Suppose for simplicity we start off with starting amount $S = 1$ and we wish to reach target amount $T$. To do this we sequentially wager a certain amount and then win that amount with probability $p$ and lose that amount with probability $1-p$ ($p$ is known). We wish to reach target amount $T$ in the minimum expected number of plays. At each stage we can wager a different amount. To make the expectation simpler to compute we can assume that if the current amount goes below $S$ we incur a large number of additional plays, say $M$, and then our amount returns to $S$. I have a feeling that the optimal strategy for this situation depends on $p$ (and possibly $T$ and $M$) but I can't figure out how to show the optimal strategy.