Mathematical investigation of the Gerber–Shiu function in the case of dependent inter-claim time and claim size

Mihálykó, Éva Orbán; Mihálykó, Csaba

doi:10.1016/j.insmatheco.2011.01.005

Cited by 9 publications

(9 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…[24] considered the Gerber-Shiu function in the compound Poisson risk model perturbed by diffusion with the dependence between claim sizes and interclaim times. For more details about the dependence structure, please refer to [2], [5], [1], [6] and [22] among others.…”

Section: Wuyuan Jiangmentioning

confidence: 99%

The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence

Jiang¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

We consider a compound Poisson risk process perturbed by a Brownian motion through using a potential measure where the claim sizes depend on inter-claim times via the Farlie-Gumbel-Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for the distribution of the maximum surplus before ruin. This distribution can be calculated through the probability that the surplus process attains a given level from the initial surplus without first falling below zero. The explicit expressions for this distribution are derived when the claim amounts are exponentially distributed.

show abstract

Section: Wuyuan Jiangmentioning

confidence: 99%

The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence

Jiang¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

show abstract

“…Folytonos eloszlások esetén nagyáttörést hozott a Gerber-Shiu-féle diszkontált büntetőfüggvény [3] bevezetése, mivelígy egyetlen függvény segítségével lehet számolni a tönkremenés valószínűségétés a tönkremenési idő várható ertékét, sőt a tönkremenés idejének szórását is. A Gerber-Shiu-függvény matematikai tulajdonságainak elemzésével, speciális esetekben kiszámolásával számos publikáció foglalkozik ( [3], [1], [8], [9]).…”

Section: Bevezetésunclassified

Diszkrét kockázati modell általános befizetési ráta mellett

Győrfi-Bátori¹,

Mihálykó²,

Orbán³

2019

AML

Self Cite

View full text Add to dashboard Cite

A publikációban egy Sparre Andersen-kockázati folyamatot vizsgálunk. Egy diszkrét modellel foglalkozunk, amelybenáltalános befizetési rátát feltételezünk. A Gerber-Shiu-függvény egy diszkrét változatáraáltalános esetben differenciaegyenletet adunk meg, a differenciaegyenlet megoldásának egyértelműségére feltételt adunk. Az egyenletet negatív binomiális káreloszlás esetén egy másik egyenletté transzformáljuk, amelynek megadjuk a megoldását. Eredményünket példával illusztráljuk.

show abstract

“…T ðξÞ = ξ), then (4.11) has a unique solution. For the details regarding the use of contracting mapping principle in the actuarial literature, see, for example, Mihálykó & Mihálykó (2011) and Albrecher & Haas (2011).…”

Section: Lemmamentioning

confidence: 99%

A plan of capital injections based on the claims frequency

Woo

Han

et al. 2018

Ann. actuar. sci.

View full text Add to dashboard Cite

In this work, we propose a capital injection strategy which is periodically implemented based on the number of claims in the classical Poisson risk model. Especially, capital injection decisions are made at a predetermined accumulated number of claim instants, if the surplus is lower than a minimum required level. There appears to be a similar problem found in reliability theory such that preventive maintenance policies are performed at certain shock numbers. Assuming a combination of exponentials for the claim severities, we first derive an explicit expression for the discounted density of the surplus level after a certain number of claims if ruin has not yet occurred. Utilising this result, we study the expected total discounted capital injection until the first ruin time. To solve the differential equation associated with this quantity, we analyse an extended Lundberg’s fundamental equation. Similarly, an expression for the Laplace transform of the time to ruin is also explicitly found. Finally, we illustrate the applicability of the present capital injection strategy and methodologies through various numerical examples. In particular, for exponential claim severities, some optimal capital injection strategy which minimises the expected capital spending per unit time is numerically studied.

show abstract

Mathematical investigation of the Gerber–Shiu function in the case of dependent inter-claim time and claim size

Cited by 9 publications

References 11 publications

The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence

The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence

Diszkrét kockázati modell általános befizetési ráta mellett

A plan of capital injections based on the claims frequency

Contact Info

Product

Resources

About