The Lundberg exponent at half Kelly

For $p = 0.55$, $b = 1$: (a) Compute $\theta$ for $f = 0.05$ (half Kelly) numerically and compare with the small-bet approximation $2/c - 1$. (b) Give the bound on the probability of ever halving and compare with the continuous-time value $0.5^3$. (c) Explain why simulations over a finite number of bets give a slightly smaller probability than the bound. (d) Prove that $1/W_n$ is a martingale for the Kelly bettor and that $W_n(f)/W_n(f^*)$ is a martingale for every fixed $f \in [0, 1)$. Reconcile this with Proposition 3.3(b), which says the same ratio tends to 0 almost surely.