Extremal holomorphic diffusion processes

Extremal holomorphic diffusion processes are studied. We formulate a stochastic control problem for the extremal function of a set. We then characterize the extremal holomorphic diffusion processes as the optimal diffusion processes of the problem. By making use of SDE representation for the processes, we show that they move on an integral submanifold of the coefficients vector fields of the SDE passing through the starting point.