3 A solution of the problem-A

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), whereF= (F_t)_t≥0 is a filtration, anF-adapted continuous increasing processΛand a positive P-Flocal martingaleN such that Z_t := N_te^−Λt ≤ 1, t ≥ 0, we construct a model of default time, i.e., a probability measure Q^Z and a random time τ on an extension of (Ω,F,P), such that Q[τ > t|F_t] = Z_t, t ≥ 0.

The probability Q^Z 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 Q^Z from others. We will make a particular attention on the complete separation property. As usual the model is equipped with an enlarged filtration G = (G_t)_t≥0 withG_t =F_t∨σ(τ ∧t). We will show that all P-F martingales areQ^Z-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 Fare denoted by(F_t)_0≤t≤∞withF_∞=∨_0≤t<∞F_t. 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 = 0and 0≤N_te^−Λt ≤1 for all 0≤t <∞. Let Z_t:=N_te^−Λt, t≥0. We note that the process Z is a class (D) supermartingale whose Doob-Meyer decomposition, thanks to the continuity of Λ, is dZ_s =e^−ΛsdN_s−Z_sdΛ_s. The random variableR_∞

0 Z_sdΛ_s is integrable and R_·

0e^−ΛsdN_s is a BMO martingale.

Let ( ˆΩ,A,ˆ F,ˆ 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

1Notice that in this paper callingaa positive number means thata≥0and callingf(t), t∈R,an increasing function meansf(s)≤f(t)fors≤t.

Fˆ are denoted by ( ˆF_t)_t≥0. Let π be a map from Ωˆ into Ω. We say that ( ˆΩ,A,ˆ F,ˆ Q, π) is an extension of (Ω,F,P) ifFˆ_t =π⁻¹(F_t) for all 0≤ t≤ ∞, and P= Q|_Fˆ∞◦π⁻¹. In this paper, if extension holds, we shall simply identify F,P with F,ˆ Q|_Fˆ

∞ and equally we shall consider a random variableY on(Ω,F_∞)as the random variableY◦πon( ˆΩ,Fˆ_∞). With this identification an F-adapted process onΩis aP-F martingale if and only if it is aQ-Fˆ martingale.

In this paper we study the following problem:

Problem-AConstruct an extension ( ˆΩ,A,ˆ F,ˆ Q, π)of (Ω,F,P) and a random timeτ on ( ˆΩ,A)ˆ such that Q[τ > t|F_t] =N_te^−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 ≤ Z_t < 1,0 ≤ Z_t− < 1, for 0 < t < ∞, and almost surely, the random measure _1−Z^dΛt_t is 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 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 factorsN 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 solutionτ is even notF_∞-measurable.

1.2 Motivation

This question arises from credit risk modelling. Usually, in that setting, the starting point is a given increasing continuous processΛ, adapted with respect to some information filtration F= (F_t)_t≥0. The processΛcontains the only parameters to be calibrated from the market data.

Then, one constructs a random timeτ on some probability space such that(H_t−Λ_t∧τ, t≥0)is a G-local martingale, whereH_t= 11_τ≤t, andG= (G_t)_t≥0 withG_t=F_t∨σ(τ∧t). The "canonical"

construction is the Cox model, where τ := inf{t : Λ_t ≥ 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|F_t] is equal to e^−Λt, whilst it is known that, for a general τ, for 11_{τ≤t} −Λ_t∧τ to be a G-local martingale, a necessary and sufficient condition is that Q[τ > t|F_t] has a multiplicative decomposition N_te^Λt, 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 formN 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 lim_t→∞N_t= 0, the authors have established the multiplicative decomposition formula

P[g > t|F_t] = N_t

S_t =N_te^−ln(St), 0≤t <∞,

whereS_t= sup_0≤s≤tN_s andg= sup{t≥0 :N_t=S_t}. It is a particular case of the problem-A

withΛ_t= ln(S_t). It is particular also because the solution(P, g)is living on the initial spaceΩ.

With the hypothesisHy(N,Λ) our situation is radically different. In fact none of our solutions can live in the initial spaceΩ.

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 productσ-fieldB[0,∞]⊗ F_∞, with the mapπ :π(s, ω) =ω and the map τ:τ(s, ω) =s, with the filtrationFˆ=π⁻¹(F) which will immediately be identified with F. In such a setting, for ([0,∞]×Ω,B[0,∞]⊗ F_∞,ˆF,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 probabilityQ. Therefore, in these cases, we shall simply say thatQis an extension, or thatQis a solution. We equip systematically the product space [0,∞]×Ω with the progressively enlarged filtration G= (G_t)_t≥0 made from F with the random time τ, i.e.G_t=F_t∨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 theQ-Gsemimartingale decomposition formula we will establish for theP-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 ofF_∞. More generally, ifN ≡1and Λ_∞=∞, again a solution exists on the product space given by the probability measureν^Λ:

ν^Λ[A∩ {s < τ ≤t}] =P[I_A Z _t

s

e^−ΛvdΛ_v], A∈ F_∞, 0< s < t <∞ (1) 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-Aon 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 measureP, we use the notationP[X]forEP[X]

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

Q^F[F] =P[

Z _∞

0

F(s,·)Z_sdΛ_s]

Under this measure, the F_t-conditional survival probability of τ is effectively Z_t. In order to be a solution of the problem-A,Q^F must be an extension of(Ω,F,P). This is equivalent to the condition: R_∞

0 Z_sdΛ_s ≡ 1 (see Nikeghbali [10] for more discussion on such situation), because the restriction ofQ^F onto F_∞ is calculated by

Q^F[A] =P[I_A Z _∞

0

Z_sdΛ_s], A∈ F_∞.

We note that in such a case the Föllmer’s measure Q^F 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 theF-predictableσ-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 ≤ ∞|F_t] =Q[11_{t≤u<∞}Z_udΛ_u+Z_∞δ_∞(du)|F_t]

But few information on Q[τ ∈ du,0 ≤ τ ≤ t|F_t] can be drawn from the supermartingale Z.

That is the main difficulty of our problem.

2.3 Time change

We can believe that ifΛ_t=tfor all 0≤t <∞, the problem could be easier. The following result clarifies the situation. Set ι(t) =t,0≤t≤ ∞and

C_t= inf{s≥0 : Λ_s> t}, 0≤t <∞

Then, (C_t, t ≥ 0) is an increasing family of F-stopping times. Since Λ is a finite increasing process,C_∞= lim_t↑∞C_t=∞. We consider the time change induced by (C_t, t≥0). We recall that Z_Ct = N_Cte^−t∧Λ∞, t ≥ 0, is an F_C-supermartingale with 0 ≤ Z_Ct ≤ 1 (we use the fact Λ_Ct = t∧Λ_∞) and N_C = (N_Ct, t ≥ 0) is a true F_C-martingale. The following result can be checked straightforwardly :

Theorem 2.1 Assume the hypothesisHy(N,Λ). If(Q, τ)is a solution of the problem-Aon the space(Ω,F,P) with parameters N and Λ, (Q,Λ_τ) is a solution of the Problem-A on (Ω,F_C,P) with parametersN_C andι∧Λ_∞. Conversely if(Q, τ)is a solution of the problem-Aon(Ω,F_C,P) with parametersN_C andι,(Q, C_τ−)is a solution of the problem-Aon(Ω,F,P)with parameters N andΛ.

2.4 Probability change

Suppose Λ_∞ = ∞. Let ν^Λ be Cox measure (see the formula (1)). A natural idea to solve the problem-Ais to look for solutions among the probability measuresQon the product space, which are absolutely continuous with respect toν^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

densityξ = _dν^dQ_Λ. Recall that, under the probability measure ν^Λ, 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= (L_t, 0≤t <∞) of positiveB[0, t]⊗ F_tmeasurable functions, such that

1. Lsatisfies the equation :

(L) : N_te^−Λt + Z _t

0

L_t(s,·)e^−ΛsdΛ_s= 1, ∀t≥0.

2. for any 0< s <∞, the process L_t(s), t≥sis a càdlàg ν^Λ-Gmartingale 3. for 0≤t <∞,ν^Λ[ξ|G_t] =N_tI_{t<τ}+L_t(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 onΩfor the problem-Awith parameterN ≡1because then random variable Λ_τ is independent ofF_∞ (see [2], p.136).

3 A solution of the problem-A

In this section, we assume HypothesisHy(N,Λ). We construct a solution of the problem-A which is absolutely continuous with respect to the measure defined by

P[I_A Z

]s,t]

(dΛ_v+δ_∞(dv))], A∈ F_∞,0≤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≤ ∞:

J_t=J_t(θ) := (1−Z_t) exp

½

− Z _t

θ

Z_s 1−Z_sdΛ_s

¾

Then, the processJ = (J_t, θ≤t≤ ∞) is a P-F uniformly integrable martingale.

Proof : Notice thatHy(N,Λ) guarantees the well definiteness ofJ. Recall that the processZ is a supermartingale with the Doob-Meyer decomposition dZ_s =e^−ΛsdN_s−Z_sdΛ_s. Applying integration by parts formula onJ, for θ≤t <∞,

dJ_t = −exp

½

− Z _t

θ

Z_s dΛ_s 1−Z_s

¾

e^−ΛtdN_t

we see that J is aP-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 E_t(θ) = J_t(θ)

1−Z_θ = 1−Z_t 1−Z_θexp

½

− Z _t

θ

Z_s dΛ_s 1−Z_s

¾

, θ≤t≤ ∞.

The processes (E_t(θ), θ ≤ t ≤ ∞) is a P-F uniformly integrable martingale ; and, for t ≥ θ, E[E_t(θ)|F_θ] = 1.

Note that, in different situation,E_t(θ)may be denoted in different forms such asE_t(θ, ω), or E_t^N,Λ(θ). We need to know the behaviour ofE_t(θ)when tis fixed andθ→0.

Lemma 3.3 Let 0 < t ≤ ∞ be fixed. There is a sequence (θ_n) decreasing to 0 such that lim_n(1−Z_θn)E_t(θ_n) = 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 andZ₀ = 1, we can write

limθ↓0P[(1−Z_θ)E_t(θ)] = lim

θ↓0P[(1−Z_θ)] = 0.

But everything being positive, the sequence(1−Z_θ)E_t(θ)converges in L¹ to zero. The lemma follows.

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

θ

Z_vE_t(v)dΛ_v = (1−Z_t)−(1−Z_θ)E_t(θ),

and, forθ≤t≤ ∞, Z _t

0

Z_vE_t(v)dΛ_v = 1−Z_t Proof :We compute for 0< θ <∞, θ≤t <∞,

Z _t

θ

Z_vE_t(v)dΛ_v = Z _t

θ

Z_v 1−Z_t 1−Z_v exp

½

− Z _t

v

Z_s dΛ_s 1−Z_s

¾ dΛ_v

= (1−Z_t) exp

½

− Z _t

v

Z_s dΛ_s 1−Z_s

¾¯¯

¯¯

v=t v=θ

= (1−Z_t)−(1−Z_θ)E_t(θ)

Letθ goes to zero along the sequence(θ_n) in the preceding lemma. We obtain Z _t

0

Z_vE_t(v)dΛ_v = 1−Z_t. Passing to the limit, this identity holds fort=∞.

Remark 3.1 Suppose thatN is continuous. We have the identity

E_t(θ) = exp{−

Z _t

θ

e^−sdN_s 1−Z_s −1

2 Z _t

θ

e^−2sdhNi_s (1−Z_s)² }

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

1−Z_t= (1−Z_θ) exp{

Z _t

θ

Z_udΛ_u 1−Z_u}E_t(θ)

It is the multiplicative decomposition of the positive submartingale(1−Z_t, t≥θ) whereE_t(θ) is the martingale factor and(1−Z_θ) exp{R_t

θ ZudΛu

1−Zu }is the bounded variation factor. Notice that we keep a θ >0 in the decomposition, because in a neighborhood ofθ= 0the integrability of

1−Z1 θ is troublesome.

3.2 The Construction Define

u(θ) =u^Z(θ, ω) :=













0 θ= 0

Z_θ(ω)E_∞(θ, ω) 0< θ <∞

Z_∞(ω) θ=∞

Notice that, for eachθ, the random functionu(θ)onθ∈[0,∞]is non negative and it is càdlàg inθ over(0,∞).

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

Z _∞

0

u(θ)dΛ_θ+u(∞) = 1. Proof.According to Lemma 3.4, we can write

Z _∞

0

u(v)dΛ_v+u(∞) = Z _∞

0

Z_vE_∞(v)dΛ_v+Z_∞= (1−Z_∞) +Z_∞= 1

Now we construct a probability Q^Z on the product space by:

Q^Z[A∩ {t < τ ≤t⁰}] =P[I_A Z

]t,t⁰]

u(θ)(dΛ_θ+δ_∞(dθ))] (2)

for A∈ F_∞,0≤t < t⁰ ≤ ∞. Applying Lemma 3.5, we see that Q^Z is an extension of(Ω,F,P), i.e. Q^Z[A] =P[A], for A ∈ F_∞. Compute now the conditional survival probability under Q^Z. Let0< t <∞ andA∈ F_t. We have

Q^Z[A∩ {τ > t}] =P[I_A( Z _∞

t

u(v)dΛ_v+u(∞))] =P[I_A( Z _∞

t

Z_vE_∞(v)dΛ_v+u(∞))]

From Lemma 3.4, it follows

Q^Z[A∩ {τ > t}] =P[I_A((1−Z_∞)−(1−Z_t)E_∞(t) +Z_∞)] =P[I_A−I_A(1−Z_t)E_∞(t)]

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

Q^Z[A∩ {τ > t}] =P[I_A−I_A(1−Z_t)] =P[I_AZ_t] =Q^Z[I_AZ_t] i.e., Q^Z[τ > t|F_t] =Z_t. We can now state the following theorem.

Theorem 3.1 Suppose Hy(N,Λ). Then, the probability measure Q^Z 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 E_t(θ) ≡ 1,0 < θ ≤ t ≤ ∞. If moreover Λ_∞ = ∞ so that Z_∞ = 0, the random function u^e−Λ(θ) is simply e^−Λθ and the probability measureQ^Z becomes

Q^e−Λ[A∩ {s < τ ≤t}] =P[I_A Z _t

s

e^−ΛθdΛ_θ] =ν^Λ[A∩ {s < τ ≤t}]

whereν^Λ is the Cox measure (see the formula (1)).

Return back to a general Z =N e^−Λ. We know by the construction that Q^Z is absolutely continuous with respect toν^Λ. Now we look at its density process.

Theorem 4.1 Suppose Hy(N,Λ) and Λ_∞=∞ so that Z_∞= 0. Set L(s, ω) =

½ N_s(ω)E_∞(s, ω), 0< s <∞, ω ∈Ω

0 s= 0 or s=∞

(in other words, L=N_τE_∞(τ) ν^Λ-a.s.). Then, L is the probability density of Q^Z with respect to ν^Λ: Q^Z = L·ν^Λ, and the G-density process L_t = ν^Λ[L|G_t], 0 ≤ t < ∞, is given by L_t = N_t∧τE_t∨τ(τ), 0≤t≤ ∞.

Proof. The first part of the theorem is a direct consequence of the construction. For the assertion onG-density process(L_t,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|G_a](s, ω) =G(s, ω)I_{s≤a}+J(ω)I_{a<s}

where

G(s, ω)I_{s≤a}=P[F(s,·)|F_a](ω)I_{s≤a} ν^Λ-almost surely J(ω) =e^Λa(ω)P[R_∞

a F(s,·)e^−ΛsΛ(ds)|F_a](ω) P-almost surely In particular, we have

ν^Λ[N_τE_∞(τ)|G_a] =N_a∧τE_a∨τ(τ)

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

P[N_sE_∞(s)|F_a] =N_sE_a(s) =N_a∧sE_a∨s(s), s≤a.

ItsJ(ω) part is computed by,s > a, e^ΛaP[R_∞

a N_vE_∞(v)e^−ΛvΛ(dv)|F_a] = e^ΛaP[R_∞

a Z_vΛ(dv)|F_a]

= e^ΛaZ_a(recall thatZ_∞= 0)

= N_a∧sE_a∨s(s) Combining these two results we obtain the lemma.

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

5 Characterizations of Q

The probability measureQ^Z possesses many properties and can be characterized in different ways as we present now. We assume the hypothesisHy(N,Λ) andΛ_∞=∞.

5.1 Definition of new Properties

We introduce the following properties. Let K to be the set of all families (χ_t,0 ≤ t < ∞) such that each χ_t=χ_t(s, ω)is a function defined on [0, t]×Ωand isB[0, t]⊗ F_t-measurable.

•Partial SeparationAn elementχ= (χ_t,0≤t <∞)inKis said to be in a partially separable form, if, for any 0 < v < ∞, there exist two F-adapted càdlàg processes (X_t^(v))_t≥v,(Y_t^(v))_t≥v (defined only fort≥v) such that

χ_t(s,·) =X_t^(v)Y_s^(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= (χ_t,0≤t <∞) inK is said to be in a completely separable form, if there exist twoF-adapted càdlàg processes (X_t)_t≥0,(Y_t)_t≥0 such that

χ_t(s,·) =X_tY_s,

for all0≤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 Qon the product space.

¦ Local Density Hypothesis³ We say that Qsatisfies the local density hypothesis, if there exists a positive element α= (α_t,0≤t <∞)inK such that, for0≤t <∞ andB ∈ B[0, t],

Q[τ ∈B|F_t] = Z

B

α_t(s,·)e^−ΛsdΛ_s

3To be distinguished from the density hypothesis introduced in Karoui et al. [3]

The family α will be called a family of local density functions. To such a family α, we can ascribe different properties (We will see later the implications of these properties):

? Decreasing conditionR_t

0α_t(u,·)e^−ΛudΛ_u >0 for any 0< t <∞, and, for 0≤u <∞, the map

t→ α_t(u,·) R_t

0 α_t(s,·)e^−ΛsdΛ_s is a continuous decreasing function int∈[u,∞).

? Partial separation formα is in partially separable form.

? Complete separation formα is in a completely separable form.

? Canonical formThe familyα takes the formα_t(s,·) =N_sE_t(s) for 0< s≤t <∞.

¦ Conditional law InvarianceWe say thatQ satisfies the conditional law invariance hypo- thesis, if, for any0< a≤b, for any A, B∈ B[0, a], we have

Q[τ ∈A|F_a]

Q[τ ∈B|F_a] = Q[τ ∈A|F_b] Q[τ ∈B|F_b] wheneverQ[τ ∈B|F_b]>0.

¦ Proportionality Hypothesis LetQ be a probability measure on (Ω,A). We say that the conditional proportionality hypothesis is satisfied, ifQ[τ ≤t|F_t]>0 for any0< t <∞, and, for any 0< u <∞, the map

t→ Q[τ ≤u|F_t] Q[τ ≤t|F_t]

has a version which is a continuous decreasing function int∈[u,∞).

Remark 5.1 Chronologically it was the proportionality hypothesis which leaded us to find our solution Q^Z to the problem-A.

5.2 Q^Z in canonical form

Theorem 5.1 The probabilityQ^Z 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.Let0≤t <∞, A∈ F_t, B∈ B[0, t], by construction ofQ^Z, we can write Q^Z[A∩ {τ ∈B}] =Q^Z[I_A

Z

B

N_sE_∞(s)e^−ΛsdΛ_s] =Q^Z[I_A Z

B

N_sE_t(s)e^−ΛsdΛ_s]

This proves, forQ^Z, the local density hypothesis with canonical form. The uniqueness is obvious.

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

t↓0(1−Q[τ > t]) = lim

t↓0(1−Q[Z_t]) = 0.

Lemma 5.1 For any 0< t <∞, for A∈ B[t,∞], Q[τ ∈A|F_t] =Q[

Z

A

N_ue^−ΛudΛ_u+I_{∞∈A}Z_∞|F_t] Proof.Foru≥t,

Q[τ > u|F_t] =Q[Z_u−Z_∞+Z_∞|F_t] =Q[

Z _∞

u

N_se^−ΛsdΛ_s+Z_∞|F_t] That is what should be proved.

Lemma 5.2 Suppose the proportionality hypothesis. Then, for any 0≤t <∞, forA∈ B[0, t], Q[τ ∈A|F_t] =

Z

A

N_sE_t(s)e^−ΛsdΛ_s i.e., the local density hypothesis holds in canonical form.

Proof.Let0< s <∞be fixed. Let r_t:= Q[τ ≤s|F_t]

Q[τ ≤t|F_t] = Q[τ ≤s|F_t]

1−Z_t , s≤t <∞.

By hypothesis (r_t, t≥s) is a continuous decreasing function. It is clearlyF-adapted. Consider the identity

Q[τ ≤s|F_t] = (1−Z_t)r_t. (3)

The left-hand side of this identity is an F-martingale. Therefore, the right-hand side should have its drift null, i.e.,

r_tZ_tdΛ_t+ (1−Z_t)dr_t= 0, s≤t <∞.

Solving this equation leads to

r_t= exp{−

Z _t

s

Z_udΛ_u

1−Z_u}, s≤t < η,

where η := inf{t ≥s :r_t = 0}. But the above expression shows that necessarily η = ∞. We substitute this expression of r_t into the identity (3) to get

Q[τ ≤s|F_t] = (1−Z_t) exp{−

Z _t

s

Z_udΛ_u 1−Z_u} Lets↓0. We see that

exp{−

Z _t

0

Z_udΛ_u 1−Z_u}= 0

Using these facts, we write finally

Q[τ ≤s|F_t] = (1−Z_t) exp{−

Z _t

s

Z_udΛ_u 1−Z_u}

= Z _s

0

(1−Z_t) exp{−

Z _t

u

Z_vdΛ_v 1−Z_v} Z_u

1−Z_udΛ_u = Z _s

0

N_uE_t(u)e^−ΛudΛ_u That ends the proof.

Lemma 5.3 We assume that, for any 0< v < a <∞, on the set {Q[v < τ ≤a|F_a]>0} ∈ F_a the map

t→r^(v)_t = Q[v < τ ≤a|F_t] Q[v < τ ≤t|F_t]

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 parameterv >0. Notice

limv↓0Q[v < τ ≤a|F_a] =Q[τ ≤a|F_a] = 1−Z_a>0 Let0< v < a <∞ and consider

r_t=r^(v)_t := Q[v < τ ≤a|F_t] Q[v < τ ≤t|F_t].

By hypothesis, on the set A_v ={Q[v < τ ≤ a|F_a]>0} ∈ F_a, the function (r_t, a ≤t <∞) is strictly positive, continuous and decreasing, and clearlyF-adapted. Rewrite this fact in another form

Q[v < τ ≤a|F_t] = (m_t−Z_t)r_t, t≥a, (4) where m_t = Q[v < τ|F_t] (a càdlàg version). On the set A_v, 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.

r_tZ_tdΛ_t+ (m_t−Z_t)dr_t= 0, t≥a.

Let S_n= inf{t≥a:m_t−Z_t≤ ¹_n}, η_v = sup_nS_n,

Becausem_a−Z_a>0on the setA_v, one has η_v > a. Leta≤t < η_v. We can write

−∞<ln(r_t) = Z _t

a

(m_t−Z_u) (m_t−Z_u)

dr_u r_u =−

Z _t

a

Z_udΛ_u (m_t−Z_u) where we use the fact thatln(r_a) = 0. In other words,

r_t= exp{−

Z _t

a

Z_udΛ_u

m_u−Z_u}, a≤t < η_v.

We have now another equation forQ[v < τ ≤a|F_b]on the setA_v: Q[v < τ ≤a|F_t] = (Q[v < τ|F_t]−Z_t) exp{−

Z _t

a

Z_udΛ_u

Q[v < τ|F_u]−Z_u}, a≤t < η_v. Lettingv↓0we getA_v ↑Ω,η_v ↑ ∞(because of Hy(N,Λ)) andQ[v < τ ≤a|F_t]↑Q[τ ≤a|F_t].

We obtain

Q[τ ≤a|F_t] = (1−Z_t) exp{−

Z _t

a

Z_udΛ_u 1−Z_u} Leta↓0. We see that

exp{−

Z _t

0

Z_udΛ_u 1−Z_u}= 0 We have finally

Q[τ ≤a|F_t]

= (1−Z_t) exp{−R_t

a ZudΛu

1−Zu}

= R_a

0(1−Z_t) exp{−R_t

s ZudΛu

1−Zu }_1−Z^Zs_sdΛ_s

= R_a

0 N_sE_t(s)e^−ΛsdΛ_s 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 thatQ^Z 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 Q^Z.

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|F_b], such that, outside of a negligible set, for all 0≤a≤b <∞, we have the limits

lima⁰↑bf(a⁰, b) =f(b, b), lim

b⁰↓bf(a, b⁰) =f(a, b) (5) Proof.The implication1.⇒2.has been proved in Lemma 5.2.

To see the implication 2.⇒3.we need only to write, for 0< v < s < t, N_sE_t(s) = N_s^1−Z_1−Z^t_sexp{−R_t

s ZwdΛw

1−Zw }

= N_s^1−Z_1−Z^t

sexp{−R_t

v ZwdΛw

1−Zw +R_s

v ZwdΛw

1−Zw }

= ^1−Zt

exp{R_t

vZwdΛw 1−Zw }

Nsexp{R_s

v ZwdΛw 1−Zw } 1−Zs

This proves the partial separation form with

X_t^(v) = 1−Z_t exp{R_t

v ZwdΛw

1−Zw }, Y_s^(v)= N_sexp{R_s

v ZwdΛw

1−Zw } 1−Z_s

The condition 3. is proved.

Let us see the implication3.⇒2.. Let0< v < a < b <∞. IfQ[v < τ ≤a|F_a]>0, we have R_a

v Y_s^(v)e^−ΛsdΛ_s >0. Since, for a≤t <∞,X_t^(v)>0 by hypothesis, we can write r^(v)_t := Q[v < τ ≤a|F_t]

Q[v < τ ≤t|F_t] = R_a

v Y_s^(v)e^−ΛsdΛ_s R_t

vYs^(v)e^−ΛsdΛ_s

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→ N_sE_t(s)

R_t

0N_vE_t(v)e^−ΛvdΛ_v = N_sE_t(s)

1−Z_t = N_s 1−Z_sexp

½

− Z _t

s

Z_v 1−Z_vdΛ_v

¾

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|F_t] = 1−Z_t is strictly positive. Notice also that

Q[τ ≤t|F_t] =R_t

0α_t(v,·)e^−ΛvdΛ_v= 1−Z_t Q[τ ≤s|F_t] =R_s

0 α_t(v,·)e^−ΛvdΛ_v

So, Q[τ ≤s|F_s]

Q[τ ≤t|F_t] = 1 1−Z_t

Z _s

0

α_t(v,·)e^−ΛvdΛ_v. By hypothesis,

t→ α_t(v,·) 1−Z_t

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

that Q[τ ≤s|F_t]

Q[τ ≤t|F_t]

has a version which is a continuous decreasing function int∈[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],

Q[τ ∈B|F_b] = Z

B

X_b^(v)Y_u^(v)e^−ΛudΛ_u

IfQ[τ ∈B|F_b]>0,R

BY_u^(v)e^−ΛudΛ_u >0. Recall that, by hypothesis, X_a^(v) >0, X_b^(v)>0. These facts yield, forA,B∈ B[v, a],

Q[τ ∈A|F_a] Q[τ ∈B|F_a] =

R

AYu^(v)e^−ΛudΛ_u R

BY_u^(v)e^−ΛudΛ_u = Q[τ ∈A|F_b] Q[τ ∈B|F_b] Such an identity being true for all v >0, we have actually

Q[τ ∈A|F_a]

Q[τ ∈B|F_a] = Q[τ ∈A|F_b] Q[τ ∈B|F_b]

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

f(a, b) = Z _a

0

N_uE_b(u)e^−ΛudΛ_u

and we check directly thatf(a, b) is a version of Q[τ ≤a|F_b]which satisfies the property (5).

The implication from condition 1-4 to the condition 5. is proved.

We show finally the implication 5.⇒1.. Letf(a, b),0≤a≤b <∞, be the function in the condition 5. According to the conditional law invariance hypothesis, we can write

f(a, b)

f(b, b) = f(a, b⁰)

f(b, b⁰), a≤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:

limb⁰↑b

f(a, b⁰) f(b⁰, b⁰) = lim

b⁰↑b

f(a, b)

f(b⁰, b) = f(a, b) f(b, b)

(notice that there is no problem of zero denominatorf(b⁰, b), because f(b, b) = 1−Z_b >0 so that forb⁰ close to b,f(b⁰, b)>0.) and

limb⁰↓b

f(a, b⁰) f(b⁰, b⁰) = lim

b⁰↓b

f(a, b⁰)

1−Z_b⁰ = 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, b⁰)

f(b, b⁰) ≥ f(a, b⁰)

f(b⁰, b⁰), a≤b≤b⁰ <∞ The condition 1. is proved

6 Complete Separation Condition

We assumeHy(N,Λ)andΛ_∞=∞. 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(T_n)_n≥1 of F-stopping times such that lim_n→∞T_n =∞ and, for any n ≥1,Q is a probability measure on any of G_Tn and P|_FTn =Q◦π⁻¹|_FTn, andQ[τ > T_n∧t|F_Tn∧t] =Z_Tn∧tfor all 0≤t <∞.

Notice that the above two conditions on Q are well defined because they concern only the behaviour ofQover σ-field G_Tn.