On a risk measure inspired from the ruin probability and the expected deficit at ruin

In this paper, we study a risk measure derived from ruin theory defined as the amount of capital needed to cope in expectation with the first occurrence of a ruin event. Specifically, within the compound Poisson model, we investigate some properties of this risk measure with respect to the stochastic ordering of claim severities. Particular situations where combining risks yield diversification benefits are identified. Closed form expressions and upper bounds are also provided for certain claim severities.     

On a multi-threshold compound Poisson surplus process with interest

We consider a multi-threshold compound Poisson surplus process. When the initial surplus is between any two consecutive thresholds, the insurer has the option to choose the respective premium rate and interest rate. Also, the model allows for borrowing the current amount of deficit whenever the surplus falls below zero. Starting from the integro-differential equations satisfied by the Gerber–Shiu function that appear in Yang et al. (2008), we consider exponentially and phase-type(2) distributed claim sizes, in which cases we are able to transform the integro-differential equations into ordinary differential equations. As a result, we obtain explicit expressions for the Gerber–Shiu function.