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First crossing time, overshoot and Appell-Hessenberg type functions

Ignatov, Z. G. & Kaishev, V. K. (2016). First crossing time, overshoot and Appell-Hessenberg type functions. Stochastics: An International Journal of Probability and Stochastic Processes, 88(8), pp. 1240-1260. doi: 10.1080/17442508.2016.1230613


We consider a general insurance risk model with extended flexibility under which claims arrive according to a point process with independent increments, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous, real valued function. Point processes with independent increments are in general non-stationary, allowing for an arbitrary (possibly discontinuous) claim arrival cumulative intensity function which is appealing for insurance applications. Under these general assumptions, we derive a closed form expression for the joint distribution of the time to ruin and the deficit at ruin, which is remarkable, since as we show, it involves a new interesting class of what we call Appell-Hessenberg type functions. The latter are shown to coincide with the classical Appell polynomials in the Poisson case and to yield a new class of the so called Appell-Hessenberg factorial polynomials in the case of negative binomial claim arrivals. Corollaries of our main result generalize previous ruin formulas e.g., those obtained for the case of stationary Poisson claim arrivals.

Publication Type: Article
Additional Information: This is an Accepted Manuscript of an article published by Taylor & Francis in Stochastics: An International Journal of Probability and Stochastic Processes on 19 Sep 2016, available online:
Publisher Keywords: Appell polynomials, risk process, ruin probability, first crossing time, overshoot, point process.
Subjects: H Social Sciences > HF Commerce
Departments: Bayes Business School > Actuarial Science & Insurance
SWORD Depositor:
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