Parisian ruin for a refracted Lévy process

Lkabous, Mohamed Amine; Czarna, Irmina; Renaud, Jean‐François

doi:10.1016/j.insmatheco.2017.03.005

Cited by 21 publications

(27 citation statements)

References 17 publications

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“…[1]. For a refracted Lévy process and with a fixed delay for the Parisian ruin time, Lkabous et al [13] determined the Laplace transform of the Parisian ruin time τ r possibly killed when the process goes above a given level.…”

Section: Introductionmentioning

confidence: 99%

Discounted penalty function at Parisian ruin for Lévy insurance risk process

Loeffen¹,

Palmowski²,

Surya³

2018

Insurance: Mathematics and Economics

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In the setting of a Lévy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold r. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level b), which generalises known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the q-potential measure of the process killed at Parisian ruin. The results have semi-explicit expressions in terms of the q-scale function and the distribution of the Lévy process.

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Section: Introductionmentioning

confidence: 99%

Discounted penalty function at Parisian ruin for Lévy insurance risk process

Loeffen¹,

Palmowski²,

Surya³

2018

Insurance: Mathematics and Economics

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“…To deal with the limit as b → ∞, we apply the same machinery using identity (31). We can also compute the limit directly using (10) and (11) and the fact that Then,…”

Section: 1mentioning

confidence: 99%

“…We present a probabilistic analysis and simple resulting expressions for the two-sided exit problem, Laplace transform and the probability of Parisian ruin under a hybrid observation scheme in terms of the delayed scale functions introduced by Lkabous and Renaud [13]. Our approach is based on the expression of the Gerber-Shiu distribution at Parisian ruin with exponential implementation delays obtained in [2], combined with some results in [11].…”

Section: Introductionmentioning

confidence: 99%

A note on Parisian ruin under a hybrid observation scheme

Lkabous

2019

Statistics & Probability Letters

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In this paper, we study the concept of Parisian ruin under the hybrid observation scheme model introduced by Li et al. [9]. Under this model, the process is observed at Poisson arrival times whenever the business is financially healthy and it is continuously observed when it goes below 0. The Parisian ruin is then declared when the process stays below zero for a consecutive period of time greater than a fixed delay. We improve the result originally obtained in [9] and we compute other fluctuation identities. All identities are given in terms of second-generation scale functions.

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“…The authors of [21] revisited the Parisian ruin probability and provided an expression which is considerably simpler than that of [9], and unifies the results for spectrally negative Lévy processes of bounded and of unbounded variation. In [20], the result of [21] was further extended to refracted Lévy processes. The Parisian-ruin-related dividend optimization problem was investigated in [10], where the barrier dividend strategy turned out to be the optimal strategy.…”

Section: Introductionmentioning

confidence: 99%

Draw-down Parisian ruin for spectrally negative Lévy processes

Wang

Zhou²

2020

Adv. Appl. Probab.

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Draw-down time for a stochastic process is the first passage time of a draw-down level that depends on the previous maximum of the process. In this paper we study the draw-down-related Parisian ruin problem for spectrally negative Lévy risk processes. Intuitively, a draw-down Parisian ruin occurs when the surplus process has continuously stayed below the dynamic draw-down level for a fixed amount of time. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit problems via excursion theory. We also find an expression for the potential measure for the process killed at the draw-down Parisian time. As applications, we obtain new results for spectrally negative Lévy risk processes with dividend barrier and with Parisian ruin.

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Parisian ruin for a refracted Lévy process

Cited by 21 publications

References 17 publications

Discounted penalty function at Parisian ruin for Lévy insurance risk process

Discounted penalty function at Parisian ruin for Lévy insurance risk process

A note on Parisian ruin under a hybrid observation scheme

Draw-down Parisian ruin for spectrally negative Lévy processes

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