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

Abstract

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.

The time to ruin for a class of Markov additive risk process with two-sided jumps

“…The simple model (1) with a renewal process for the arrivals of one-sided claims has been studied recently in a number of papers: a particular case was discussed by Dickson and Hipp [7], while Avram and Usábel [3] obtained general distribution results concerning the time to ruin and the undershoot using a method entirely different from that of [11] and the present paper. For earlier work, also see [1,Chapter 5,.…”

“…It is worth emphasizing that, for Theorem 1, the solutions to the Cramér-Lundberg equations with strictly negative real parts are required. The work of [3] and [2] involved special cases of the model (4) and used the solutions with positive real parts.…”

风险投资和大额索赔下更新模型的破产概率

“…机微分方程 (2) 的解如下: 概率, 例如, Asmussen 和 Højgaard [1] , Avram 和 Usábel [2] , Cossette 等 [3] , Stanford 等 [4] , Thorin [5] , 等等; 同时, 大家对带有资本投资和大额索赔情形下风险模型的研究也很重视, 参 见文献 [6][7][8][9][10].…”

Μελέτη Μη Ανανεωτικών Στοχαστικών Μοντέλων Στη Θεωρία Κινδύνου

“…)£(-«■:)) 1 Λ ^ -e))To,i(s) ai -Xi ((1 -ε)ϊί{-εγ) 1 ^ w|î/0,i(s)|£, όπου για την δεύτερη ανίσωση χρησιμοποιούμε το γεγονός ότι fi i (-ε) > 1 για ε > 0. Ακόμη, από το γεγονός ότι1 + > (1ε)/ί(-ε) έπεται ότι ισχύει (ε + Χβηχ > ή ' -**• / λ \ 1/îlj Xi flε)/ί(-ε), και συνεπώς ε + λ* > λ* ( (1 -ε)/ί(-ε) ) Συνεπώς, η τελευταία εξίσωση γράφεται ισοδύναμα ως \%} (s) -uXinifi(s)\ > aiu\v0j{s)\ = \uaiki>0j(s)\, Vi e E,k=l,k^i απ' όπου'αποδεικνύεται το det Aq(0, 1)(s,u) φ 0 για s e (7ε με D? (s) <0.…”

“…2=1όπου {JV(i)}~0) (Xifêi όπως ορίσθηκαν παραπάνω και (Τ(ί)})Τ0 μια στοχαστική διαδικα σία που εκφράζει το άθροισμα όλων των υπο-ζημιών Yj που εμφανίζονται πριν από το χρόνο t. Ακόμη, θεωρούμε ότι οι διαδικασίες {5(i)}g0 και {-Β(ί)1 …”