On Cramér-like asymptotics for risk processes with stochastic return on investments
type: Item
instance of: scholarly article
title: On Cramér-like asymptotics for risk processes with stochastic return on investments
zbMATH Open document ID: 1019.60041
author: Jostein Paulsen
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: The author studies a risk process following the model \(Y_t=y+P_t+\int_0^t Y_{s-} dR_s\) where \(P\) and \(R\) are Lévy processes starting off at zero. In a recent paper [Stochastic Processes Appl. 75, 135--148 (1998; Zbl 0932.60044)] the author has investigated the ruin probabilities for this process. The aim of the present paper is to develop asymptotic formulas for the ruin probability \(\psi(y)\) as the initial capital \(y\) gets large in those cases where the ruin is not certain. It is shown that \(\psi \in \text{RV}_{-\kappa_0},\) i.e., is regularly varying of order \(-\kappa_0\) in case where \(R\) dominates, here \(\kappa_0\) is the zero of \(\log E(e^{-k\widetilde{R}_1})\) with \(\widetilde{R}\) being the logarithm of the Doolean-Dade functional of \(R ,\) or in case where \(P\) dominates, here it is assumed that the Lévy measure of \(P\) is in \(\text{RV}_{-\kappa_0}\) and some further conditions are satisfied.
reviewed by: Ulrich Stadtmüller
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 91B30
zbMATH DE Number: 1906153
zbMATH Keywords: risk processes
zbMATH Keywords: ruin probability
zbMATH Keywords: Lévy process
zbMATH Keywords: stochastic difference equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1037125862
OpenAlex ID: W2031874744
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DOI: 10.1214/AOAP/1037125862
description: scientific article; zbMATH DE number 1906153