Asymptotic expansions of defective renewal equations with applications to perturbed risk models and processor sharing queues

Blanchet, José; Zwart, Bert

doi:10.1007/s00186-010-0321-6

Cited by 10 publications

(4 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The main disadvantage of such approximations is that they provide a good fit only at the tail of the ruin probability, especially for small safety loading. Another stream of research focuses on corrected diffusion approximations for the ruin probability [11,38]. A disadvantage of such asymptotic techniques is the requirement of finite higher moments for the claim size distribution.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Vatamidou¹,

Adan

Vlasiou

et al. 2013

Insurance: Mathematics and Economics

View full text Add to dashboard Cite

Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phasetype approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical experiments.

show abstract

Section: Introductionmentioning

confidence: 99%

“…However, the term "perturbation" in this area is used to denote the superposition of two risk processes. Contrary to other asymptotic techniques that use perturbation analysis to approximate the ruin probability [11,38], our approach is different; we apply perturbation to the claim sizes rather than the arrival rate.…”

Section: Introductionmentioning

confidence: 99%

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Vatamidou¹,

Adan

Vlasiou

et al. 2013

Insurance: Mathematics and Economics

View full text Add to dashboard Cite

show abstract

“…Expansions of the type given in Equation (1.3) and similar types of exponential expansions have also been given for ruin probabilities in perturbed risk models, see for example Silvestrov (2000, 2008), Englund (2001), Blanchet and Zwart (2010), Ni (2011Ni ( , 2014, and Petersson (2014).…”

Section: Introductionmentioning

confidence: 89%

Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time

Petersson

2016

Methodol Comput Appl Probab

View full text Add to dashboard Cite

We consider a discrete time semi-Markov process where the characteristics defining the process depend on a small perturbation parameter. It is assumed that the state space consists of one finite communicating class of states and, in addition, one absorbing state. Our main object of interest is the asymptotic behaviour of the joint probabilities of the position of the semi-Markov process and the event of non-absorption as time tends to infinity and the perturbation parameter tends to zero. The main result gives exponential expansions of these probabilities together with an recursive algorithm for computing the coefficients in the expansions.

show abstract

“…Closely related to the previous approximations is the Edgeworth series expansion [38], which is a refinement of the central limit theorem. Asymptotic results for the ruin probability are given in [9] and these approximations can be useful in applications where moments are computable, but the distribution is not.…”

Section: Introductionmentioning

confidence: 99%

On the accuracy of phase-type approximations of heavy-tailed risk models

Vatamidou

Adan

Vlasiou

et al. 2012

Scandinavian Actuarial Journal

View full text Add to dashboard Cite

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a prespecified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.

show abstract

Asymptotic expansions of defective renewal equations with applications to perturbed risk models and processor sharing queues

Cited by 10 publications

References 19 publications

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Quasi-Stationary Asymptotics for Perturbed Semi-Markov Processes in Discrete Time

On the accuracy of phase-type approximations of heavy-tailed risk models

Contact Info

Product

Resources

About