Maximizing the discounted survival probability in Vardi's casino

We are in a subfair casino, with fortune , and we want to turn it into a fortune of size 1 in discounted time. We may stake any amount if our fortune is f at any time, , and at any odds, r>0. We assume that every such gamble has a pay-off with fixed expected value cs, with . The problem is to determine the maximum expected discounted pay-off , , where the maximum is taken over all choices of stakes and odds, S are stakes, and R are odds, and we get a positive pay-off only if our final fortune is 1. We determine the explicit recurrence satisfied by the unique optimal strategy and discounted pay-off. The optimal function is piecewise smooth in intervals , ], , between critical fortunes .

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