A semi-markovian game of economic survival

Abstract

This paper is devoted to a semi-markovian generalization of a well-known game of economic survival: a firm has an initial capital x, the profits resulting from the successive time intervals (n, n + 1) are random variables; at each instant n the firm may pay dividends to the shareholders, his only goal being to maximize the expected discounted value of all dividends payed before the ruin. The existence of an optimal stationary strategy is proved; the only “impatient” optimal stationary strategy is a “band-strategy”. In the last part of the paper we construct an algorithm producing the optimal band-strategy after a finite number of iterations, and we show how to calculate the ruin-probabilities associated with this strategy.