The submartingale assumption in risk theory

Abstract

The main consequences of the submartingale assumption are examined in a discrete time model. After we have shown how the submartingale decomposition theorem can be given an actuarial meaning, we formulate the ruin theory by classifying the gain processes according to the properties of the set of the safety indexes of their increments. Inequalities for ruin probabilities are derived for embeddable submartingales and for P-submartingales. As an example, we describe a rather general risk model with an adjustable gain process and we show how it can be modified to obtain examples of embeddable submartingales and P-submartingales. Optional gain processes, in which the number of policies in the portfolio is depending on the charged premiums, are also shown to satisfy the submartingale assumption; suitable demand functions can be introduced. We shortly discuss the construction of decision models to define the pricing policy of the insurance company and we provide a simple example of a pricing model with stochastic demand function.