Which leverage grows fastest?

An index follows $dS = \mu S\,dt + \sigma S\,dW$ with $\mu = 8\%$ and $\sigma = 20\%$. Interest rates, fees and borrowing costs are zero. A leveraged fund keeps exactly $L$ times its value in the index, rebalancing continuously.

(a) What SDE does the fund value $V$ follow, and at what rate does its median grow?

(b) Put $100 in the unlevered index and $100 in a 3x fund. Compare the mean and the median of each after 10 years.

(c) Which $L$ maximizes the median growth rate, and what is that rate?