On ruin probabilities in risk models with interest rate

Abstract

An explicit formula for the finite-time ruin probability in a discrete-time collective ruin model with constant interest rate is found under the assumption that claims follow a generalised hyperexponential distribution. The formula can be used for finding approximations for finite-time ruin probabilities in the case when claim sizes follow a heavy-tailed distribution e.g. Pareto. We also provide theoretical bounds for the accuracy of approximations of the finite-time ruin probabilities in terms of a distance between the distribution of claims and its approximation. Results of numerical comparisons with asymptotic formulas and simulations are presented.

Appendix

Proof of Proposition 2

The proof is based on the following lemma which was proved in [7].

Lemma. Let the Markov chain V _n be a solution of the

(16)

Then

(17)

1. Proof of (14)

Set

then

For n=1 we have

We use the induction method, let us assume that

Next,

Thus, the statement 1 is proved.

2. Proof of (15)

Set

where random variables ω _i , i=1, 2 follow the uniform distribution on [0, 1],

Then by induction we have

Note that in view of results in [8], we obtained

where infimum is taken over all joint distributions that have the same marginal distributions as that of V _n and $V_n^*$ . This completes the proof of statement 2.