Necessary conditions for continuous parameter stochastic optimization problems

A maximum principle is derived for the problem where the system is the Ito equation \[dx = f(x,u,t)dt + \sigma (x,t)dz,\quad 0 \leqq t \leqq T,\] and the cost is of the form $Ex_0 (T) + Eh(x(T))$, where $x_0 (T)$ is the zeroth component of $x(T)$, and T is a real number. There are constraints of the type $E\tilde r_0 (x(0)) = 0$, $Er_i (x(t_i ),Ex(t_i )) = 0$, $i = 1, \cdots ,k$, $E\tilde q_i (x(t_i ),Ex(t_i )) \leqq 0$, $i = 0,1 \cdots ,k$, where $t_i $ are given real numbers. The paper adapts the general maximum principle of Neustadt to the above stochastic problem.