Working through some probability problems from Introduction to Probability, Blitzstein:

Kelly makes a series of n bets, each of which she has probability p of winning, independently. Initially, she has x0 dollars. Let Xj be the amount she has immediately after her jth bet is settled. Let f be a constant in (0, 1), called the betting fraction. On each bet, Kelly wagers a fraction f of her wealth, and then she either wins or loses that amount. For example, if her current wealth is 100 dollars and f = 0.25, then she bets 25 dollars and either gains or loses that amount. (A famous choice when p > 1/2 is f = 2p − 1, which is known as the Kelly criterion.) Find E($X_n$) (in terms of n, p, f, x0).

Hint: First find $E(X_{j+1}|X_j)$.

How can I use the hint? I found $E(X_1|X_0) = x_0(2pf-f+1)$, by plugging in the probability of winning (p) * winnings ($x_0+fx_0$) plus the probability of losing (1-p) * new winnings ($x_0-fx_0$)

But I do not know how to turn this into a recursive formula; any hints on the hint?


$E(X_{j+1}|X_j)$ = $X_j * p * (1+f) + X_j * ( 1 - p) * (1-f)$

$E(X_{j+1}|X_j)$ = $X_j *[2pf−f+1]$

now given that we can recursively write

$E(X_{j+1}|X_j, X_{j-1})$ = $X_j *[2pf−f+1]$

= $X_{j-1} *[2pf−f+1] *[2pf−f+1]$

= $X_{j-1} *[2pf−f+1]^2$

If we keep doing it we will get:

$E(X_n)$ = $X_0 * [2pf−f+1] ^ n$

The Kelly Criterion mentioned in the question maximizes the logarithm of this expected value.

Hope it helps

  • $\begingroup$ Yes it does - I got stuck on subscripts for my random variables, but also have a hard time conceptualizing recursive formulas. This makes sense though. Thanks @ofya $\endgroup$ – user603569 Nov 30 '18 at 20:04
  • $\begingroup$ You're welcome, don't forget to like and mark the answer :)) $\endgroup$ – Ofya Nov 30 '18 at 20:10

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