Conditional expectation and martingales of random sets

摘要

The aim of this paper is to present a self-contained introduction to the theory of martingales of random sets (also called `set-valued martingales’ or `multivalued martingales'). For this purpose, we first present a method for constructing the integral and the conditional expectation for random sets with compact convex values in Rd. This method was used by Debreu for constructing the set-valued integral. It is based on the approximation of random sets by simple random sets, i.e., assuming only a finite number of values. It allows us to recover the classical properties of the conditional expectation of real-valued random variables and to deduce new applications, such as an extension of the Brunn-Minkowski inequality. Afterwards, multivalued martingales are introduced, as well as multivalued submartingales and supermartingales. We prove several convergence results, and we give examples and applications. Further, strong laws of large numbers for sequences of independent random sets with compact values are briefly examined. Another approach for defining the integral and the conditional expectation of random sets is also sketched, and an existence theorem for martingale selections is stated. The last section is devoted to the definition of the integral for non convex random sets. Finally, in a short annex, we present a criterion for the conditional expectation of a random set to be single-valued.