Decomposition and Characterization of Risk with a Continuum of Random Variables

Cet article examine la représentation et la caractérisation des risques générés par un continuum de variables aléatoires. Le résultat principal caractérise une classe de processus en terme de la décomposition du risque et en terme de l'approximation du continuum par des ensembles finis de variables aléatoires. L'article applique ce résultat au problème de décision et aux jeux avec un continuum de joueurs. The paper studies the representation and characterization of risks generated by a continuum of random variables. The main Theorem is a characterization of a broad class of continuum processes in terms of the decomposition of risk into aggregate and idiosyncratic components, and in terms of the approximation of the continuum process by finite collections of random variables. This characterization is used to study decision making problems with anonymous and state-independent payoffs. An Extension Theorem shows that if such a payoff function is defined on simple processes, then it has a unique continuous extension to the class of processes characterized in this paper. This extension is formulated without reference to sample realizations and with minimal restrictions on the patterns of correlation between the random variables. As an application, the thery is used to develop a new model of large games which emphasizes the explicit description of the players' randomizations. This model is used to study the class of environments is which Schmeidler's (1973) representation of strategic uncertainty in large games is valid.