A new approach to Poissonian two-armed bandit problem

We consider a continuous time two-armed bandit problem in which incomes are described by Poissonian processes. We develop Bayesian approach with arbitrary prior distribution. We present two versions of recursive equation for determination of Bayesian piece-wise constant strategy and Bayesian risk and partial differential equation in the limiting case. Unlike the previously considered Bayesian settings our description uses current history of the process and not evolution of the posterior distribution.