2) The process Λ is stopped at time τ, i.e. Λt = Λt∧τ.
To see that, we observe that the process (11{τ≤t}, t ≥0) is stopped at τ, and that (11{τ≤t} −Λt∧τ, t≥0) is a martingale. Then both predictable increasing processes (Λt, t≥0) and (Λt∧τ, t≥0) areundistinguishable.
3) The process Λ is continuous if and only ifτ is a totally inaccessibleG-stopping time ([23] p.151), that is, for anyG-predictable stopping time σ, P(τ =σ) = 0.
The jumps of Λ occur at predictable stopping times u such that ∆Λu ≤1. More-over, if ∆Λu = 1, thenu=τ,a.s..
4) TheG-survival processS, defined bySt = 11{τ >t} = 1−11{τ≤t}, is a right-continuous supermartingale, which satisfies the following equation
dSt =St−(−dNt−dΛt) since St−= 11{τ≥t}. This equation will be discussed later.
1.2 Classical frameworks with closed formulae for the com-pensator processes
One important objective now is to calculate the compensator process Λ ofτ. We now present two examples where this computation is done in an explicit way.
1.2.1 The smallest filtration generated by τ
In this example, the filtration is generated by the process associated with random vari-able 0< τ <∞. This is a classical but important case which has been largely studied in the single credit modelling (e.g. Elliott, Jeanblanc and Yor [31]) whereτ represents the default time of one credit.
Let D = (Dt)t≥0 be the usual augmentation ([22] p.183) of the right-continuous fil-tration generated by the process (11{τ≤t}; t ≥ 0). Then D is the smallest filtration satisfying the usual conditions such thatτ is a D-stopping time. Any random variable X is Dt-measurable if and only if
X =x11e {τ >t}+x(τ)11{τ≤t} (1.1) where xe is a constant and x is some Borel function. In particular, the restriction of Dt on {τ > t} is trivial. Predictable processes, stopped at time τ are deterministic function oft∧τ.
To explicitely calculate ΛD, theD-compensator ofτ, we use the cumulative distribution functionF of τ, i.e. F(t) =P(τ ≤t) and the survival functionG(t) =P(τ > t).
The following property has been discussed in [31] and [55]:
Proposition 1.2.1 Let r be the first time such that G(r) = 0 (orF(r) = 1). For any sufficiently small positive real number , we have on [0, τ ∧(r−)],
dΛeDs = dF(s)
1−F(s−) =−dG(s) G(s−).
Proof. By definition of the compensator process, for any bounded Borel functionh, E
h
h(τ)11[0,τ∧(r−)](τ)i
=E h Z
[0,r−]
h(s)11{τ≥s}dΛDsi
(1.2)
= Z
[0,r−]
h(s)dF(s) = Z
[0,r−]
h(s)G(s−)dF(s)
G(s−) =Eh Z
[0,r−]
h(s)11[0,τ](s)dF(s) G(s−)
i,
The last equality is becauseG(s−) =P(τ ≥s). 2
1.2.2 Conditional independance and (H)-hypothesis An example of stopping time
An important example of default time is given below, which has been discussed by many authors (see Lando [59], Sch¨onbucher and Schubert [74] or Bielecki and Rutkowski [9]
for example). The financial interpretation of this model is from the structural approach, at the same time, by introducing a barrier which is independent with the filtration gen-erated by the background process, the intensity of the default time can be calculated.
So the two credit modelling approaches are related in this model. Moreover, it has become a standard construction of the default time when given an F-adapted process whereF= (Ft)t≥0 is an arbitrary filtration on the probability space (Ω,G,P).
Let (Ω,G,P) be a probability space and F be a filtration of G. Let Φ be an F -adapted, continuous, increasing process with Φ0 = 0 and Φ∞ := lim
t→+∞Φt = +∞. Letξ be a G-measurable random variable following exponential law with parameter 1 which is independent ofF∞. We define the random timeτ by
τ = inf{t≥0 : Φt≥ξ}. Then we can rewrite τ as τ = inf
t ≥ 0 : e−Φt ≤ U where U = e−ξ is a uniform random variable on [0,1]. So, the conditional distribution ofτ given F∞ is given by
P(τ > t|F∞) =P(Φt≤ξ|F∞) =e−Φt =P(τ > t|Ft) =:Gt (1.3) Let us introduce the new filtration G = (Gt)t≥0 as the minimal extension of F for which τ is a stopping time, that is Gt = Ft ∨ Dt, where Dt = σ(11{τ≤s}, s ≤ t). In particular, any Gt-measurable random variable coincides with an Ft-measurable r.v.
on{τ > t}. Based on this observation, we obtain the following characterization of the G-compensator process ofτ. Note that Φ is not necessarily continuous here.
Corollary 1.2.2 Let G = F∨D be the filtration previously defined. Let us assume that the conditional distribution of τ given F∞ is given by P(τ > t|F∞) =e−Φt where Φ is anF-adapted increasing process. Then, the G-compensator of τ is the process
ΛGs = -measurable random variable, the G-martingale property of Nt = 11{τ≤t} −ΛGt may be expressed as, for anyAFt ∈ Ft, Using the conditional distribution of τ given F∞, we have
Eh
In order to reintroduce the indicator function 11{τ≥t} whoseF∞conditional expectation ise−Φt−, we rewrite the right-hand side of the above equality as
E
By uniqueness of the dual predictable decomposition, we know that the corollary holds.
2
The processGdefined byGt=P(τ > t|Ft) is called theF-survival process. In this framework, Gis a decreasing F-adapted process. However, this is not true in general, as we will see in the following section. We note that this example is from the “filtration expansion” point of view, that is, the filtrationGis set to be the “minimal expansion”
ofF as been discussed in Mansuy and Yor [62].
About the (H)-Hypothesis.
We now introduce the (H)-hypothesis first introduced in filtering theory by many authors, see Br´emaud and Yor [12] for instance.
Hypothesis 1.2.3 We say that the (H)-hypothesis holds for (F,G), or equivalently, we say that a sub-filtrationFof Ghas themartingale invariant property with respect to the filtration Gor that F is immersed in G, if any F square-integrable martingale is a G-martingale.
For the credit modelling purpose, it has been studied by Kusuoka [58] and Jeanblanc and Rutkowski [55]. Note that the (H)-hypothesis is often supposed in the credit mod-elling, since it is equivalent to conditional independence between theσ-algebrasGt and F∞ givenFt (see [22]) as in the above example.
We give some equivalent forms below (cf. [22]). The (H)-hypothesis for (F,G) is equivalent to any of the following conditions:
(H1) for anyt≥0 and any boundedGt-measurable r.v. Y we haveE[Y|F∞] =E[Y|Ft];
(H2) for any t ≥ 0, any bounded Gt-measurable r.v. Y and any bounded F∞ -measurable r.v. Z, we haveE[Y Z|Ft] =E[Y|Ft]E[Z|Ft].
Remark 1.2.4 The main disadvantage of the (H)-hypothesis is that it may fail to hold under a change of probability. Kusuoka [58] provided a counter example to show this property. One can also refer to Bielecki and Rutkowski [9] and Jeanblanc and Rutkowski [55] for a detailed review.