Calculating multivariate ruin probabilities via Gaver–Stehfest inversion technique

Usabel, Miguel

doi:10.1016/s0167-6687(99)00029-3

Cited by 18 publications

(13 citation statements)

References 37 publications

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“…Finally, one can compute formula (21), and therefore formula (20), and Laplace invert in order to get the CDPs. In this article, we use the Gaver-Stehfest algorithm to do the Laplace inversions [see Usabel (1999) for a presentation of the algorithm together with a nice application in insurance].…”

Section: Endogenously Determined Historical Default Probabilitiesmentioning

confidence: 99%

Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model

Courtois

Quittard-Pinon

2007

Asia-Pacific Finan Markets

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This paper focuses on historical and risk-neutral default probabilities in a structural model, when the firm assets dynamics are modeled by a double exponential jump diffusion process. Relying on the Leland [(1994a) Journal of Finance, 49, 1213–1252; (1994b) Bond prices, yield spreads, and optimal capital structure with default risk. Working paper no. 240, IBER, University of California, Berkeley] or Leland and Toft [(1996) Journal of Finance, 51(3), 987–1019] endogenous structural approaches, as formalized by Hilberink and Rogers [(2002) Finance and Stochastics, 6(2), 237–263], this article gives a coherent construction of historical default probabilities. The risk-neutral world where evolve the firm assets, modeled by a class of geometric Lévy processes, is constructed based on the Esscher measure, yielding useful and new analytical relations between historical and risk-neutral probabilities. We do a complete numerical analysis of the predictions of our framework, and compare these predictions with actual data. In particular, this new framework displays an enhanced predictive power w.r.t. current Gaussian endogenous structural models. Copyright Springer Science+Business Media, LLC 2006Cumulative default probability, Structural model, Jump-diffusion, Endogenous capital structure, Esscher transform, Kou processes, G32, G33,

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Section: Endogenously Determined Historical Default Probabilitiesmentioning

confidence: 99%

Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model

Courtois

Quittard-Pinon

2007

Asia-Pacific Finan Markets

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“…This approach was described by Davies (2002, chapter 19) and by Usabel (1999). Finally, we note that according to the Markov inequality, the asymptotic probability of default is bounded by the following limit:…”

Section: Estimating the Probabilities Of Defaultmentioning

confidence: 91%

Evaluation and default time for companies with uncertain cash flows

Hainaut

2015

Insurance: Mathematics and Economics

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“…Then the original ruin probability function can be approximated using Laplace transform inversion techniques, see Usábel (1999Usábel ( , 2001 and Thorin (1970, 71, 73, 77, 82), Thorin and Wikstad (1973), Wikstad (1971, 77) who used the Piessens (1969) inversion method of the Laplace transform, Bohman (1971, 74, 75) who used the Fourier transform and Seal (1971, 74, 77) who dealt with both Laplace and Fourier numerical inversions.…”

Section: (Via a Single Laplace Inversion)mentioning

confidence: 99%

“…so that the multivariate finite time ruin probability can be expressed very conveniently as a sum of phase-type distributions of order k. Usábel (1999) (1971) …”

Section: 1 In Asmussen (1992) Can Be Also Applied To Obtain J M Usinmentioning

confidence: 99%

Ruin Probabilities and Deficit for the Renewal Risk Model with Phase-type Interarrival Times

Avram¹,

Usabel²

2004

ASTIN Bull.

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This paper shows how the multivariate finite time ruin probability function, in a phase-type environment, inherits the phase-type structure and can be efficiently approximated with only one Laplace transform inversion.From a theoretical point of view, we also provide below a generalization of Thorin's formula (1971) for the double Laplace transform of the finite time ruin probability, by considering also the deficit at ruin; the model is that of a Sparre Andersen (renewal) risk process with phase-type interarrival times.In the case when the claims distribution is of phase-type as well, we obtain also an alternative formula for the single Laplace transform in time (or "exponentially killed probability''), in terms of the roots with positive real part of the Lundberg's equations, which complements Asmussen's representation (1992) in terms of the roots with negative real part 1 . KEYWORDSFinite time ruin probability, exponentially killed ruin probability, deficit at ruin, Sparre Andersen Model, phase-type distributions.

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Calculating multivariate ruin probabilities via Gaver–Stehfest inversion technique

Cited by 18 publications

References 37 publications

Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model

Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model

Evaluation and default time for companies with uncertain cash flows

Ruin Probabilities and Deficit for the Renewal Risk Model with Phase-type Interarrival Times

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