Prophet regions for independent random variables with simultaneous costs and discountings

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Let Y = (Y1,Y2,…) be a sequence of independent [li,ri]-valued random variables, where (li), (ri), (di):=(ri-li) are non-increasing sequences, and all di>0. Considering the class of all such sequences, a complete comparison is made between M(Y), the expected gain of a prophet (an observer with complete foresight), and V(Y), the maximal expected gain of a gambler (unsing only non-anticipatory stopping times). The solution of this problem is a set in , the so-called prophet region, which is determined for finite and infinite sequences. These regions generalize earlier results [4], [6], [1] in a natural manner. Especially, they yield a variety of prophet inequalities, e.g. all results derived in [5], [6] and [1].