΛvdΛv

= QZ [Iτ≤a|Ft], a ≤ t < ∞.

Here we have used the fact that exp− ∫ t

0Zs dΛs1−Zs

= 0 (see the proof of Lemma 5.3). As Q

and QZ coincide on F∞, they must coincide on the whole space. The theorem is proved.

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