We revisit the two-armed bandit (TAB) problem where both arms are driven by diffusive stochastic processes with a common instantaneous reward. We focus on situations where the Radon-Nikodym derivative between the transition probability densities of the first arm with respect to the second is explicitly known. We calculate how the corresponding Gittins' indices behave under such a change of probability measure. This general framework is used to solve the optimal allocation of a TAB problem where the first arm is driven by a pure Brownian motion and the second is driven by a centered super-diffusive non-Gaussian process with variance quadratically growing in time. The probability spread due to the super-diffusion introduces an extra risk into the allocation problem. This drastically affects the optimal decision rule. Our modeling illustrates the interplay between the notions of risk and volatility.