Continuous-time Kelly from a discrete step

(a) Over a short interval $\Delta$, let the risky return be $R = \mu\Delta + \sigma\sqrt\Delta Z$ with $Z \sim N(0,1)$, and let wealth change by $1 + r\Delta + f(R - r\Delta)$. Show that $E\ln(1 + r\Delta + f(R - r\Delta)) = [r + f(\mu - r) - \frac12 f^2\sigma^2]\Delta + o(\Delta)$. (b) With $\mu = 11\%$, $r = 6.5\%$, $\sigma = 20\%$ for the Nifty, a ₹20 lakh account and Nifty futures at 24,000 with lot size 65, how many lots does full Kelly prescribe? Half Kelly? Comment on lot granularity.