  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.
