RPA pathwise derivative estimation of ruin probabilities
type: Item
instance of: scholarly article
title: RPA pathwise derivative estimation of ruin probabilities
zbMATH Open document ID: 1017.91051
DOI: 10.1016/S0167-6687(99)00022-0
author: Felisa J. Vázquez-Abad
published in: Insurance Mathematics \& Economics
publication date: 2003-08-26T00:00:00Z
review text: The surplus process of an insurance portfolio, defined as the wealth obtained by the premium payments minus the reimbursements made at the times of claims, is studied in the case where the wealth available is invested at a continuously compounded rate \(\delta\): \[ dU(t) = (c + \delta U(t)) dt - dS(t); \quad U(0) = u. \] Here \(U(t) = u + ct + S(t)\) and \(S(t) = \sum_{i=1}^{N(t)} {Y_i}\) where \(N(t)\) is a Poisson process with rate \(\lambda\) that models the epochs when claims are made and \(Y_i\) (the corresponding amounts) are i.i.d. random variables; premiums being received at constant rate \(c\) [cf., e.g., \textit{H. Gerber}, An introduction to mathematical risk theory. Monograph 8, Univ. Pennsylvania, Philadelphia (1979; Zbl 0431.62066)]. In general, for the ruin probability: \[ \psi(u, \lambda) = P\{ \text{min} \{t: U(t) < 0\} < \infty \} \] there is no analytical expression available. The paper under review focuses on the estimation of the sensivities of \(\psi(u, \lambda)\) to the arrival rate \(\lambda\) via the rare perturbation analysis method applied to related: regenerative storage process as well as to importance sampling. For phantom and virtual estimators recursion formulas are provided to program these methods efficiently. There are also found computer simulation results for the exponential claim distribution, in order to compare the two estimation methods presented.
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 91B70
Mathematics Subject Classification ID: 60K20
Mathematics Subject Classification ID: 62P05
zbMATH DE Number: 1525147
zbMATH Keywords: insurance
zbMATH Keywords: ruin probabilities
zbMATH Keywords: estimation
zbMATH Keywords: surplus process
zbMATH Keywords: storage process
zbMATH Keywords: importance sampling
zbMATH Keywords: rare perturbation analysis
reviewed by: Bogdan A. Choczewski
MaRDI profile type: Publication
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full work available at URL: s0167-6687(99)00022-0
OpenAlex ID: W2088713649
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description: scientific article; zbMATH DE number 1525147