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Snell's optimization problem for sequences of convex compact valued random sets

Krupa G.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 1

77--91

A random set analogue of the Snell problem is presented. In the original Snell’s problem one observes a sequence of random variables (ξn), say a gambler’s capital at successive games. If the gambler leaves the game at a random time ν, his expected capital at this time is Eξν. The objective is to stop at time ν (using information available up to this moment) such that the expected gambler’s fortune Eξν is maximal. Here a multivalued analogue of this problem will be studied. Given a Banach space and a sequence of convex weakly or strongly compact valued random sets (Zn) in that space, the existence of a stopping time ν such that EZν is maximal is investigated.