Let us assume there is a game in which there is a p probability of success and 1-p of failure and v dollars are bet on each round and the total bankroll is k dollars. The un-invested portion is allowed to compound at some rate $r_0 $. A successful bet produces a return of r; otherwise, lose the v.
The growth rate of capital is thus given by:
$g=\dfrac{(k-v)(1+r_o)+vp(1+r)-v(1-p)} k-1$
The expected capital over many rounds is $ke^{gt}$
My question is, how do i convert this to a log-normal distribution so I can find the probability of having a certain amount of capital assuming many rounds are played.
I know the expected value of the log-normal distribution is: $$ e^{u + {\sigma^2\over2} } $$
So how do I know what to choose for sigma and mu
My guess for $\sigma$ is it is ${\sqrt t}{v\over k}$