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On the absolute ruin problem in a Sparre Andersen risk model with constant interest. (English) Zbl 1235.91100

Summary: We extend the work of I.-R. Mitric and K. P. Sendova [“On a multi-threshold compound Poisson surplus process with interest”, Scand. Actuar. J. 2011, No. 2, 75–95 (2011; doi:10.1080/03461231003603054)], which considered the absolute ruin problem in a risk model with debit and credit interest, to renewal and non-renewal structures. Our first results apply to MAP processes, which we later restrict to the Sparre Andersen renewal risk model with interclaim times that are generalized Erlang (\(n\)) distributed and claim amounts following a matrix-exponential (ME) distribution. Under this scenario, we present a general methodology to analyze the Gerber-Shiu discounted penalty function defined at absolute ruin, as a solution of high-order linear differential equations with non-constant coefficients. Closed-form solutions for some absolute ruin related quantities in the generalized Erlang (2) case complement the results obtained under the classical risk model by Mitric and Sendova [loc.cit.].

MSC:

91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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