We discuss the problem of allocating effort among several competing projects, the states of which evolve according to one-dimensional diffusion processes. It is shown that the "play-the-leader" policy of continuing the project with the leading Gittins index is optimal, and very explicit computations of the index are offered. The question of constructing the diffusions according to the above policy is also addressed.
Ioannis Karatzas. "Gittins Indices in the Dynamic Allocation Problem for Diffusion Processes." Ann. Probab. 12 (1) 173 - 192, February, 1984. https://doi.org/10.1214/aop/1176993381