Hazard processes and martingale hazard processes
aa r X i v : . [ q -f i n . R M ] J u l HAZARD PROCESSES AND MARTINGALE HAZARDPROCESSES
DELIA COCULESCU AND ASHKAN NIKEGHBALI
Abstract.
In this paper, we provide a solution to two problems whichhave been open in default time modeling in credit risk. We first showthat if τ is an arbitrary random (default) time such that its Az´ema’ssupermartingale Z τt = P ( τ > t |F t ) is continuous, then τ avoids stoppingtimes. We then disprove a conjecture about the equality between thehazard process and the martingale hazard process, which first appearedin [14], and we show how it should be modified to become a theorem. Thepseudo-stopping times, introduced in [21], appear as the most generalclass of random times for which these two processes are equal. We alsoshow that these two processes always differ when τ is an honest time. Introduction
Random times which are not stopping times have recently played anincreasing role in the modeling of default times in the hazard-rate ap-proach of the credit risk. Following [14], [9], [3], a hazard rate model maybe constructed in two steps. We begin with a filtered probability space(Ω , F , F = ( F t ) , P ) satisfying the usual assumptions. The default time τ isdefined as a random time (i.e., a nonnegative F -measurable random vari-able) which is not an F - stopping time). Then, a second filtration G = ( G t )plays an important role for pricing. This is obtained by progressively en-larging the filtration F with the random time τ : G is the smallest filtrationsatisfying the usual assumptions, containing the original filtration F , and forwhich τ is a stopping time, such as explained in [15], [17]. The filtration G isusually considered as the relevant filtration to consider in credit risk models:it represents the information available on the market. The enlargement offiltration provides a simple formula to compute the G -predictable compen-sator of the process τ ≤ t , which is a fundamental process in the modelingof default times. Note that an alternative and more direct hazard-rate ap-proach, which historically appeared first, consists in introducing one singleglobal filtration G from the start, where the default time is a totally inacces-sible stopping time with a given intensity. Major papers using the intensitybased framework are [13], [12], [19], [20], [8]. Mathematics Subject Classification.
Key words and phrases.
Default modeling, credit risk models, random times, enlarge-ments of filtrations, hazard process, immersed filtrations, pseudo-stopping times, honesttimes.
Both hazard-rate approaches mentioned above, i.e., the direct approachor the one based on two different sets of filtrations, model the occurringof the default as a surprise for the market, that is, the default time is atotally inaccessible stopping time in the global market filtration G . Thetechnique of enlargements of filtrations appears to be a useful tool, since itallows to compute easily the price of a derivative, using the hazard process.It allows as well an explicit construction of the default compensator (insection 3, we shall give simple ”universal” formulae for the compensator ofpseudo-stopping times and honest times ). For instance, one can take intoaccount the link between the default-free and the defaultable assets, or theincomplete information about the firm fundamentals, and thus construct thecompensator in an endogenous manner ([7], [18], [9], [11], [4], [10]).We now briefly justify the use of non stopping times for default times(see [5] for a more detailed analysis, where no-arbitrage conditions are alsostudied).Defaultable claims are defined by their maturity date, say T, and theirpromised stream of cash flows through time. Typically these consist of apromised face value, to be paid at maturity and a stream to be paid duringthe lifetime of the contract. We may suppose that the promised claim isan F T -measurable random variable, denoted by P , since the intermediarypayments may be invested in the default-free money market account. Inaddition, there is a random time τ at which the default occurs, and when arecovery payment R = P is made, in replacement of the promised one. Thedefaultable payoff is of the form: X = P τ>T + R τ ≤ T . (1.1)When constructing a model for the pricing of defaultable claims issued bya particular firm, say XYZ, one can proceed in two steps. First, one needsto model the value of the promised claim P , as well as the recovery claim R at intermediary times 0 ≤ t ≤ T . For this, one can use the traditionaldefault-free evaluation technique. For instance, the promised claim can bea fixed amount of dollars or commodity. The question of the recovery, eventhough more complicated, depends on the value of the contract’s collateral(for instance a physical asset), which can be assumed to be default-free. Inthis case, default-free techniques may be applied. Another possibility is toestimate recovery rates from historical default data. Without regard of thetechnique chosen, we denote by F the information available to the modelerafter the first step, i.e, the estimation of the promised and recovery assets,as well as the other available market information. We exclude informationabout the assets issued by the firm XY Z , even if it is available, since thisshould be the output of our evolution procedure, rather than the input. Forinstance, we consider that the filtration F does not contain information aboutthe price of a defaultable bond issued by the firm XYZ, even though thisbond might be traded. Usually, this construction leads to the situation where τ is not an F -stopping time. For instance, in the classical Cox framework, AZARD PROCESSES AND MARTINGALE HAZARD PROCESSES 3 the default time is defined as: τ := inf { t | Λ t > Θ } , where Λ is F predictable and increasing, and Θ is an exponential randomvariable independent from F . This situation is also common in default mod-els with incomplete information.In a second step, we define the global filtration G (i.e., the one to use forpricing claims of the type (1.1)) in such a way that τ becomes a stoppingtime. We are thus in the progressive enlargements of filtrations setting.When the random time is not a stopping time, several quantities play animportant role in the analysis of the model. The most fundamental objectattached to an arbitrary random time τ is certainly the supermartingale Z τt = P ( τ > t |F t ), chosen to be c`adl`ag, called the Az´ema’s supermartingaleassociated with τ ([1]). In the credit risk literature, very often the randomtime τ is given with extra regularity assumptions, such as continuity ormonotonicity of Z τt . However, these assumptions were not translated intoproperties of the random time τ . We shall try to clarify the link betweenthe assumptions about the process Z τt and the properties of the default time τ , since it is crucial for the modeler to select the properties of the randomtime which appear to be the most sensible.Two more processes, closely related to the Az´ema supermartingale Z τ and the G predictable compensator of τ ≤ t , are often used in the evaluationof defaultable claims: the hazard process and the martingale hazard process,which we now define. Definition 1.1. (1) Let τ be a random time such that Z τt >
0, for all t ≥ τ is not an F -stopping time). The nonnegativestochastic process (Γ t ) t ≥ defined by:Γ t = − ln Z τt , is called the hazard process .(2) Let D t = τ ≤ t . An F -predictable right-continuous increasing processΛ is called an F - martingale hazard process of the random time τ ifthe process f M t = D t − Λ t ∧ τ is a G martingale.We see that the martingale hazard process is only defined up to time τ and that the stopped martingale hazard process is the G -predictable com-pensator of the process D . This has two implications. First, several martin-gale hazard processes might exist for a default time, even if the predictablecompensator is unique. Secondly, this representation allows the martingalehazard process to be F -adapted as stated in the definition even if, obviously,the compensator is only G -adapted. In the next section we will characterizethe situation where the martingale hazard process is unique.Another important problem is to know under which conditions the hazardprocess and the martingale hazard processes coincide: this was object of aconjecture made in [14]: DELIA COCULESCU AND ASHKAN NIKEGHBALI
Conjecture:
Suppose that the process Z τt is decreasing. If Λ is continuous,then Λ = Γ.We shall show that the problem was not well posed and we shall seehow it should be phrased in order to have the equality between the hazardprocess and the martingale hazard process under some general conditions.More generally, the aim of this paper is to show that the general theory ofstochastic processes provides a natural framework to pose and to study themodeling of default times, and that it helps solve in a simple way some ofthe problems raised there.The paper is organized as follows:In section 2, we recall some basic facts from the general theory of stochasticprocesses that will be relevant for this paper.In section 3, we show that if Z τt is continuous, then τ avoids stopping times.We also see under which conditions the martingale hazard process and thehazard process coincide: the pseudo-stopping times, introduced in [21], ap-pear there as the most general class of random times for which these twoprocesses are equal. Moreover, we prove that for honest times, which formanother remarkable class of random times, the hazard process and the mar-tingale hazard process always differ. Acknowledgments . We wish to thank Monique Jeanblanc for very help-ful conversations and comments that improved the first drafts of this paper.2.
Basic facts
Throughout this paper, we assume we are given a filtered probabilityspace (Ω , F , F , P ) satisfying the usual assumptions. Definition 2.1.
A random time τ is a nonnegative random variable τ :(Ω , F ) → [0 , ∞ ].When dealing with arbitrary random times, one often works under thefollowing conditions: • Assumption ( C ): all ( F t )-martingales are continuous (e.g: the Brow-nian filtration). • Assumption ( A ): the random time τ avoids every ( F t )-stopping time T , i.e. P [ ρ = T ] = 0.When we refer to assumptions ( CA ), this will mean that both the conditions( C ) and ( A ) hold.We also recall the definition of the Az´ema’s supermartingale as well assome important processes related to it: • the ( F t ) supermartingale Z τt = P [ τ > t | F t ] (2.1)chosen to be c`adl`ag, associated to τ by Az´ema ([1]); AZARD PROCESSES AND MARTINGALE HAZARD PROCESSES 5 • the ( F t ) dual optional and predictable projections of the process1 { τ ≤ t } , denoted respectively by A τt and a τt ; • the c`adl`ag martingale µ τt = E [ A τ ∞ | F t ] = A τt + Z τt . We also consider the Doob-Meyer decomposition of (2.1): Z τt = m τt − a τt . We note that the supermartingale ( Z τt ) is the optional projection of [0 ,τ [ .Let us also define very rigourously the progressively enlarged filtration G .We enlarge the initial filtration ( F t ) with the process ( τ ∧ t ) t ≥ , so thatthe new enlarged filtration ( G t ) t ≥ is the smallest filtration (satisfying theusual assumptions) containing ( F t ) and making τ a stopping time, that is G t = K t + , where K t = F t _ σ ( τ ∧ t ) . A very common situation encountered in default times modeling is the ( H )hypothesis framework: every F -local martingale is also a G -local martingale.For instance, this property is always satisfied when the default time is a Coxtime.However, it is possible to introduce more general random times. We recallthe definition of pseudo-stopping times which extend the ( H ) hypothesisframework and which will play an important role in the study of hazardprocesses and martingale hazard processes. Definition 2.2 ([21]) . We say that τ is a ( F t ) pseudo-stopping time if forevery ( F t )-martingale ( M t ) in H , we have E M τ = E M . (2.2) Remark.
It is equivalent to assume that (2.2) holds for bounded martingales,since these are dense in H . It can also be proved that then (2.2) also holdsfor all uniformly integrable martingales (see [21]).The following characterization of pseudo-stopping times will be often usedin the sequel: Theorem 2.3 ([21]) . The following four properties are equivalent:(1) τ is a ( F t ) pseudo-stopping time, i.e (2.2) is satisfied;(2) µ τt ≡ , a.s (3) A τ ∞ ≡ , a.s (4) every ( F t ) local martingale ( M t ) satisfies ( M t ∧ τ ) t ≥ is a local ( G t ) martingale. If, furthermore, all ( F t ) martingales are continuous, then each ofthe preceding properties is equivalent to DELIA COCULESCU AND ASHKAN NIKEGHBALI (5) ( Z τt ) t ≥ is a decreasing ( F t ) predictable processRemark. Of course, every stopping time is a pseudo-stopping time by thethe optional sampling theorem. But there are many examples or families ofpseudo-stopping which are not stopping times (see [21]). Similarly, all ran-dom times which ensure that the ( H ) hypothesis holds are pseudo-stoppingtimes. But there are pseudo-stopping times for which the ( H ) hypothesisdoes not hold (in particular those which are F ∞ -measurable; see [21] forconstruction and further characterizations of pseudo-stopping times).The following classical lemma will be very helpful: it indicates the prop-erties of the above processes under the assumptions ( A ) or ( C ) (for moredetails or references, see [6] or [23]). Lemma 2.4.
Under condition ( A ) , A τt = a τt is continuous.Under condition ( C ) , A τ is predictable (recall that under ( C ) the pre-dictable and optional sigma fields are equal) and consequently A τ = a τ .Under conditions ( CA ) , Z τ is continuous. We give a first application of theorem 2.3 and lemma 2.4 to illustrate howthe general theory of stochastic processes shed a new light on default timemodeling. It is very often assumed in the literature on default times that τ is a random time whose associated Az´ema supermartingale is continuousand decreasing. Proposition 2.5.
Let τ be a random time that avoids stopping times. Then ( Z τt ) is continuous and decreasing if and only if τ is a pseudo-stopping time.Proof. If τ is a pseudo-stopping, then from theorem 2.3, Z τt = 1 − A τt . If τ avoids stopping times, then it follows from lemma 2.4 that A τ is continuousand consequently Z τ is continuous.Conversely, if Z τ is continuous, and if τ avoids stopping times, then fromthe uniqueness of the Doob-Meyer decomposition, Z τt = 1 − a τt . But since τ avoids stopping times, we have a τt = A τt from lemma 2.4 and hence Z τt =1 − A τt . Consequently, from theorem 2.3, τ is a pseudo-stopping time. (cid:3) Remark.
We shall see a slight reinforcement of this theorem in the next sec-tion: indeed, we shall prove that if Z τ is continuous, then τ avoids stoppingtimes. 3. Main theorems
First, we clarify a situation concerning the hazard process. Indeed, inthe credit risk literature, the G martingale L t ≡ τ>t e Γ t plays an importantrole (see [14] or [3]). But from definition 1.1, the hazard process is definedonly when Z τt > t ≥
0. We wish to show that nevertheless, themartingale ( L t ) is always well defined. For this, it is enough to show that on AZARD PROCESSES AND MARTINGALE HAZARD PROCESSES 7 the set { τ > t } , Γ t = − log Z τt is always well defined. This is the case thanksto the following result from the general theory of stochastic processes: Proposition 3.1 ([15], [6], p.134) . Let τ be an arbitrary random time. Thesets { Z τ = 0 } and (cid:8) Z τ − = 0 (cid:9) are both disjoint from the stochastic interval [0 , τ [ , and have the same lower bound T , which is the smallest stopping timelarger than τ . The next proposition gives general conditions under which Γ is continuous,which is generally taken as an assumption in the literature on default times:indeed, when computing prices or hedging, one often has to integrate withrespect to Γ (see [14], [9] or [3]).
Proposition 3.2.
Let τ be a random time.(i) Then under ( CA ) , (Γ t ) is continuous and Γ = 0 .(ii) If τ is a pseudo-stopping time and if ( A ) holds, then (Γ t ) is a contin-uous increasing process, with Γ = 0 .Proof. This is a consequence of Lemma 2.4 and theorem 2.3. (cid:3)
Now, what can one say about the random time τ if one assumes that itsassociated Az´ema’s supermartingale is continuous? It seems to have beenan open question in the literature on credit risk modeling for a few yearsnow. The next proposition answers this question: Proposition 3.3.
Let τ be a finite random time such that its associatedAz´ema’s supermartingale Z τt is continuous. Then τ avoids stopping times.Proof. It is known that Z τt = o ( [0 ,τ ) ) , that is Z τt is the optional projection of the stochastic interval [0 , τ ). Now,following Jeulin-Yor [17], define e Z t as the optional projection of the stochas-tic interval [0 , τ ]: e Z t = o ( [0 ,τ ] ) . It can be shown (see [17]) that e Z + = Z τ and e Z − = Z τ − . Since Z τ is continuous, we have e Z + = e Z − = Z τ , and consequently, for any stopping time T : E [ τ ≥ T ] − E [ τ>T ] = 0 , which means that P [ τ = T ] = 0 for all stopping times T . (cid:3) As an application, we can state the following enforcement of proposition2.5:
DELIA COCULESCU AND ASHKAN NIKEGHBALI
Corollary 3.4.
Let τ be a random time. Then ( Z τt ) is a continuous anddecreasing process if and only if τ is a pseudo-stopping time that avoidsstopping times. Now we recall a theorem which is useful in constructing the martingalehazard process.
Theorem 3.5 ([16]) . Let H be a bounded ( G t ) predictable process. Then H τ τ ≤ t − Z t ∧ τ H s Z τs − da τs is a ( G t ) martingale. Corollary 3.6.
Let τ be a pseudo-stopping time that avoids F stoppingtimes. Then the G dual predictable projection of τ ≤ t is log (cid:16) Z τt ∧ τ (cid:17) .Let g be an honest time (that means that g is the end of an F optionalset) that avoids F stopping times. Then the G dual predictable projection of g ≤ t is A gt .Proof. Let τ be a random time; taking H ≡
1, in Theorem 3.5 we find that R t ∧ τ Z τs − dA τs is the G dual predictable projection of τ ≤ t .When τ is a pseudo-stopping time that avoids F stopping times, wehave from Theorem 2.3 that the G dual predictable projection of τ ≤ t is − log ( Z τt ∧ τ ) since in this case A τt = 1 − Z τt is continuous.The second fact is an easy consequence of the well known fact that themeasure dA gt is carried by { t : Z gt = 1 } (see [1]). (cid:3) As a consequence, we have the following characterization of the martingalehazard process:
Proposition 3.7.
Let τ be a random time. Suppose that Z τt > , ∀ t . Then,there exists a unique martingale hazard process Λ t , given by: Λ t = Z t da τu Z u − , where recall that a τt is the dual predictable projection of τ ≤ t .Proof. We suppose there exist two different martingale hazard processes Λ and Λ and denote T ( ω ) = inf (cid:8) t : Λ t ( ω ) = Λ t ( ω ) (cid:9) .T is an ( F t )-stopping time hence a G stopping time. Due to the uniquenessof the predictable compensator we must have for all t ≥ t ∧ τ = Λ t ∧ τ a.s. Hence,
T > τ a.s. and hence Z τt = 0, ∀ t ≥ T . By assumption, this isimpossible, hence Λ = Λ a.s. (cid:3) AZARD PROCESSES AND MARTINGALE HAZARD PROCESSES 9
It is conjectured in [14] that if τ is any random time (possibly a stoppingtime) such that P ( τ ≤ t |F t ) is an increasing process, and if the martingalehazard process Λ is continuous, then Λ = Γ, where Γ is the hazard process.We now provide a counterexample to this conjecture. Indeed, let τ be atotally inaccessible stopping time of the filtration F . Then of course P ( τ ≤ t |F t ) = τ ≤ t is an increasing process. Let now ( A t ) be the predictablecompensator of τ ≤ t . It is well known (see [1] or [15] for example) that A t is a continuous process (that satisfies A t = A t ∧ τ ) and hence Λ t = A t iscontinuous. But clearly Γ t = Λ t .We propose the following theorem instead of the above conjecture (recallthat the fact that Az´ema’s supermartingale is continuous and decreasingmeans that τ is a pseudo-stopping time): Theorem 3.8.
Let τ be a pseudo-stopping time. Assume further that Z τt > for all t .(i) Under (A) , Γ is continuous and Γ t = Λ t = − ln Z t .(ii) Under (C) , if Λ is continuous, then Γ t = Λ t = − ln Z t .Proof. (i) follows from lemma 2.4, Theorem 2.3 and proposition 3.7.(ii) Assume (C) holds. Since Λ is assumed to be continuous, it followsfrom proposition 3.7 (2) that a τt is continuous. Hence τ avoids all pre-dictable stopping times. But under (C) , all stopping times are predictable.Consequently τ avoids all stopping times and we apply part (i). (cid:3) It has been proved in [14] that in general, even under the assumptions (CA) , the hazard process and the martingale hazard process may differ.The example they used was g ≡ sup { t ≤ W t = 0 } , where W denotesas usual the standard Brownian Motion. This time is a typical example ofan honest time (i.e. the end of an optional set). We shall now show thatthis result actually holds for any honest time g and compute explicitly thedifference in this case. We shall need for this the following characterisationof honest times given in [22]: Theorem 3.9 ([22]) . Let g be an honest time. Then, under the conditions (CA) , there exists a unique continuous and nonnegative local martingale ( N t ) t ≥ , with N = 1 and lim t →∞ N t = 0 , such that: Z gt = P ( g > t | F t ) = N t Σ t , where Σ t = sup s ≤ t N s . The honest time g is also given by: g = sup { t ≥ N t = Σ ∞ } = sup { t ≥ t − N t = 0 } . (3.1) Proposition 3.10.
Let g be an honest time. Under (CA) , assume that P ( g > t |F t ) > . Then there exists a unique strictly positive and continuouslocal martingale N , with N = 1 and lim t →∞ N t = 0 , such that: Γ t = ln Σ t − ln N t whilst Λ t = ln Σ t , where Σ t = sup s ≤ t N s . Consequently, Λ t − Γ t = ln N t , and Γ = Λ .Proof. From theorem 3.9, there exists a unique strictly positive continuouslocal martingale N , such that N = 1 and lim t →∞ N t = 0, such that: Z gt = P ( g > t | F t ) = N t Σ t . Now an application of Itˆo’s formula yields: P ( g > t | F t ) = 1 + Z t d N s Σ s − Z t N s Σ s dΣ s . But on the support of (dΣ s ), we have Σ t = N t and hence: P ( g > t | F t ) = 1 + Z t d N s Σ s − ln Σ t . From the uniqueness of the Doob-Meyer decomposition, we deduce that thedual predictable projection of g ≤ t is ln Σ t . Now,applying proposition 3.7,we have: Λ t = Z t d(ln Σ s ) P ( g > s | F s ) = Z t Σ s Σ s N s dΣ s = ln Σ t , where we have again used the fact that the support of (dΣ s ), we have Σ t = N t . The result of the proposition now follows easily. (cid:3) We shall now outline a nontrivial consequence of Theorem 3.9 here. In[2], the authors are interested in giving explicit examples of dual predictableprojections of processes of the form L ≤ t , where L is an honest time. Indeed,these dual projections are natural examples of increasing injective processes(see [2] for more details and references). With Theorem 3.9, we have acomplete characterization of such projections, which are also very importantin credit risk modeling: Corollary 3.11.
Assume the assumption (C) holds, and let ( C t ) be anincreasing process. Then C is the dual predictable projection of g ≤ t , forsome honest time g that avoids stopping times, if and only if there exists acontinuous local martingale N t , with N = 1 and lim t →∞ N t = 0 , such that C t = ln Σ t . Proof.
This is a consequence of theorem 3.9 and the fact, established in theproof of proposition 3.10, that the dual predictable projection of g ≤ t isln Σ t . (cid:3) AZARD PROCESSES AND MARTINGALE HAZARD PROCESSES 11
References [1]
J. Az´ema : Quelques applications de la th´eorie g´en´erale des processus I , Invent. Math. (1972) 293–336.[2] J. Az´ema, T. Jeulin, F. Knight, M. Yor : Quelques calculs de compensateursimpliquant l’inj´ectivit´e de certains processus croissants , S´em.Proba. XXXII, LectureNotes in Mathematics , (1998), 316–327.[3]
C. Blanchet-Scaillet, M. Jeanblanc
Hazard rate for credit risk and hedgingdefaultable contingent claims , Finance Stochast., , 145–159 (2004).[4] D. Coculescu, H. Geman and M. Jeanblanc
Valuation of default-sensitive claimsunder imperfect information , Finance Stochast., (2), 195-218 (2008).[5] D. Coculescu, M. Jeanblanc and A. Nikeghbali
No arbitrage conditions indefault models , in preparation.[6]
C. Dellacherie, B. Maisonneuve, P.A. Meyer : Probabilit´es et potentiel ,Chapitres XVII-XXIV: Processus de Markov (fin), Compl´ements de calcul stochas-tique, Hermann (1992).[7]
Duffie, D. and D. Lando (2001): Term Structures of Credit Spreads with Incom-plete Accounting Information,
Econometrica , (3), 633-664.[8] Duffie, D. and K. Singleton (1999): Modelling Term Structures of DefaultableBonds,
The Review of Financial Studies , (4), 687-720.[9] R.J. Elliott, M. Jeanblanc, M. Yor : On models of default risk , Math. Finance, , 179–196 (2000).[10] R. Frey and T. Schmidt : ”Pricing Corporate Securities under Noisy Asset Infor-mation”, 2007. Forthcoming in Mathematical Finance.[11]
Guo, X., R.A. Jarrow and Y. Zeng (2005) : Credit risk models with incompleteinformation, Mathematics of Operations Research, To appear..[12]
Jarrow, R.A. Lando, D. and S.M. Turnbull (1997): A Markov model for theterm sructure of credit risk spreads,
Review of Financial Studies , (2), 481-523.[13] Jarrow, R.A. and S.M. Turnbull (1995): Pricing Options on Derivative SecuritiesSubject to Credit Risk,
Journal of Finance , (1), 53-85.[14] M. Jeanblanc, M. Rutkowski : Modeling default risk: Mathematical tools , FixedIncome and Credit risk modeling and Management, New York University, SternSchool of business, Statistics and Operations Research Department, Workshop (2000).[15]
T. Jeulin : Semi-martingales et grossissements d’une filtration , Lecture Notes inMathematics , Springer (1980).[16]
T. Jeulin, M. Yor : Grossissement d’une filtration et semimartingales: formulesexplicites , S´em.Proba. XII, Lecture Notes in Mathematics , (1978), 78–97.[17]
T. Jeulin, M. Yor (eds) : Grossissements de filtrations: exemples et applications ,Lecture Notes in Mathematics , Springer (1985).[18]
Kusuoka, S. (1999): A Remark on Default Risk Models,
Advances on MathematicalEconomics , , 69-82.[19] Lando, D. (1998): On Cox Processes and Credit Risky Securities,
Review of Deriva-tives Research , (2/3), 99-120.[20] Madan, D. and H. Unal (1998): Pricing the risks of default,
Review of DerivativesResearch , (2/3), 121-160.[21] A. Nikeghbali, M. Yor : A definition and some characteristic properties of pseudo-stopping times , Ann. Prob. , (2005) 1804–1824.[22] A. Nikeghbali, M. Yor : Doob’s maximal identity, multiplicative decompositions andenlargements of filtrations , Illinois Journal of Mathematics, (4) 791–814 (2006).[23] A. Nikeghbali : An essay on the general theory of stochastic processes , Prob. Surveys, , (2006) 345–412. ETHZ, Departement Mathematik, R¨amistrasse 101, Z¨urich 8092, Switzer-land.
E-mail address : [email protected] Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse 190,CH-8057 Z¨urich, Switzerland
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