(too late for me to edit my post, so I'll reply to it) The logic I posted above will show you that the log of your money X converges in probability to some val

(too late for me to edit my post, so I'll reply to it) The logic I posted above will show you that the log of your money X converges in probability to some value x that you can optimize. Optimizing x will optimize a monotonic utility f(x), which seems useful, but isn't. The problem is that just because X converges in probability to x, doesn't mean that f(X) converges in probability to f(x). Probability and convergence are weird.