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## Intensity process and compensator: A new filtration expansion approach and the Jeulin–Yor theorem. The Annals of Applied Probability (2007)

Citations: | 16 - 2 self |

### Citations

355 | On Cox Processes and Credit Risky Securities - Lando - 1998 |

316 | Term structure of credit spreads with incomplete accounting information.
- Duffie, Lando
- 2001
(Show Context)
Citation Context ...≥0, stopping time τ . The notion of an intensity process (λt)t≥0 of a stopping time is of essential interest in credit risk, especially in the information-based approach pioneered by Duffie and Lando =-=[14]-=-. First, if τ is the default time of a firm and is a stopping time relative to some filtration G = (Gt)t≥0, then under appropriate technical conditions (such as those in Aven [1]), λt is the instantan... |

308 |
Point Processes and Queues: Martingale Dynamics
- Bremaud
- 1981
(Show Context)
Citation Context ...ngale.) The intensity process (λt)t≥0 of τ is then defined as the Radon–Nikodym derivative (dAt/dt)t≥0, provided that A is a.s. absolutely continuous with respect to the Lebesgue measure. See Brémaud =-=[5]-=-, Chapter II, D7, T12 and T13. 1.1. Problems and previous work. Problem 1: Existence and characterization of (λt)t≥0. Given a stopping time τ with respect to a given filtration, its intensity (λt)t≥0 ... |

153 |
Semimartingale Theory and Stochastic Calculus,
- He, Wang, et al.
- 1992
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Citation Context ...s. of finite variation on (S,T ]. [(S,T ]:=∅ on {S ≥ T }.] In particular, Zt is of finite variation on (S,T ]. Another useful lemma can be proved by modifying the technique for jumping filtrations in =-=[20]-=-, pages 160–173, to local jumping filtrations. LEMMA 2.2. Let S and T be a pair of F-stopping times. If a filtration F jumps locally from S to T according to equation (4), then for any F-stopping time... |

98 |
Multivariate Point Processes: Predictable Projection, Radon–Nikodym Derivatives,
- Jacod
- 1975
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Citation Context ...xamples in Section 3. (c) In equation (5), the term PX (g(V1,z)∈dt−a) S |a=S,z=V2 PX (g(V1,z)≥t−a) can be re-written as S P(T ∈ dt|FS) P(T ≥ t|FS) . This F-intensity of T is a generalization of Jacod =-=[23]-=-, Proposition 3.1, under the local jumping filtration. 2.3. Proof of Theorem 1.3. First, recall that under the completed natural filtration (F X t )t≥0, any stopping time T is totally inaccessible if ... |

88 | Recursive valuation of defaultable securities and the timing of resolution of uncertainty. - Due, Schroder, et al. - 1996 |

75 |
On models of default risk.
- Elliott, Jeanblanc, et al.
- 2000
(Show Context)
Citation Context ...ed into computing the compensator of Zt = E{1{τ>t}|Ft} under the original filtration F,sinceZ is an F-supermartingale. Unfortunately, the latter problem is not necessarily easier than the former. And =-=[16]-=- considered the very special case when Z is continuous and decreasing, which is also known as satisfying the G-hypothesis. In this case, the F-compensator ofsINTENSITY PROCESS AND COMPENSATOR 123 Z is... |

58 |
On some relations between the harmonic measure and the Levy measure for a certain class of Markov processes.
- Ikeda, Watanabe
- 1962
(Show Context)
Citation Context ...rnal clock for X.IfX is a continuous-time, time-homogeneous Markov chain, K(x,dy) is its generator matrix and Ut = t.IfX is a Lévy process, K(x,dy)= ν(dy−x),where ν is the Lévy measure and Ut = t.(See=-=[21]-=-, Section 2, [34]and[32], Chapter VI.)s126 X. GUO AND Y. ZENG Organization of the paper. Section 2 provides detailed proofs of the main results, as well as discussion and comparison with the existing ... |

57 |
Capacités et Processus Stochastiques
- Dellacherie
- 1972
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Citation Context ...he existence problem of intensity has been studied in some cases by identifying specific filtrations and stopping times and by explicit calculations. Fors122 X. GUO AND Y. ZENG instance, Dellacherie (=-=[11]-=-, Chapter V, T56) calculated the intensity of an arbitrary positive random variable τ under the natural filtration of 1{τ≤t}. Çetin, Jarrow, Protter and Yildirim [7] studied this problem with a filtra... |

57 |
Stochastic Integration and Differential
- Protter
- 2005
(Show Context)
Citation Context ...a semi-martingale under filtration expansion. However, in our paper the G-semi-martingale 1{τ≤t} is NOT adapted to F, hence, no longer a semi-martingale under F. (See Stricker’s theorem on page 53 of =-=[30]-=- for necessary and sufficient conditions for a semi-martingale to remain a semi-martingale under filtration shrinkage.) This leads to another interesting question: does expanding the original filtrati... |

51 | Modeling credit risk with partial information, Working paper, Cornell University, forthcoming, Annals of Applied Probability.
- Cetin, Jarrow, et al.
- 2003
(Show Context)
Citation Context ...ND Y. ZENG instance, Dellacherie ([11], Chapter V, T56) calculated the intensity of an arbitrary positive random variable τ under the natural filtration of 1{τ≤t}. Çetin, Jarrow, Protter and Yildirim =-=[7]-=- studied this problem with a filtration generated by the sign of a Brownian motion where the stopping time τ was chosen to be the first time a Brownian motion doubles in absolute magnitude after remai... |

51 |
Grossissement d’une filtration et semi-martingales: theoremes generaux. In:
- Yor
- 1978
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Citation Context ...nding a dominating process for the approximation. Problem 1B: Computing the compensator (At)t≥0 via the Jeulin–Yor theorem. Under a different progressive filtration expansion approach, Jeulin and Yor =-=[27]-=- suggested that calculating the compensator (and its intensity) of 1{τ≤t} under the enlarged filtration G can be translated into computing the compensator of Zt = E{1{τ>t}|Ft} under the original filtr... |

31 | Default and information,”
- Giesecke
- 2006
(Show Context)
Citation Context ...the first hitting times of a Brownian motion under the natural filtration are not. However, total inaccessibility is not sufficient for the existence of the intensity of τ ;seeDavis [10] and Giesecke =-=[17]-=-, Proposition 6.1, for such an example. As any stopping time can be decomposed into a totally inaccessible part and an accessible part, Zeng [36] recently showed that the totally inaccessible part can... |

30 | Additional logarithmic utility of an insider.
- Amendinger, Imkeller, et al.
- 1998
(Show Context)
Citation Context ...he analogous necessity result for the local jumping filtration. However, it remains an interesting mathematical problem to extend their sufficiency result to the local jumping filtration. Finally, in =-=[2, 3, 9, 22]-=-, the notion of “information drift” was introduced and analyzed by tools of Malliavin calculus for the additional information induced by the filtration expansion. Furthermore, this “information drift”... |

28 | The Shannon information of filtrations and the additional logarithmic utility of insiders.
- Ankirchner, Dereich, et al.
- 2006
(Show Context)
Citation Context ...he analogous necessity result for the local jumping filtration. However, it remains an interesting mathematical problem to extend their sufficiency result to the local jumping filtration. Finally, in =-=[2, 3, 9, 22]-=-, the notion of “information drift” was introduced and analyzed by tools of Malliavin calculus for the additional information induced by the filtration expansion. Furthermore, this “information drift”... |

26 |
A general framework for pricing credit risk.
- Belanger, Shreve, et al.
- 2004
(Show Context)
Citation Context ...iscount factor rt is replaced by rt + λt. (See also Duffie, Schroder and Skiadas [15] for the reduced-form pricing approach via (λt)t≥0, and Jeanblanc and Rutkowski [25] and Bélanger, Shreve and Wong =-=[6]-=- for generalizations.) Mathematically, the intensity process (λt)t≥0 of a stopping time τ is associated with the compensator A of τ with respect to an appropriate filtration. Let (�, F ,P) be a probab... |

26 |
Malliavin’s calculus in insider models: additional utility and free lunches.
- Imkeller
- 2003
(Show Context)
Citation Context ...he analogous necessity result for the local jumping filtration. However, it remains an interesting mathematical problem to extend their sufficiency result to the local jumping filtration. Finally, in =-=[2, 3, 9, 22]-=-, the notion of “information drift” was introduced and analyzed by tools of Malliavin calculus for the additional information induced by the filtration expansion. Furthermore, this “information drift”... |

23 |
A theorem for determining the compensator of a counting process,
- Aven
- 1985
(Show Context)
Citation Context ... by Duffie and Lando [14]. First, if τ is the default time of a firm and is a stopping time relative to some filtration G = (Gt)t≥0, then under appropriate technical conditions (such as those in Aven =-=[1]-=-), λt is the instantaneous likelihood of default at time t conditioned on Gt, the information at time t. Thatis, 1 λt = lim h↓0 h P(t<τ ≤ t + h|Gt) (1) a.s. Received February 2006; revised March 2007.... |

17 |
Additional utility of insiders with imperfect dynamical information
- Corcuela, Imkeller, et al.
- 2004
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Citation Context |

17 | Modelling of Default Risk: An Overview
- Jeanblanc, Rutkowski
- 2000
(Show Context)
Citation Context ...-free counterpart except that the discount factor rt is replaced by rt + λt. (See also Duffie, Schroder and Skiadas [15] for the reduced-form pricing approach via (λt)t≥0, and Jeanblanc and Rutkowski =-=[25]-=- and Bélanger, Shreve and Wong [6] for generalizations.) Mathematically, the intensity process (λt)t≥0 of a stopping time τ is associated with the compensator A of τ with respect to an appropriate fil... |

16 |
Systèmes de Lévy des processus de Markov
- Benveniste, Jacod
- 1973
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Citation Context ... By the definition of a Lévy system, it is easy to see for any positive function Y(t,ω,x) that is P ⊗ B(E)-measurable [P is the predictable σ -field and B(E) is the Borel σ -field on E], we have (see =-=[4]-=-) Ex � � Y(s,Xs) 0<s<∞ � �� ∞ � = Ex dUs 0 � � K(Xs,dy)Y(s,y) . In particular, by setting Y(s,y)= f(y)Hs1[0,τ�](s)1{ρ(y,Xs−)>ε},weget � Note �� ∞ Ex Ht d1{τ�ε≤t} 0 �� ∞ = Ex 0 � � Hs1[0,τ�](s)dUs K(Xs... |

11 |
On discontinuous additive functionals and Lévy measures of a Markov process
- Watanabe
- 1964
(Show Context)
Citation Context ...IfX is a continuous-time, time-homogeneous Markov chain, K(x,dy) is its generator matrix and Ut = t.IfX is a Lévy process, K(x,dy)= ν(dy−x),where ν is the Lévy measure and Ut = t.(See[21], Section 2, =-=[34]-=-and[32], Chapter VI.)s126 X. GUO AND Y. ZENG Organization of the paper. Section 2 provides detailed proofs of the main results, as well as discussion and comparison with the existing literature. Secti... |

10 | Modeling the Recovery Rate in a Reduced Form Model
- Guo, Jarrow, et al.
(Show Context)
Citation Context ...|y| ψ(η,t,y)= P = 1 − √ 2πs3 e−(y−ηs)2 /(2s) ds inf 0≤s≤t W(η) s >y 0 for y<0, and ψt is the derivative of ψ with respect to t. This is the same as the result in [18]. Example of Guo, Jarrow and Zeng =-=[19]-=-. Consider a jump diffusion process X with Xt = X0e (μ−1/2σ 2 )t+σWt � 0<s≤t,�ε(s)�=0 ξε(s). Here W and ε are the same as in the previous example. Assume further the generator matrix of ε is Q = (qij)... |

9 | Probabilities and
- Dellacherie, Meyer
- 1978
(Show Context)
Citation Context ...DIX A: PROOF OF THEOREM 1.1: THE EXTENDED JEULIN–YOR THEOREM Section A.1 provides the necessary terminology and background. For a more comprehensive presentation of related materials, please refer to =-=[12]-=-, Chapter VI or [20], Chapter V. A.1. Preliminaries. Suppose (�, F ,(Ft)t≥0,P) is a complete filtered probability space such that the filtration F = (Ft)t≥0 satisfies the usual hypotheses. The predict... |

8 |
Partial information and hazard process
- Jeanblanc, Valchev
- 2005
(Show Context)
Citation Context ...stence of Z ′ (t)] on stochastic intervals. As an illustration of this method, special cases are presented to reproduce the work in [16] when Z is continuous and decreasing, as well as the results in =-=[14, 26, 18, 19]-=-. Finally, for the very special case of the full filtration where filtration expansion is unnecessary, we provide explicit expressions for (λt)t≥0 in terms of Lévy systems for Hunt processes. Main res... |

6 |
Jumping Filtrations and Martingales with Finite Variation, Séminaire de Probabilités
- Jacod, Skorohod
- 1994
(Show Context)
Citation Context ...G. To overcome the technical difficulty without the G-hypothesis, the notion of local jumping filtrations is introduced, and an analogous characterization result for martingales of Jacod and Skorohod =-=[24]-=- under this filtration is established. These ensure the finite variation property of Zt = E{1{τ>t}|Ft} [hence, the existence of Z ′ (t)] on stochastic intervals. As an illustration of this method, spe... |

3 | Filtration shrinkage by level-crossings of a diffusion
- Sezer
(Show Context)
Citation Context ... a Brownian motion doubles in absolute magnitude after remaining below zero for a certain amount of time. This result is based on Azéma’s martingale and the excursion theory of Brownian motion. Sezer =-=[33]-=- considered a similar but more general problem with a filtration generated by the process (R(Xt))t≥0,where X is a one-dimensional diffusion process and R is a Borel function of the form � Ni=1 i1(xi−1... |

2 |
Credit risk with incomplete information. (Earlier title: “Information reduction in credit risk models
- Guo, Jarrow, et al.
- 2005
(Show Context)
Citation Context ...r approach is based on the Meyer’s Laplacian approximation for the compensator of a point process and a dominated convergence type of result by Aven [1]. Following this approach, Guo, Jarrow and Zeng =-=[18, 19]-=- studied and computed (λt)t≥0 for first passage times of general Markov processes, including diffusion processes with jumps, and Markov modulated processes. The filtration is of the delayed type, and ... |

1 |
A note on optimal stopping times with filtration expansion
- CHAKRABARTY, X
- 2007
(Show Context)
Citation Context ... the original filtration in a minimal way by the nonadapted point process 1{τ≤t} lead any added value/utility in some way? This is an interesting project that goes beyond the scope of this paper (see =-=[8]-=-). APPENDIX A: PROOF OF THEOREM 1.1: THE EXTENDED JEULIN–YOR THEOREM Section A.1 provides the necessary terminology and background. For a more comprehensive presentation of related materials, please r... |

1 |
Hazard rates in the credit grades model. Unpublished manuscript
- DAVIS
- 2002
(Show Context)
Citation Context ...accessible, while the first hitting times of a Brownian motion under the natural filtration are not. However, total inaccessibility is not sufficient for the existence of the intensity of τ ;seeDavis =-=[10]-=- and Giesecke [17], Proposition 6.1, for such an example. As any stopping time can be decomposed into a totally inaccessible part and an accessible part, Zeng [36] recently showed that the totally ina... |

1 |
Existence du processus croissant natural associé à un potentiel de la classe (D
- DOLÉANS
- 1968
(Show Context)
Citation Context ...tinuous, o X and p X are not necessarily so ([12], page 113, 50(d)). Another important type of projection processes is the dual projections of an increasing process. (See Dellacherie [11] and Doléans =-=[13]-=-.) The main perspective is that an increasing process induces a measure on the product σ -field B(R+)⊗F , and the measurability properties of the increasing process are characterized by those of the i... |

1 |
On compensators of stopping times. Ph.D. dissertation, Cornell Univ
- ZENG
- 2005
(Show Context)
Citation Context ...ence of the intensity of τ ;seeDavis [10] and Giesecke [17], Proposition 6.1, for such an example. As any stopping time can be decomposed into a totally inaccessible part and an accessible part, Zeng =-=[36]-=- recently showed that the totally inaccessible part can be further uniquely decomposed into an absolute continuous part (i.e., with an intensity) and a singular part. Still, the question of the existe... |

1 |
Itô Calculus. Cambridge Univ
- Rogers, Williams
- 2000
(Show Context)
Citation Context ...us-time, time-homogeneous Markov chain, K(x,dy) is its generator matrix and Ut = t. If X is a Lévy process, K(x,dy) = ν(dy − x), where ν is the Lévy measure and Ut = t. (See [21], Section 2, [34] and =-=[32]-=-, Chapter VI.) Organization of the paper. Section 2 provides detailed proofs of the main results, as well as discussion and comparison with the existing literature. Section 3 illustrates some old and ... |

1 |
On compensators of stopping times
- Zeng
- 2005
(Show Context)
Citation Context ...R 3 of the intensity of τ; see Davis [10] and Giesecke [17], Proposition 6.1, for such an example. As any stopping time can be decomposed into a totally inaccessible part and an accessible part, Zeng =-=[36]-=- recently showed that the totally inaccessible part can be further uniquely decomposed into an absolute continuous part (i.e., with an intensity) and a singular part. Still, the question of the existe... |