Infinite dimensional open-loop linear quadratic stochastic optimal control problems and related games

We investigate the linear quadratic stochastic optimal control problems in infinite dimension without Markovian restriction for coefficients. The necessary and sufficient conditions for open-loop optimal controls are presented. We prove the Fr\'echet differentiable of the cost functional with respect to the control variable, and the Fr\'echet derivatives are characterized in detail by operators derived from dual analysis, which are proven to be the stationary conditions. Transposition methods are adopted to deal with the adjoint equations. As applications, we employ the results to study open-loop Nash equilibria for two-person stochastic differential games.