Ignatov, Z. G. and Kaishev, V. K. (2011). Finite Time NonRuin Probability Formulae for Erlang Claim Interarrivals and Continuous Interdependent Claim Severities. Stochastics: An International Journal of Probability and Stochastic Processes, 84(4), pp. 461485. doi: 10.1080/17442508.2011.615932
Abstract
A closed form expression, in terms of some functions which we call exponential Appell polynomials, for the probability of nonruin of an insurance company, in a finitetime interval is derived, assuming independent, nonidentically Erlang distributed claim interarrival times, τi ∼ Erlang (gi, λi) , i = 1, 2, . . ., any continuous joint distribution of the claim amounts and any nonnegative, nondecreasing real function, representing its premium income. In the special case when τi ∼ Erlang (gi, λ) , i = 1, 2, . . . it is shown that our main result yields a formula for the probability of nonruin expressed in terms of the classical Appell polynomials. We give another special case of our nonruin probability formula for τi ∼ Erlang (1, λi) , i = 1, 2, . . ., i.e., when the interarrival times are nonidentically exponentially distributed and also show that it coincides with the formula for Poisson claim arrivals, given in [18], when τi ∼ Erlang(1, λ), i = 1, 2, . . .. The main result is extended further to a risk model in which interarrival times are dependent random variables, obtained by randomizing the Erlang shape or/and rate parameters. We give also some useful auxiliary results which characterize and express explicitly (and recurrently) the exponential Appell polynomials which appear in our finite time nonruin probability formulae.
Publication Type:  Article 

Additional Information:  This is an Accepted Manuscript of an article published by Taylor & Francis in Stochastics on 29 Sep 2011, available online: http://www.tandfonline.com/10.1080/17442508.2011.615932 
Publisher Keywords:  finitetime (non) ruin probability; risk process; Erlang claim interarrival times; dependent claim amounts; exponential Appell polynomials; divided difference 
Subjects:  Q Science > QA Mathematics 
Departments:  Cass Business School > Actuarial Science & Insurance 
URI:  https://openaccess.city.ac.uk/id/eprint/15207 

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