I'm working through John Kelly's original paper that proves the "Kelly Criterion", but I'm getting hung up on one of the general proofs. Here's a PDF of the paper.
http://turtletrader.com/kelly.pdf
On page 6 of the PDF, Kelly shows the following:
The section I'm having trouble with in Kelly's paper
I'm having a hard time figuring out how he solves for the maximum value of the first time.
Definitions: p(s,r) = joint probability of the s'th transmitted and r'th received symbol
a(s/r) the fraction of a gambler's capital that he decides to bet on the occurrence of the s'th transmitted symbol after observing the r'th received symbol
p(s,r) and a(s/r) are subject to constraints:
a(s/r) > 0
the sum of a(s/r) over s = 1
p(s,r) > 0
Thanks for any help anyone can provide. I think I'm missing something simple, but I'm stumped.
