On a Non-Linear Semi-Group Attached to Stochastic Optimal Control

In [6] we introduced a non-linear semi-group attached to the stochastic control of diffusion type, by the following way. Let T be a Cf-compact subset of R, called by a control region. Let a triple (, B, U) be an admissible system where Q is a probability space, B is an ⁻dimensional Brownian motion on Q and U is a /-valued 23-non-anticipative process on J2. For an admissible system (, B, U) we consider the following 72-dimensional stochastic differential equation