# Risk processes with dependence and premium adjusted to solvency targets

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## Abstract

This paper considers risk processes with various forms of dependence between waiting times and claim amounts. The standing assumption is that the increments of the claims process possess exponential moments so that variations of the Lundberg upper bound for the probability of ruin are in reach. The traditional point of view in ruin theory is reversed: rather than studying the probability of ruin as a function of the initial reserve under fixed premium, the problem is to adjust the premium dynamically so as to obtain a given ruin probability (solvency requirement) for a fixed initial reserve (the financial capacity of the insurer). This programme is carried through in various models for the claims process, ranging from Cox processes with i.i.d. claim amounts, to conditional renewal (Sparre Andersen) processes.

## Keywords

Lundberg upper bound Martingalizing premium rate General risk process## Mathematics Subject Classification

62P05 91B30## Notes

### Acknowledgments

The authors thank the BNP Paribas Cardif Chair *“Management de la modélisation”* for financial support. The views expressed in this document are the authors’ own and do not necessarily reflect those endorsed by BNP Paribas Cardif. C.C. gratefully acknowledges financial support from the Swiss National Science Foundation Project 200021-124635/1. The paper has benefited significantly from feedback from editors and referees (received three weeks after submission).

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