This is precise the argument at the penultimate time-step in the dynamic programming solution of the multi-round case. The other interesting aspect is that the expected returns are logarithmic, i.e, with y = (2p -1) x
p log(x+y) + (1-p) log(x-y) = log(x) + C
where C is the Shannon capacity of the binary symmetric channel with cross-over probability p.
By the same argument, the expected wealth after T rounds will be
log(x) + T C
So, in addition to the optimal strategy, we have also derived the rate of growth of wealth. This is also in tune with the motivation of Kelly's paper where he was showing a relationship between Shannon capacity and optimal gambling (without using a dynamic programming argument)
By the same argument, the expected wealth after T rounds will be
So, in addition to the optimal strategy, we have also derived the rate of growth of wealth. This is also in tune with the motivation of Kelly's paper where he was showing a relationship between Shannon capacity and optimal gambling (without using a dynamic programming argument)