Aspects of prospective mean values in risk theory

The present paper deals with conditional mean values for analysing prospective events in risk theory, mainly related to reserve evaluation. In some (Markov) cases, for instance the classical life insurance set-up, Kolmogorovʹs backward differential equations suffice as a constructive tool, together with basic martingale relations. However, in many important (Markov) cases we need more refined martingale techniques. We shall mainly focus on cases with random time horizon defined as an exit time. The martingale results are carried out in a marked point process set-up, by use of the important concept of an intensity measure.