The relations among the three kinds of conditional risk measures

Abstract

Let (Ω, E, P) be a probability space, F a sub-σ-algebra of E, L ^p(E) (1 ⩽ p ⩽ +∞) the classical function space and L ^p _F (E) the L ⁰(F)-module generated by L ^p(E), which can be made into a random normed module in a natural way. Up to the present time, there are three kinds of conditional risk measures, whose model spaces are L ^∞(E), L ^p(E) (1 ⩽ p < +∞) and L ^p _F (E) (1 ⩽ p ⩾ +∞) respectively, and a conditional convex dual representation theorem has been established for each kind. The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems. We first establish the relation between L ^p(E) and L ^p _F (E), namely L ^p _F (E) = H _cc (L ^p(E)), which shows that L ^p _F (E) is exactly the countable concatenation hull of L ^p(E). Based on the precise relation, we then prove that every L ⁰(F)-convex L ^p(E)-conditional risk measure (1 ⩽ p ⩽ +∞) can be uniquely extended to an L ⁰(F)-convex L ^p _F (E)-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter, which shows that the study of L ^p-conditional risk measures can be incorporated into that of L ^p _F (E)-conditional risk measures. In particular, in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L ⁰-convex conditional risk measures.