Equations for probability distributions of local occupation time on a surface for diffusion processes and control problems

We deduce equations for the probability distribution of a value similar to the local occupation time of a diffusion process on a hypersurface. A stochastic control problem for the local occupation time is considered. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 96–118. Translated by S. Yu. Pilyugin.