Some diffusion models for the Mabinogion sheep problem of Williams

The 'Mabinogion sheep' problem, originally due to D. Williams, is a nice illustration in discrete time of the martingale optimality principle and the use of local time in stochastic control. The use of singular controls involving local time is even more strikingly highlighted in the context of continuous time. This paper considers a class of diffusion versions of the discrete-time Mabinogion sheep problem. The stochastic version of the Bellman dynamic programming approach leads to a free boundary problem in each case. The most surprising feature in the continuous-time context is the existence of diffusion versions of the original discrete-time problem for which the optimal boundary is different from that in the discrete-time case; even when the optimal boundary is the same, the value functions can be very different.