Ruin probability in the Cramér-Lundberg model with risky investments
type: Item
instance of: scholarly article
title: Ruin probability in the Cramér-Lundberg model with risky investments
zbMATH Open document ID: 1236.91087
author: Sheng Xiong
author: Wei-Shih Yang
published in: Stochastic Processes and their Applications
publication date: 2011-06-15T00:00:00Z
review text: Let \(Y\) be a geometric Brownian motion. That is, the solution to the stochastic differential equation \(d Z_t = a Z_t \;d t + \sigma Z_t \;d W_t\), where \(W\) is a standard Brownian motion. By \(P\) we denote an independent compound Poisson process \(P_t = \sum_{k=1}^{N(t)} \xi_k\), where \(N\) is a Poisson process with rate \(\lambda\) and \(\{\xi_k\}\) are iid positive random variables. \(N\), \(\{\xi_k\}\) and \(W\) are assumed to be independent. \(Z\) models the price of a risky asset and \(P\) models the aggregate claim of an insurance portfolio. If all the surplus is invested in the risky asset, the surplus process of the insurance portfolio is \[ X_t = X_0 Z_t + \int_0^t \frac{Z_t}{Z_s} c_s \;d s - \sum_{k=1}^{N_t} \xi_k \frac{Z_t}{Z_{T_k}}\;, \] where \(\{c_s\}\) is the premium rate and \(\{T_k\}\) are the occurrence times of the Poisson process. The premium rate is assumed to be bounded, \(0 \leq c_s \leq c < \infty\) for some \(c > 0\). The case \(2 a > \sigma^2\) has been considered in several papers, see for instance [\textit{A. Frolova, Y. Kabanov} and \textit{S. Pergamenshchikov}, Finance Stoch. 6, No. 2, 227--235 (2002; Zbl 1002.91037); \textit{V. Kalashnikov} and \textit{R. Norberg}, Stochastic Processes Appl. 98, No. 2, 211--228 (2002; Zbl 1058.60095); \textit{J. Paulsen}, Stochastic Processes Appl. 75, No. 1, 135--148 (1998; Zbl 0932.60044); \textit{H. Nyrhinen}, Stochastic Processes Appl. 92, No. 2, 265--285 (2001; Zbl 1047.60040); \textit{S. Pergamenshchikov} and \textit{O. Zeitouny}, Stochastic Processes Appl. 116, No. 2, 267--278 (2006); erratum ibid. 119, No. 1, 305--306 (2009; Zbl 1088.60076)]. They show that, even for light-tailed claim size distributions, the ruin probability decays at a polynomial rate. In [Pergamenshchikov and Zeitouny, loc.\,cit.] it was proved, under the assumption that the claim size distribution has unbounded support, that the ruin probability is \(\psi(u) = P[\inf_t X_t < 0\mid X_0 = u] = 1\) if \(2 a \leq \sigma^…
reviewed by: Hanspeter Schmidli
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 5908026
zbMATH Keywords: Cramér--Lundberg model
zbMATH Keywords: geometric Brownian motion
zbMATH Keywords: ruin probability
zbMATH Keywords: risky investments
zbMATH Keywords: martingale methods
MaRDI profile type: Publication
full work available at URL: j.spa.2011.01.008
OpenAlex ID: W1995789919
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DOI: 10.1016/J.SPA.2011.01.008
description: scientific article; zbMATH DE number 5908026