Random optimization on random sets (Q2304911)

The authors study the portfolio processes which are minimal with respect to the random preorders. A general setting is considered where the random preorder is either defined by a random set, which is not necessarily convex, or is defined by a random countable multi-utility representation. The notion of essential minimum of a family of vector-valued random variables with respect to the random preorder is introduced. The main contribution of the paper is the following: it is proved that the minimal elements exist, i.e., the essential minimum is not empty, under mild conditions. Finally, the authors illustrate the main result by two applications. The first one characterizes the minimal portfolio processes super-replicating a European claim. The second one is a classical problem in economics that is considered in a random environment. Precisely, a random cost function on a random set is minimized.