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)