Ruin models with investment income

Abstract

This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integro-differential equations, exact and numerical solutions, asymptotic results, bounds on the ruin probability and also the possibility of minimizing the ruin probability by investment and possibly reinsurance control. The main emphasis is on continuous time models, but discrete time models are also covered. A fairly extensive list of references is provided, particularly of papers published after 1998. For more references to papers published before that, the reader can consult [47].Comment: Published in at http://dx.doi.org/10.1214/08-PS134 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic Ruin Probabilities for a Bivariate Lévy-Driven Risk Model with Heavy-Tailed Claims and Risky Investments

“…See Paulsen (1998aPaulsen ( , 1998bPaulsen ( , 2002Paulsen ( , 2008 for detailed explanations. This model does not mean that the surplus must be completely invested in a risky asset.…”

Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model

“…This assumption on price processes is widely used in mathematical …nance. We refer the reader to the monograph of Cont and Tankov (2004) and a recent survey paper of Paulsen (2008). See also Paulsen (1993Paulsen ( , 2002 As usual, we assume that all sources of randomness, fX 1 ; X 2 ; : : :g, fN t ; t 0g, and fR t ; t 0g, are mutually independent.…”

Mathematical Model of Banking Operation

“…The studies [24,25] contain chapters devoted to this subject. The paper [26] reviews the achievements in continuous models, which in some sense are the limit of informative discrete models. The risk process considered therein consists of terms one of which describes arrival of premiums proportionally to time with some fluctuation presented by Brownian motion, the second term, which describes compensation of claims, is presented by a compound Poisson process.…”

“…This process is a homogeneous Markov process, and the presented technique for the analysis of bankruptcy probability in this model implies analyzing the asymptotics of solutions of integro-differential equations which it satisfies. In [26], the asymptotics of such solutions is obtained and the results of numerical analysis are presented. Finding any estimates is rather difficult since they either are problematic or cannot be evaluated.…”