Necessary Conditions for Continuous Parameter Stochastic Optimization Problems

A maximum principle is derived for the problem where the system is the Itô equation \[dx = f(x,u,t)dt + σ (x,t)dz, 0 t T,\] and the cost is of the form Ex₀ (T) + Eh(x(T)), where x₀ (T) is the zeroth component of $x(T)$, and T is a real number. There are constraints of the type E r₀ (x(0)) = 0, Erᵢ (x(tᵢ ),Ex(tᵢ )) = 0, i = 1, ⋯ ,k, E qᵢ (x(tᵢ ),Ex(tᵢ )) 0, i = 0,1 ⋯ ,k, where tᵢ are given real numbers. The paper adapts the general maximum principle of Neustadt to the above stochastic problem.