Summary of key results · M03 Kelly and ruin

Result	Formula
R-multiple, expectancy	$R = \Pi/1R$ ; $E = \bar R = p\bar W - (1-p)\bar L$
SQN	$\sqrt{\min(N,100)}\,\bar R/s_R$ (capped $t$ -statistic)
Binary Kelly	$g(f) = p\ln(1+bf) + q\ln(1-f)$ ; $f^* = p - q/b$ ; $g(f^*) = p\ln p + q\ln q + \ln(1+b) - q\ln b$
Asymptotic optimality	$\frac1n\ln W_n(f) \to g(f)$ a.s.; $W_n(f^*)/W_n(f) \to \infty$
Overbetting	$b = 1$ : $g(2f^) < 0$ ; general small bets: $g(cf^)/g(f^*) \approx 2c - c^2$
Continuous Kelly	$g(f) = r + f(\mu - r) - \frac12 f^2\sigma^2$ ; $f^* = (\mu-r)/\sigma^2$ ; $g^* = r + S^2/2$ ; $g(2f^*) = r$
Brownian hitting	$P(\inf(mt + sB_t) \le -d) = e^{-2md/s^2}$
Fractional Kelly	growth share $2c - c^2$ ; vol share $c$ ; $P(\text{ever} \le xW_0) = x^{2/c-1}$
Estimation risk	$E[g] - r = k\mu^2/\sigma^2 - k^2(\mu^2 + \text{se}^2)/(2\sigma^2)$ ; $k^* = t^2/(1+t^2)$
Multi-asset Kelly	$f^* = \Sigma^{-1}(\mu - r\mathbf 1)$ ; $g^* = r + \frac12(\mu - r\mathbf 1)^\top\Sigma^{-1}(\mu - r\mathbf 1)$
Gambler's ruin	$P_a = (\rho^a - \rho^N)/(1 - \rho^N)$ , $\rho = q/p$ ; $N = \infty$ : $\rho^a$ if $p > q$
Duration	$D_a = a/(q-p) - (N/(q-p))(1-\rho^a)/(1-\rho^N)$ ; $p = q$ : $a(N-a)$
Lundberg	$E(1+fX)^{-\theta} = 1$ ; $P(\text{ever} \le xW_0) \le x^\theta$ ; $\theta = 1$ at Kelly; $\theta \approx 2/c - 1$
Stationary drawdown	$P(D_\infty > d) = e^{-2md/s^2}$
Expected MDD (large $T$ )	$\frac{2s^2}{m}(\frac14\ln\frac{m^2T}{2s^2} + 0.49088)$
Excursion depth	$\pi_k = \rho^k(1-\rho)/(1-\rho^{k+1})$
Position size	$Q = \ell\lfloor\lfloor fE_q/\lvert P_e - P_s\rvert\rfloor/\ell\rfloor$

Notation for later modules

$f$ fraction of capital at risk (or invested, in continuous time); $f^*$ Kelly fraction; $c = f/f^*$ Kelly multiple; $g(f)$ growth rate of log wealth; $\theta$ Lundberg exponent; $R$ R-multiple; $1R$ initial risk in rupees; $S$ Sharpe ratio.