Coherent risk measures on general probability spaces. (English) Zbl 1020.91032

Delbaen, Freddy

Sandmann, Klaus (ed.) et al., Advances in finance and stochastics. Essays in honour of Dieter Sondermann. Berlin: Springer. 1-37 (2002).

The paper extends the definition of coherent risk measures, as introduced by Artzner, Delbaen, Eber and Heath, to general probability spaces. It is shown how to define such measures on the space of all random variables. Examples that relates the theory of coherent risk measures to game theory and to distorted probability measures are given. The mathematics are based on the characterization of closed convex sets \({\mathcal P}_{\sigma}\) of probability measures that satisfy the property that every random variable is integrable for at least one probability measure in the set \({\mathcal P}_{\sigma}\).
For the entire collection see [Zbl 0986.00085].

Reviewer: Georgy Osipenko (St.Peterburg)

MSC:

91B30	Risk theory, insurance (MSC2010)
91A10	Noncooperative games
91B82	Statistical methods; economic indices and measures

Keywords:

capital requirement; coherent risk measure; capacity theory; convex games; insurance premium principle; measure of risk; Orlicz spaces; quantile; scenario; shortfall; subadditivity; submodular functions; value at risk