Generalized Cox Model for Default Times

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Generalized Cox Model for Default Times

Djibril Gueye, Monique Jeanblanc

To cite this version:

Djibril Gueye, Monique Jeanblanc. Generalized Cox Model for Default Times. 2021. �hal-03264864�

Generalized Cox Model for Default Times ^∗

Djibril Gueye

Monique Jeanblanc

June 18, 2021

Contents

1 Introduction 1

2 Generalities 2

2.1 Facts on stochastic processes . . . 2

2.2 Default times and characteristics . . . 3

2.3 Conditional densities . . . 4

3 Generalized Cox model 5 3.1 Case whereK is continuous . . . 6

3.2 Case whereK is c`adl`ag . . . 6

3.2.1 Doob-Meyer decomposition ofZ . . . 7

3.2.2 Multiplicative decomposition . . . 8

3.2.3 Conditional densities . . . 9

3.2.4 Other computations . . . 10

3.2.5 Particular case: K predictable . . . 10

3.3 Examples of c`adl`ag processesK . . . 10

3.3.1 Brownian filtration . . . 10

3.3.2 Jiao & Li Model . . . 11

3.3.3 Subordinator . . . 12

3.3.4 Marked point process . . . 13

3.3.5 Shot noise . . . 13

3.4 Case whereK is c`agl`ad . . . 17

4 Pricing of Defaultable Bonds 18

1 Introduction

In this paper, we revisit the standard model of default time based on Cox process, introduced by Lando in [19], in which the default time is the first time when an increasing process adapted to a given filtration F, absolutely continuous with respect to Lebesgue’s measure, hits a level which is a positive random variable independent of the given filtration. It follows that this default time

∗This research benefited from the support of the ’Chaire March´es en Mutation’, French Banking Federation and ILB, Labex ANR 11-LABX-0019.

†Institut de Recherche en Math´ematique Avanc´ee, 7 rue Ren´e Descartes, 67084 Strasbourg Cedex e-mail:

djibril.gueye@aims-senegal.org

‡Universit´e Paris-Saclay, CNRS, Univ Evry, Laboratoire de Math´ematiques et Mod´elisation d’Evry, 91037, Evry, France; e-mail: monique.jeanblanc@univ-evry.fr

avoids all F-stopping times. We relax the assumption that the increasing process that hits the threshold level is absolutely continuous and only assume that this process is adapted, increasing and continuous on right with limits on left (or continuous on left with limits on right). This leads us to a random time which does not avoid F-stopping times. We compute the characteristics of this random time, i.e., its compensator, the associated Az´ema supermartingales and the conditional distribution. We also study existence of conditional densities, in the sense of Jacod [13], Jiao & Li [17] and the extended Jacod’s hypothesis introduced in Li & Rutkowski [21]. One of the advantages of our construction is that one can fix in advance the sequence ofF-stopping times not avoided by the random time. A first attempt to such a generalisation can be found in B´elanger et al. [3] where the increasing process is predictable and right-continuous. Some related works are those of Jiao &

Li [17, 18], Gehmlich & Schmidt [9] and Fontana & Schmidt [8]. We have chosen, for ease of the reader, to give elementary proofs, instead of making use of general results based on dual predictable (resp. optional) projections of the default process, as it is done in Jeanblanc & Li [14].

In the first section, we introduce notation and basic notions. In the second section, we present our model. In the third section, we give many examples of our construction and we pay a particular attention to shot noise modeling. In the fourth section, we give closed form expression for the price of some defaultable claims and their dynamics.

2 Generalities

In this section, we recall well known facts about stochastic calculus and models of default times.

2.1 Facts on stochastic processes

In this first subsection, we recall, for the ease of the reader, some classical results, notation and def- initions which will be used in the paper. We refer to [1, Chapter 1] for related proofs (or references for them) and more information.

In this subsection, we work on a probability space (Ω,G,K,P) endowed with a filtrationK, com- plete and continuous on right. We denote byO(K) (resp. P(K)) the K-optional (resp. predictable) σ-algebra on Ω×R+ and byB(R+) the Borelian sets of R+:={x :x≥0}.

For a c`adl`ag process X = (Xt)_t≥0, we denote by X₋ = (X_t−)_t≥0 its left limit process and

∆Xt=Xt−Xt−,∀t≥0 its jump at timet(withX0−= 0).

A processK = (Kt)t≥0 is increasing (resp. decreasing) if, for 0≤s < t, one hasKs≤Kt, a.s.

(resp. K_t≤K_s, a.s.). We setK₀₋= 0. For an increasing (or decreasing) c`adl`ag processK, we note K_t^c =K_t−P

s≤t∆K_s,∀t≥0 its continuous part.

The Stieljes integral of a bounded c`adl`ag process ϕwith respect to a c`adl`ag increasing process Kis, for 0≤s < t, denotedRt

sϕ_udK_u:=R

]s,t]ϕ_udK_u. Note thatR

]0,t]dK_u=K_t−K₀,∀t >0 and thatR

[0,t]dKu=Kt−K0+ ∆K0=Kt,∀t≥0.

We recall that any c`adl`agK-supermartingale¹ (resp. submartingale)Y admits a unique Doob- Meyer decomposition, i.e.,Y =M^Y−A^Y (resp. Y =M^Y+A^Y) whereM^Y is aK-martingale andA^Y an increasing K-predictable process with A^Y₀ = 0. Any strictly positive c`adl`ag K-supermartingale

1If the supermartingale is not c`adl`ag, one has to use its Doob-Meyer-Mertens-Gal’cˇuk decomposition, see, e.g., Th.

1.2 in [14]. We shall not enter in this kind of computations here and we refer the reader to [14].

Y admits a unique multiplicative decomposition as

Y =N C (2.1)

whereN is a K-local martingale satisfyingN0= 1 andC a decreasing K-predictable process (see, e.g., [1, Pro. 1.32, Page 15]).

TheK-compensator of aK-adapted processX with bounded variation is the K-predictable process with bounded variationX^com,K such thatX−X^com,K is aK-martingale andX₀^com,K = 0. If X is increasing (resp. decreasing), so isX^com,K.

If H is a filtration satisfying H ⊂ K, and Y is a K-adapted process such that Yϑ11_{ϑ<∞} is integrable for anyH-stopping timeϑ, theH-optional projection ofY is theH-optional process^o,HY such thatE[Yϑ11_{ϑ<∞}|Hϑ] = ^o,HYϑ11_{ϑ<∞}, for anyH-predictable stopping time ϑ. This optional projection satisfiesE[Y_t|Ht] = ^o,HY_t, for all t≥0. IfY is a c`adl`agK-martingale, then ^o,HY is an H-martingale.

Likewise, the H-predictable projection of the process Y such that Y_ϑ11_{ϑ<∞} is integrable for any predictableH-stopping timeϑ, is the uniqueH-predictable process^p,HY such that

E[Y_ϑ11_{ϑ<∞}|H_ϑ−] = ^p,HY_ϑ11_{ϑ<∞}, for anyH-predictable stopping timeϑ.

AK-stopping timeϑis said to beK-predictable if there exists an increasing sequence (ϑn)_n≥1 of K-stopping times converging toϑ, such thatϑn< ϑon the set{ϑn>0}, for alln≥1. AK-stopping timeϑis totally inaccessible if it avoids K-predictable stopping times (i.e., P(ϑ=S <∞) = 0 for anyK-predictable stopping timeS). Arandom time τ is a non-negative random variable, itsgraph is the subset [[τ]] ofR+×Ω defined as [[τ]] ={(t, ω) : τ(ω) =t}.

For two filtrationsFandKsatisfyingF⊂K, one says thatFisimmersedinKif anyF-martingale is aK-martingale (see, e.g., Chapter 3 in [1] and the references therein).

2.2 Default times and characteristics

We work now on a filtered probability space (Ω,G,F,P) whereF= (F_t)_t≥0 is a complete and con- tinuous on right filtration, where F₀ is trivial, and G a σ-algebra satisfying F_∞ ⊂ G. In what follows, the filtrationFis always taken to be the reference filtration and in order to reduce notation, whenever there is no confusion, we will not explicitly write the dependence on the filtrationFwhen writing the projections onFand referring to a predictable, optional or adapted process (e.g., X is a predictable process meansX is anF-predictable process).

We are given a random time τ defined on (Ω,G). The law of the random timeτ is denoted by η, i.e.,E[h(τ)] =R

R+h(u)η(du) for any bounded Borel functionhdefined onR⁺. We introduce the indicator default process ofτ, denoted by A, as the right-continuous increasing process defined by At= 11_{τ≤t},∀t≥0.

We denote by Z the c`adl`ag Az´ema supermartingale associated with τ, which is the optional projection of 1−A and satisfies

Z_t=P(τ > t|F_t),∀t≥0, (2.2)

and the optional Az´ema supermartingaleZewhich is the optional projection of 1−A−, and satisfies

Zet=P(τ≥t|Ft),∀t≥0. (2.3)

Note thatZ =Ze+ (see [1, page 20]) andZt>0 on the set{t < τ}andZ_t−>0 on the set{t≤τ}

(see [1, Lemma 1.51]). The Doob-Meyer decomposition ofZisZ =M−A^p, whereM is a martingale and A^p an increasing predictable process². We recall that the random time τ is said to avoid all F-stopping times (resp. all predictableF-stopping times) if P(τ =ϑ <∞) = 0 for any F-stopping time (resp. for any predictableF-stopping time)ϑ.

The filtration G is the smallest complete and right-continuous filtration that contains F and turnsτ into aG-stopping time. Thecompensator ofA (we shall also say compensator ofτ) is the G-predictable increasing process Λ^G such thatA−Λ^Gis a G-martingale (in fact, Λ^G =A^com,G). It is well known that there exists anF-predictable increasing process Λ such that Λ^G_t = Λ_t∧τ,∀t≥0 (see e.g., Pro. 2.11 b), Page 36 in [1]) and that Λ^G_t11_{t≤τ}= 11_{t≤τ}Rt

0 dA^p_s

Zs− (see, e.g., Pro. 2.15, page 37 in [1]). The process Λ is not uniquely defined afterτ (except ifZ₋>0) and, hereafter, we choose

dΛ_t=dA^p_t Zt−

11_{Zt−>0}, ∀t≥0,Λ₀= 0. (2.4)

We shall call Λ theF-predictable reduction of the compensator ofτ.

TheF-conditional cumulative function ofτ is defined, for any (t, u)∈R²+ by

Ft(u) =P(τ≤u|Ft). (2.5)

The family (F(u), u ∈ R+) is a family of F-martingales, valued in [0,1], increasing w.r.t. the parameteru (i.e., Ft(u) ≤Ft(v), a.s.for u < v,∀t ≥0). The family of processes (G(u), u ∈ R+) defined asG(u) = 1−F(u) is the family of conditional survival processes.

2.3 Conditional densities

In this subsection, we recall some definitions on conditional densities. Later on, we shall examine if, in our model, these conditional densities exist.

Definition 2.1 The random timeτ admits

• a conditional density in the sense of Jacod [13, Condition (A)](we shall say a J-conditional density) if there exist a non-negativeO(F)⊗ B(R+)-measurable map(ω, t, u)→pt(ω, u)c`adl`ag int and a non-negativeσ-finite measure ρonR+ such that

(J1) for everyu, the process(p_t(u))_t≥0 is a non-negativeF-martingale,

(J2) for every t ≥0, the measure p_t(u)ρ(du) equals P(τ ∈ du| F_t), in other words, for any Borel bounded functionh, for anyt≥0

E[h(τ)|Ft] = Z

R+

h(u)pt(u)ρ(du). In that caseA^p_t =R

[0,t]p_u−(u)ρ(du), for anyt≥0(see [1, Cor. 5.27]).

Note thatR

R+p_t(u)ρ(du) = 1,∀t≥0, a.s.

• a generalized density in the sense of Jiao & Li [17] (we shall say a JL-conditional density) if there exist a family (τi, i= 1,· · ·, n) ofF-stopping times and a non-negativeO(F)⊗ B(R+)- measurable map(ω, t, u)→αt(ω, u)c`adl`ag in tsuch that

2In the literature, the predictable part of the Doob-Meyer decomposition ofZis shown to be theF-dual predictable projection ofA, this is why we keep the notationA^p. The processZeis not c`adl`ag, hence one can not define as usual its Doob-Meyer decomposition.

(JL1) for every u, the process (αt(u))_t≥0 is a non-negative F-martingale, (JL2) for every t≥0, for any bounded Borel functionh

E[h(τ)

n

Y

i=1

11_{τ6=τi}|Ft] = Z

R+

h(u)α_t(u)π(du) whereπ is a non-negative, non atomic measure onR+.

Definition 2.2 The random time τ satisfies the extended Jacod’s hypothesis (introduced by Li &

Rutkowski in [21, Pro. 2.5], we shall say LR-condition) if there exist a non-negativeO(F)⊗ B(R+)- measurable function (ω, t, u) → mt(ω, u) c`adl`ag in t and an F-adapted increasing process D such that

(LR1) for everyu, the process(m_t(u))_t≥u is a non-negativeF-martingale, (LR2) foru≤t

Ft(u) = Z

[0,u]

mt(s)dDs.

Comments 2.3 a) If the J-conditional density exists, one can always chooseρ=η whereη is the law ofτ (see Jacod [13, Pro. 1.5]). Moreover, if η is non atomic, τ avoids allF-stopping times [7, Cor. 2.2]. If η has an atom at t^∗, then P(τ = t^∗) > 0 and the constant stopping time t^∗ is not avoided byτ.

b) If τ avoids all F-stopping times, the existence of a JL-conditional density is equivalent to the existence of a J-conditional density, and one can choseπ=η.

c) If the J-conditional density exists, then LR-condition holds. We shall see that the LR-condition may hold when J and JL-conditional densities do not exist.

d) Under LR-condition, if D is predictable, A^p = R· 0

pmu(u)dDu, where ^pm is the F-predictable projection ofm(see [21, Pro. 5.5.1]).

e) Note that, under LR-condition, the conditional cumulative distributionFt(u) given foru≤t in (LR2) can be obtained for any pair (t, u). To do that, we use the fact that for anyu, the process F(u) is a martingale. Hence, we setFt(u) = E[Fu(u)|Ft] for t < u. To check that indeed, Ft(u) is increasing w.r.t. u for any t, we note that the martingale property of F(u) implies Ft(u) = E[Fs(u)|Ft] for any s > t so that Fs(u) being increasing w.r.t. u for u < s leads to Ft(u) = E[Fs(u)|Ft]≤E[Fs(v)|Ft] =Ft(v) foru < v.

Proposition 2.4 If Fis immersed in G, then LR-condition holds.

Proof: We recall that, if F is immersed in G, then Z is a decreasing process. Under immer- sion, for u ≤ t, one has Ft(u) = 1−Zu and Z is decreasing. Hence, LR condition holds, with D= 1−Z, D0− = 0 andm(s)≡1, sinceR

[0,u]dDs=Du= 1−Zu,∀u∈R+.

3 Generalized Cox model

We now assume that theσ-algebraG is large enough to support a random variable Θ with unit ex- ponential law, independent fromF_∞. We study the generalized Cox model whereKis an increasing F-adapted process such that K₀ = 0. More precisely, if K is c`adl`ag, one defines a random time τ (called a generalized Cox time hereafter) as

τ= inf{t≥0 : Kt≥Θ} (3.1)

and, ifKis c`agl`ad, we modify the definition (we shall see why later), setting τ= inf{t≥0 : Kt>Θ}.

Note thatτ is finite if and only if K_∞− =∞.

In both cases, immersion holds between the reference filtrationFandG, its progressive enlarge- ment with τ, since, obviously Fis immersed in F∨σ(Θ) and F⊂G⊂F∨σ(Θ). The conditional survival process is

P(τ > u|Ft) = Zu, foru < t

= E[Zu|Ft], fort≤u .

In a generalized Cox model, we shall call the quadruplet Z,Ze (defined in (2.2), (2.3)), A^p and F(u) (defined in (2.5)) thecharacteristics ofτ (see [14] for the general definition of the characteris- tics of a default time).

We recall that, for anyt and anyFt-measurable r.v. ζt, one has P(ζ_t>Θ|Ft) =P(ζ_t≥Θ|Ft) =e^−ζt.

3.1 Case where K is continuous

LetKbe an increasingF-adapted continuous process, withK0= 0. We defineτ:= inf{t : Kt≥Θ}.

We obtain immediately thatZ =e^−K.Furthermore, as a particular case of the following Lemma 3.1,τavoids allF-stopping times, and, in particularZe=Z (for theF-stopping timeϑ=t, one has P(τ =t|Ft) = 0). The Doob-Meyer decomposition of Z is Z = 1−(1−e^−K), hence A^p = 1−Z.

From (2.4), the F-predictable reduction of the compensator ofτ is Λ =K, and the multiplicative decomposition (see (2.1)) ofZ isZ=e^−K=e^Λ (with a local martingale part equal to 1). The law ofτ is given byP(τ > u) =E[e^−Ku], so thatE[h(τ)] =E R

R+h(u)e^−KudK_u . Moreover, if K is absolutely continuous w.r.t. Lebesgue’s measure, i.e.,Kt=Rt

0kudu,∀t ≥0, with k a non-negative process, then, τ admits a density f w.r.t. Lebesgue’s measure satisfying f(u) =E[kue^−Ku],∀u≥0 and a J-conditional density given by (choosingρ(du) =du):

p_t(u) = k_ue^−Ku, foru < t,

= E[kue^−Ku|Ft], fort≤u .

IfKis continuous but not absolutely continuous, theJ-density may fail to exist, as we show now.

LetK be the continuous increasing process defined byKt=−ln(1−L_t∧1) + 11_{t>1}(t−1), t≥0, where L is the local time at level 0 of a standard Brownian motion. Then P(τ > t) = E[1− Lt∧1]e^(t−1)+ and, fromE[Lt] =E[|Wt|] =

√

√2t

π, we deduce thatτ has a densityf w.r.t. Lebesgue’s measure. Therefore, if the J-conditional hypothesis is satisfied then, one would have, foru < t

P(τ > u|Ft) =Zu= Z ∞

u

pt(s)f(s)ds

andZ would be absolutely continuous w.r.t. Lebesgue’s measure, which is not the case.

3.2 Case where K is c` adl` ag

LetKbe an increasingF-adapted c`adl`ag process, withK0= 0,Kt<∞for allt≥0 and³K_∞=∞.

We do not assume thatKis predictable (a basic example is a Poisson process).

We define

τ = inf{t : Kt≥Θ}. (3.2)

3IfP(K∞<∞)>0, the r.v. τ takes the value +∞with strictly positive probability. If there existsϑsuch that P(Kϑ=∞)>0, thenτ≤ϑon{K_ϑ=∞}.

Lemma 3.1 For ω fixed, the set {t : Kt ≥ Θ} is of the form [t0,∞[ with Kt₀ ≥ Θ andτ =t0, hence

{Kt<Θ}={τ > t}.

For any finiteF-stopping time ϑ, one has P(τ=ϑ|Ft) =E[e^−Kϑ−−e^−Kϑ|Ft]. In particular, P(τ= ϑ) = E[e^−Kϑ− −e^−Kϑ], hence, τ avoids all F-stopping times if and only if K is continuous. In particular, ifK has jumps and η is non-atomic, the J-conditional density does not exist.

Proof:Due to the increasing property of K, for anyω fixed, the set{t : K_t≥Θ} is of the form [t₀,∞[ or ]t₀,∞[ (wheret₀ depends onω). In the case{t : K_t≥Θ}=]t₀,∞[, one hasK_t0<Θ and K_t0+ ≥Θ for any >0. The right-continuity of K yields to K_t0 ≥Θ, which is a contradiction.

Hence{t : K_t≥Θ}= [t₀,∞[ andK_t0 ≥Θ andτ=t₀.

It follows that{s < τ ≤ t} = {Ks < Θ≤ Kt}, and, for a finite F-stopping timeϑ, one obtains, denotingϑ() = (ϑ−)∨0), that{ϑ()< τ ≤ϑ}={K_ϑ()<Θ≤Kϑ}and

P(ϑ()< τ ≤ϑ|Ft) =E[e^−Kϑ()−e^−Kϑ|Ft].

Passing to the limit when goes to 0 leads to the result. It K is continuous, then, τ avoids any F-stopping time. Ifτ avoidsF-stopping times, thenE[e^−Kϑ− −e^−Kϑ] = 0 for anyϑwhich, due to the increasing property ofKimplies thatKis continuous. Ifηhas no atoms, then the J-conditional

density does not exist ifK has jumps see Comments 2.3 a).

3.2.1 Doob-Meyer decomposition of Z From Lemma 3.1

Zt:=P(τ > t|Ft) =P(Kt<Θ|Ft) =e^−Kt,

so that, in particular, under our assumptionKt<∞,∀t≥0, one hasZt>0,∀t≥0.

Lemma 3.2 Let I be the c`adl`ag F-submartingale It = P

s≤t(1−e^−∆Ks) which admits a Doob- Meyer decomposition I = M^I +A^I. Then, Z admits a DM decomposition Z = M −A^p where dM_t=e^−Kt−dM_t^I and

dA^p_t =e^−Kt−(dK_t^c+dA^I_t), A^p₀= 0,

whereK^cis the continuous part of the increasing processK. Furthermore, theF-predictable reduction of the compensator ofτ isΛ =K^c+A^I.

Proof:FromZ =e^−K, one has, using Stieljes’ integration

dZt = −e^−Kt−dK_t^c+e^−Kt−(e^−∆Kt−1),∀t≥0, whereK^c is the continuous part ofK, i.e., K_t^c=Kt−P

s≤t∆Ks,∀t≥0.

The processK^cbeing increasing and continuous is a submartingale with Doob-Meyer decomposition with no martingale part, i.e.,K^c = 0 +K^c. The processI being increasing (indeed 1−e^−∆K≥0) is a submartingale and admits a Doob-Meyer decompositionI=M^I+A^I. Finally

dZ_t = −e^−Kt−dK_t^c−e^−Kt−dI_t (3.3)

= −e^−Kt−(dK_t^c+dM_t^I +dA^I_t)

= −e^−Kt−dM_t^I−e^−Kt−(dK_t^c+dA^I_t). (3.4) Therefore

dA^p_t =e^−Kt−(dK_t^c+dA^I_t) (3.5)

and (At−Λ_t∧τ)_t≥0 is aG-martingale where, from (2.4) and the fact thatZ₋>0, dΛ_t= 1

Z_t−dA^p_t =dK_t^c+dA^I_t.

The martingale part in the Doob-Meyer decomposition of Z is the process M given by dMt =

−Zt−dM_t^I, M0=Z0 (which is a true martingale due to the fact thatZ is bounded).

Remark 3.3 Let us go back to a general setting where τ is such that immersion holds betweenF andGwhere Gis the progressive enlargement ofFwith τ (here,τ is not necessarily obtained by a generalized Cox model). Then, the c`adl`ag Az´ema supermartingaleZ is decreasing, and there exists K, anF-adapted c`adl`ag increasing process such thatZ=e^−K. DefiningϑfromKas in (3.2) leads to a random time with Az´ema supermartingale equal to Z. Furthermore

P(τ > u|Ft) =E[Zu|Ft] =P(ϑ > u|Ft)

for all pairs (t, u) (due to immersion). Of course, the fact thatτandϑhave the same characteristics does not imply thatP(τ=ϑ) = 1, as explained in the trivial following example. Letτ= inf{t, Kt≥ Σ}whereKis anF-adapted increasing continuous process and Σ a unit exponential random variable independent fromFandϑ= inf{t, Kt≥Θ}defined as in (3.2). Ifτ=ϑ, one should haveKτ =Kϑ, hence Σ = Θ which is not requested.

Comment 3.4 If K^c and A^I,c are absolutely continuous w.r.t. Lebesgue’s measure, and if the sequence (τi, i≥1) of jumps times ofK is increasing, writingP

s≤t∆A^I_s =Rt 0

R

R+xδ(τ_i,θ_i)(ds, dx) whereδis the Dirac measure andθi= ∆A^I_τi, we recover the form of theF-predictable reduction of the compensator ofτ presented in [9, eq. (7)].

Proposition 3.5 The optional Az´ema supermartingale isZe=e^−K−.

Proof:As before, we notet() =t∨. From{t()< τ ≤t}={K_t()<Θ≤K_t}, we obtain, letting go to 0 P(τ =t|Ft) =e^−Kt()−e^−Kt and Zet =Zt+P(τ =t|Ft) =e^−Kt−. The result is also a consequence of [1, Pro.3.9a and Pro.1.46c]. The fact thatZeis predictable can also be obtained by

immersion (see [1, Th. 5.35 f]).

3.2.2 Multiplicative decomposition

Lemma 3.6 The multiplicative decomposition ofZ is⁴ E(Y)tE(−Λ)t=E(Y)te^−ΛtY

s≤t

(1−∆A^I_s)e^∆AIs,∀t≥0,

whereYt= Z t

0

1

Z_s−(1−∆A^I_s)dMs,∀t≥0, withM being the martingale part ofZ in its Doob-Meyer decomposition andA^I,Λ being given in Lemma 3.2.

4We recall that the Dol´eans-Dade exponential of a c`adl`ag semimartingaleX is

E(X)t=e^{Xt−12hX(c),X(c)it} Y

0<s≤t

(1 + ∆Xs)e^−∆Xs,

where X^(c) is the continuous martingale part ofX (see, e.g., [1, Page 8 and 14].

Proof:This result is established in full generality in [14], where the authors “guess” this form and check it. However, it seems interesting to recover that result without guessing this decomposition, and we present a proof. One knows that there exists a predictable increasing process Γ, so that, from the last equality above,Z =N e^−Γ where N is a local martingale. The integration by parts formula using Yoeurp’s lemma [1, Pro. 1.16] leads to

dZ_t = e^−ΓtdN_t+N_t−de^−Γt.

By identifying this decomposition ofZ and the DM decomposition of Z given in (3.4), one gets e^−ΓdN=dM, N₋de^−Γ=−Z₋(dK^c+dA^I),

hencedΓ^c+ 1−e^−∆Γ=dK^c+dA^I which leads to

dΓ^c=dK^c+dA^I,cand 1−e^−∆Γ= ∆A^I. Finally, using the fact that Λ =K^c+A^I =K^c+A^I,c+P

s≤·∆A^I_s, e^−Γt =e^−Γcte^{−Ps≤t∆Γs}=e^−ΛtY

s≤t

(1−∆A^I_s)e^∆AIs,∀t≥0. (3.6)

The equalitydN = e^ΓdM can be written asdN =N₋Y dM with Y = _N^eΓ

− = ^e_Z^∆Γ

− = _Z ¹

−(1−∆A^I), hence, using the fact that Λ =K^c+A^I =K^c+A^I,c+P

s≤·∆A^I_s = Γ^c+P

s≤·∆A^I_s, the proof is

done.

Comment 3.7 Note that, from (3.6) and the fact that Z is non-negative, one has 1−∆A^I ≥1.

Another proof can be done noting that ∆A^I = ^p(∆I) = ^p(1−e^−∆K) = 1− ^p(e^−∆K)≤1, where the first equality comes from [1, Pro. 1.36 b]. Note that this implies that ∆Λ≤1.

One can also check that 1 + ∆N >0. Indeed,

∆N =−e^−(K−+Γ)∆M^I =−e^−(K−+Γ)(1−e^−∆K−∆A^I)>−1. 3.2.3 Conditional densities

Proposition 3.8 If the continuous part of K is absolutely continuous w.r.t. Lebesgue’s measure, the JL-conditional density exists withαt(u) =E[kue^−Ku|Ft] andπis the Lebesgue measure.

Proof:Let (τ_i)_i≥1 be the sequence of jump times of K. For any bounded Borel function h, the process

X_t=h(t)Y

i≥1

11_{t6=τi}=h(t)Y

i≥1

(11_{t<τi}+ 11_{t>τi}) (3.7) isF-optional (indeed, 11_{t<τi} is c`ad and 11<sub>{t>τi}</sub> c`ag hence predictable therefore optional). Hence

E[h(τ)Y

i≥1

11_{τ6=τi}|F_t] = E[X_τ|F_t] =E[ Z ∞

0

X_sdZe_s|F_t] =−E[ Z ∞

0

X_sdZ_s^c|F_t]

= E[ Z ∞

0

h(s)kse^−Ksds|Ft] = Z ∞

0

h(s)E[kse^−Ks|Ft]ds .

The second equality is due to (4.1) (We shall present this well known result later in section 4), the third equality is due to the fact thatX vanishes on the discontinuities ofZe (which are the discon- tinuities ofZ, i.e., of K, since Ze_t=e^−Kt−), so thatR∞

0 X_sdZe_s =R∞

0 X_sdZ_s^c, then we use the fact that, from (3.3),Z_t^c= 1−Rt

0e^−Ks−dK_s^c= 1−Rt

0e^−KsdK_s^c = 1−Rt

0e^−Ksksds.

Remark 3.9 In the case where K^c = 0, one has [[τ]] ⊂ ∪i[[τi]], and τ is a thin time (see, e.g., definition 1.40, Page 18 in [1]). In that case, the JL density is null. This can be viewed as a consequence of the previous proposition, but this follows directly from the definition. Indeed, since P

i≥1P(τ=τi) = 1, one has

E[h(τ)Y

i≥1

11_{τ6=τi}|F_t] = 0 henceα(u)≡0.

3.2.4 Other computations

From Lemma 3.1, ifϑis an F-stopping time not avoided byτ, thenP(∆K_ϑ >0)>0 andK has a jump at timeϑ. Let (τ_i)_i≥1 be the sequence of jump times of K. The conditional probability that the default occurs at timeτi is

pⁱ_t:=P(τ=τ_i|Ft) = E

e^−Kτi−(1−e^−∆Kτi)|Ft

,∀t≥0.

Note thatpⁱ_t = pⁱ_t∧τi, a result due to immersion pointed out in [17, Pro. 5.1]. This implies that pⁱ_t∨τi =pⁱ_τi. The following result is an immediate consequence of Proposition 2.7 in [17].

Proposition 3.10 For a bounded Borel functionh, if the JL conditional density exists,

E[h(τ)|Ft] = Z ∞

0

h(u)αt(u)π(du) +X

i≥1

E[h(τi)pⁱ_τi|Ft],∀t≥0. The conditional survival probability is given by

P(τ > u|Ft) = e^−Ku, for t≥u

= Z ∞

u

α_t(y)π(dy) +X

i≥1

E[11_{τi>u}pⁱ_τ

i|Ft], for t < u 3.2.5 Particular case: K predictable

In the case whereK is c`adl`ag and predictable, the set of jump times ofK, which are all predictable, exhaust the sequence ofF-stopping times not avoided byτ. The same model was presented in [3], in a slightly more general setting where Θ is not a unit exponential r.v. (but is still independent of F). Let

K_t= Z t

0

k_sds+X

i≥1

11_{τi≤t}θ_i (3.8)

where (τi)_i≥1is an increasing sequence of predictable stopping times,kis anF-adapted non-negative process and (θi)i≥1a sequence of non-negative random variables withθi ∈ Fτ_i−. Note thatI(defined in Lemma 3.2) is predictable, henceA^I =I=P

i≥111_{τi≤·}(1−e^−θi) andM^I = 0. As a check, one can see, from the previous computations (3.5) withdA^I_t =dI_t, thatdA^p_t =−dZ_t.

Then, Λ_t=Rt

0k_sds+P

i≥111_{τi≤t}(1−e^−θi). The multiplicative decomposition obtained in Lemma 3.6 is the one obtained in [3, equation 2.4], i.e.,Zt= exp(−Λ^c_t)Q

s≤t(1−∆Λs).

3.3 Examples of c` adl` ag processes K

3.3.1 Brownian filtration

We construct a simple example whereKis c`adl`ag (but not continuous) and predictable. LetFbe a Brownian filtration, (τi)_i≥1an increasing sequence of finite stopping times (e.g.,τi= inf{t : St=bi}

where S is anF-diffusion going to +∞ when t goes to +∞ and (bi)_i≥1 an increasing sequence of real numbers withb₁ > S₀) and defineK_t=Rt

0k_sds+P

i≥111_{τi≤t}θ_i where (θ_i)_i≥1 is a sequence of non-negative random variables, withθi∈ Fτ_i, i.e.,θi=c+Rτi

0 ψs⁽ⁱ⁾dWsfor anF-optional process ψ⁽ⁱ⁾ (such that θi ≥ 0). In that case, results of Subsection 3.2.5 can be applied, and, under the conditionK_∞=∞, the random timeτ is finite.

Furthermore, predictable representation property holds inGwith respect to (W, M^G), whereM^G= A−Λ·∧τ (See [16, Rem. 6.1]).

3.3.2 Jiao & Li Model

We can recover the model of Jiao & Li [18].

Proposition 3.11 We consider (Ω,G,F,P) a filtered probability space, Γ a continuous increasing F-adapted process and(τi)_i≥1 a strictly increasing sequence of F-stopping times and we setτ0= 0.

We introduce the processAⁱ settingAⁱ_t= 11_{τi≤t},∀t≥0 and itsF-compensator Jⁱ. One denotes by Ψan increasing deterministic function such thatΨ(0) = 0 andΨ(∞) =∞. We set

Kt=X

i≥1

11_{τi≤t}[Ψ(τi)−Ψ(τi−1)] + Γt,∀t≥0.

Then,

A^p_t = Γ_t+X

i≥1

Z t

0

e^{−Ψ(τi−1)}−e^−Ψ(s)

dJ_sⁱ,∀t≥0.

Proof:Note that K is c`adl`ag, with continuous part K^c = Γ and has jumps at time τ_i with size Ψ(τ_i)−Ψ(τ_i−1).

Moreover, the support ofJⁱ is [τ_i−1, τ_i]. Indeed, the processMⁱ:=Aⁱ−Jⁱ is a martingale with null initial value and, due to the fact thatAⁱ_τi−1 = 0, one obtains

0 =E[M_τⁱ_i−1] =E[Aⁱ_τi−1−J_τⁱ_i−1] =−E[J_τⁱ_i−1]

hence J_τⁱ_i−1 = 0 which implies that J_tⁱ = 0 on {t ≤ τ_i−1}. We recall that, for any bounded F- predictable process H, the compensator of the increasing process R·

0HsdAⁱ_s is R·

0HsdJ_sⁱ. As an immediate application, theF-compensator of the increasing process

Y_t^1,i:= 11_{τi≤t}e^−Ψ(τi)= Z t

0

e^−Ψ(s)dAⁱ_s,∀t≥0, is R·

0e^−Ψ(s)dJ_sⁱ, i.e., Y_t^1,i−Rt

0e^−Ψ(s)dJ_sⁱ,∀t ≥ 0 is an F-martingale and the compensator of the increasing process

Y_t^2,i:= 11_{τi+1≤t}e^−Ψ(τi)= Z t

0

e^−Ψ(τi)dAⁱ⁺¹_s = Z t

0

e^−Ψ(τi)11_{τi<s}dAⁱ⁺¹_s ,∀t≥0 isR·

011_{τi<s}e^−Ψ(τi)dJ_sⁱ⁺¹=R·

0e^−Ψ(τi)dJ_sⁱ⁺¹, where we have used that the condition on the support of Jⁱ⁺¹ and the fact that, being adapted and c`ag, the process (11_{τi<t}e^−Ψ(τi))_t≥0 is predictable.

Let us denote byU the increasing processU_t=P

i≥111_{τi≤t}(Ψ(τ_i)−Ψ(τ_i−1)),∀t≥0. Then e^−Ut =X

i≥1

11_{{τi≤t<τi+1}}e^−Ψ(τi)=X

i≥1

(Y_t^1,i−Y_t^2,i).

The supermartingalee^−U =P

i≥1(Y^1,i−Y^2,i) admits the DM decomposition e^−Ut = Υt+X

i≥1

Z t

0

(e^−Ψ(s)−e^−Ψ(τi))dJ_sⁱ⁺¹= Υt−ζt,∀t≥0,

where Υ is a martingale andζis the predictable increasing processP

i≥0

R·

0(e^−Ψ(τi)−e^−Ψ(s))dJ_sⁱ⁺¹. Applying integration by parts formula toZ=e^−Ue^−Γ, one obtains

dZ_t=e^−ΓtdΥ_t−(Z_tdΓ_t+e^−Γtdζ_t),∀t≥0,

hencedA^p_t =ZtdΓt+e^−Γtdζt,∀t≥0 . If the stopping times (τi)_i≥1are predictable,Z is predictable, andA^p= 1−Z.

This result extends to the case where Ψ is an increasingF-adapted process andKt=P∞

i=111_{τi≤t}[Ψτ_i− Ψτi−1] + Γtwhere Γ isF-adapted increasing and continuous to give that

dA^p_t =Z_tdΓ_t+e^−Γtdζ_t,∀t≥0, withζ=P

i≥1

R·

0(e^−Ψτi−1 −e^−Ψs)dJ_sⁱ. 3.3.3 Subordinator

LetKbe a subordinator with null drift (i.e. an increasing L´evy process with null continuous part (see, e.g., Prop 3.10 and Section 4.2.2 in [6] and [15, Section 11.6]) with L´evy’s measureν, and letFbe its filtration. Then, setting Ψ(u) =R

R+(1−e^−ux)ν(dx) andκ= Ψ(1), the processnt=e^−Kt+tκ,∀t≥0 is an F-martingale and the multiplicative decomposition of Z is Zt = nte^−tκ,∀t ≥ 0, leading to dZt=−κZtdt+e^−tκdnt, hencedA^p_t =Ztκdtand Λt=tκ,∀t≥0.

This can be also obtained from the previous results noting that the processI, defined in Lemma 3.2 is also a subordinator, and from the compensation formula (see, e.g., [15, Proposition 11.2.2.3]) the process (It−tκ, t≥0) is a martingale. ThereforeA^I_t =tκ,∀t≥0 and, from Lemma 3.2, we recover the form of the process Λ.

Furthermore,using the independence and stationarity of increments of the L´evy processK, one gets P(τ > u|Ft) =E[Zu|Ft] =e^−KtE[e^−Ku−t] =e^−Kte^−κ(u−t),for allu≥t≥0.

The law ofτ is exponential with parameterκ.

Let us consider the particular case whereXis a Compound Poisson Process (CCP) with non-negative jumps, i.e.,X_t=PNt

n=1Y_n,∀t ≥0 where N is a Poisson process with intensity λwith jump times (T_n, n≥1) and (Y_n, n≥1) are non-negative random variables, i.i.d. with cumulative distribution functionF, and independent from N. The process X being a subordinator with L´evy’s measure ν(dx) =λF(dx), we obtain thatκ= Ψ(1) =λ(1−E[e^−Y1]).

We consider now the case whereFis the filtration generated by a subordinatorXand a Brownian motion W independent from X, and set Kt = Rt

0ksds+Xt,∀t ≥ 0,where k is F^W adapted and e^−Xt =nte^−κt. Then,

dZt=−ktZtdt+e^−R0tksds(−κe^−Xtdt+e^−tκdnt) =Mt−Zt(kt+κ)dt and Λt=Rt

0ksds+κt,∀t≥0.

3.3.4 Marked point process

If an increasing sequence (τ_i)_i≥1 of F-stopping times and a sequence of non-negative random vari- ables variables (θ_i ∈ Fτi, i ≥1) are given, as well as a non-negative F-adapted process k, one can construct an increasing c`adl`ag processKasK_t=Rt

0k_sds+P

i≥111_{τi≤t}θ_i. This framework covers many cases of generalized Cox model.

The associated random timeτ does not avoid the stopping random times (τ_i)_i≥1and we setτ₀= 0. We have, using the notation of section 3.2.1 whereIt=P

i≥1(1−e^−θi)11_{τi≤t}=P

i≥1γi11_{τi≤t}= M_t^I+A^I_t. We consider the marked point process (γi, τi)i≥1with jump measureµdefined as

µ(ω,[0, t], A) =X

i≥1

11_{{τi(ω)≤t}}11_{{γi(ω)∈A}},∀t≥0 and its compensatorν given by

ν(dt, dx) =X

i≥0

11_{{τi<t≤τi+1}}P(τi+1∈dt, γi+1∈dx|Fτ_i) P(τ_i+1> t|F_τi) (see Last and Brandt [20, Section 1.10]). Then

I_t= Z t

0

Z

R+

xµ(ds, dx) =M_t^I +A^I_t, whereM^Iis the martingaleR·

0

R

R+x(µ(ds, dx)−ν(ds, dx)) andA^I =R· 0

R

R+xν(ds, dx). Furthermore, dA^p_t =e^−Kt−(dK_t^c+dA^I_t).

As a particular case of marked point process, we consider the case whereK is a point process, i.e., K_t =P

i≥111_{τi≤t} for an increasing sequence of (τ_i)_i≥1 and F is the natural filtration of K.

One obtainsdA^p_t =e^Kt−dA^I_t whereI= (1−e⁻¹)K so thatA^I = (1−e⁻¹)P

i≥1Jⁱ, where, for any i, the processJⁱ is theF-compensator ofAⁱ= 11_{τi≤·}.

In the case where all theJⁱare continuous, the processZe^Λis a martingale, where Λ_t=Rt 0

dA^p_s Z_s− =A^I_t. Hence,e^{−K+(1−e−1)Pi≥1Ji} is a martingale.

One can recover the result of Giesecke [10, Pro. 3.1] that, setting ψ(u) = 1−e^−u, the process e^{−uK+ψ(u)Pi≥1Ji} is a martingale by considering the increasing processKb =uK.

Note that if all the (τi)_i≥1 are predictable,K is predictable, henceP

i≥1Jⁱ=K. Furthermore, as noticed in Subsection 3.2.5,dA^p=dI =−dZ. One has alsoA^I =I, so that Λ = (1−e⁻¹)K.

3.3.5 Shot noise

LetFbe a given filtration, (τi)i≥1 be a strictly increasing sequence ofF-stopping times withτ1>0, and (γ_i)_i≥1 a sequence of random variables withγ_i ∈ Fτi. We consider the random jump measure µof the marked point process (τ_i, γ_i)_i≥1, defined as µ(ω,[0, t], A) =P

i≥111_{{τi(ω)≤t}}11_{{γi(ω)∈A}} for A∈ B(R) andν its compensator. We denote byµethe compensated random measureµe=µ−ν. We define an increasing continuous on right processK= (K_t)_t≥0 (called a shot-noise process, see [26]) by

Kt=X

i≥1

11_{τi≤t}G(t−τi, γi) = Z t

0

Z

R

G(t−s, x)µ(ds, dx),∀t≥0, (3.9) where Gis a function R+×R →R+. Note that ∆Kτi = G(0, γi) and K_t^c =P

i≥111_{τi≤t}[G(t− τ_i, γ_i)−G(0, γ_i)],∀t≥0. We assume that

G(t, x) =G(0, x) + Z t

0

g(s, x)ds, ∀t≥0, x∈R, (3.10)

wheregis a non-negative Borel function onR+×R, so thatGis increasing with respect to its first variable. We also assume that

Z T

0

Z

R

g²(s, x)ν(ds, dx)<∞,∀T∞ (3.11) and that there exists a non negative functionϕonR+×Rsuch that

|g(s, x)| ≤ϕ(x),∀(s, x), with Z T

0

Z

R

ϕ(x)ν(ds, dx)<∞,∀T <∞. (3.12) Comment 3.12 In the case where G does not depend of t, the shot-noise process is a marked point process.

In the next lemma, we give a decomposition of the submartingale K that will be used later. Our result is similar to the one of the proof of Lemma 2 in [26]⁵.

Lemma 3.13 The shot-noise process K admits the Doob-Meyer decomposition K = M^K +A^K where theF-predictable increasing part A^K is

A^K_t = Z t

u=0

Z u

s=0

Z

R

g(u−s, x)µ(ds, dx) du+

Z t

s=0

Z

R

G(0, x)ν(ds, dx),∀t≥0 and theF-martingale partM^K has the following form :

M_t^K= Z t

0

Z

R

G(0, x)˜µ(ds, dx),∀t≥0.

Proof:We extendGtoR×RsettingG(u, x) =G(0, x) foru <0. For anya∈R⁺, we define Y_t(a) =

Z t

0

Z

R

G(a−s, x)˜µ(ds, dx),∀t≥0, one can write

K_t=Y_t(t) + Z t

0

Z

R

G(t−s, x)ν(ds, dx).

We apply the Itˆo-Ventzell formula as developed in [23, Theorem 3.1] to the process Y_t(a) with parametera, where a will be replaced by t (note that in our setting, since we integrate only de- terministic functions w.r.t. the compensated random measure, forward integrals in [23] are usual stochastic integrals). The first derivative ofYt(a) with respect to the parameterais (see [22]) , due to condition (3.12),

Y_t⁰(a) = Z t

0

Z

R

g(a−s, x)˜µ(ds, dx).

Using Theorem 3.1 of [23], it follows that, Y_t(t) =

Z t

u=0

Y_u⁰(u)du+ Z t

s=0

Z

R

G(0, x)˜µ(ds, dx),∀t≥0.

Note that, being continuous, the processR·

u=0Y_u⁰(u)du is predictable. Moreover, it is with bounded variation. Hence

Kt = Z t

u=0

Z u

s=0

Z

R

g(u−s, x)˜µ(ds, dx) du+

Z t

0

Z

R

G(0, x)˜µ(ds, dx) + Z t

0

Z

R

G(t−s, x)ν(ds, dx)

= M_t^K+ Z t

0

Z

R

G(t−s, x)ν(ds, dx) + Z t

u=0

Z u

s=0

Z

R

g(u−s, x)˜µ(ds, dx)

du,∀t≥0.

5There is a missprint in the formula given in the proof of this lemma.