Continuous-time multi-armed bandits under random intervention times

This paper examines multi-armed bandits in which actions are taken at random discrete times. The model consists of $J$ independent arms. When an arm is operated, it must remain active for a random duration, modeled by the inter-arrival time of a (possibly arm-dependent) renewal process. For arms evolving as a L\'evy process, we provide an explicit characterization of the Gittins index, which is known to yield an optimal strategy. Furthermore, when the inter-arrival times are exponential and the arms evolve as either a spectrally negative L\'evy process, a reflected spectrally negative L\'evy process, or a diffusion process, the Gittins index is explicitly characterized in terms of the scale function or diffusion characteristics, respectively. Numerical experiments are performed to support the theoretical results.