Abstract
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
.
