Monitoring dates of maximal risk

Monitoring means to observe a system for any changes which may occur over time, using a monitor or measuring device of some sort. In this paper we formulate a problem of monitoring dates of maximal risk of a financial position. Thus, the systems we are going to observe arise from situations in finance. The measuring device we are going to use is a time-consistent measure of risk. In the first part of the paper we discuss the numerical representation of conditional convex risk measures which are defined in a space Lp(F,R) and take values in L1(G,R). This will allow us to consider time-consistent convex risk measures in L1(R). In the second part of the paper we use a time-consistent convex risk measure in order to define an abstract problem of monitoring stopping times of maximal risk. The penalty function involved in the robust representation changes qualitatively the time when maximal risk is for the first time identified. A phenomenon which we discuss from the point of view of robust statistics.