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Intensity process and compensator: A new filtration expansion approach and the Jeulin–Yor theorem. The Annals of Applied Probability
, 2007
"... Let (Xt)t≥0 be a continuoustime, timehomogeneous strong Markov process with possible jumps and let τ be its first hitting time of a Borel subset of the state space. Suppose X is sampled at random times and suppose also that X has not hit the Borel set by time t. What is the intensity process of τ ..."
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Let (Xt)t≥0 be a continuoustime, timehomogeneous strong Markov process with possible jumps and let τ be its first hitting time of a Borel subset of the state space. Suppose X is sampled at random times and suppose also that X has not hit the Borel set by time t. What is the intensity process of τ based on this information? This question from credit risk encompasses basic mathematical problems concerning the existence of an intensity process and filtration expansions, as well as some conceptual issues for credit risk. By revisiting and extending the famous Jeulin–Yor [Lecture Notes in Math. 649 (1978) 78–97] result regarding compensators under a general filtration expansion framework, a novel computation methodology for the intensity process of a stopping time is proposed. En route, an analogous characterization result for martingales of Jacod and Skorohod [Lecture Notes in Math. 1583 (1994) 21–35] under local jumping filtration is derived.
Credit Risk Models with Incomplete Information
, 2007
"... Incomplete information is at the heart of informationbased credit risk models. In this paper, we rigorously define incomplete information with the notion of “delayed filtrations”. We characterize two distinct types of delayed information, continuous and discrete: the first generated by a time chang ..."
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Cited by 15 (0 self)
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Incomplete information is at the heart of informationbased credit risk models. In this paper, we rigorously define incomplete information with the notion of “delayed filtrations”. We characterize two distinct types of delayed information, continuous and discrete: the first generated by a time change of filtrations and the second by finitely many marked point processes. This notion unifies the noisy information in Duffie and Lando (2001) and the partial information in CollinDufresne et al. (2004), under which structural models are translated into reducedform intensitybased models. We illustrate through a simple example the importance of this notion of delayed information, as well as the potential pitfall for abusing the Laplacian approximation techniques for calculating the intensity process in an informationbased model. The authors are grateful to the Associate Editor and the two anonymous referees for their constructive suggestions and enlightening remarks.
Modeling the recovery rate in a reduced form model. Unpublished working paper
, 2005
"... This paper provides a model for the recovery rate process in a reduced form model. After default, a firm continues to operate, and the recovery rate is determined by the value of the firm’s assets relative to its liabilities. The debt recovers a different magnitude depending upon whether or not the ..."
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Cited by 10 (2 self)
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This paper provides a model for the recovery rate process in a reduced form model. After default, a firm continues to operate, and the recovery rate is determined by the value of the firm’s assets relative to its liabilities. The debt recovers a different magnitude depending upon whether or not the firm enters insolvency and bankruptcy. Although this recovery rate process is similar to that used in a structural model, the reduced form approach is maintained by utilizing information reduction in the sense of Guo, Jarrow and Zeng (2005). Our model is able to provide analytic expressions for a firm’s default intensity, bankruptcy intensity, and zerocoupon bond prices both before and after default. KEY WORDS: credit risk, recovery rates, reduced form model, filtration reduction ∗ Helpful comments from seminar participants at Cornell University, the Johannes
Absolutely Continuous Compensators
, 2010
"... We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener pro ..."
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We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all compensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However Çinlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and slightly of space, have such a representation. 1
Strict local martingales with jumps
, 2013
"... A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of “naturally occurring ” strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filt ..."
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A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of “naturally occurring ” strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a phenomenon we would contend is common in applications such as filtering, control, and especially in mathematical finance. We give a method for constructing such examples and analyze one particular method in detail. 1