A Variational Formula for Stochastic Controls and Some Applications

For a controlled stochastic differential equation with a Bolza type performance functional, a variational formula for the functional in a given control process direction is derived, by means of backward stochastic differential equations. As applications, some Pontryagin type maximum principles are established for optimal controls of control problems, for saddle points of open-loop two-person zero-sum differential games, and for Nash equilibria of N-person nonzero-sum differential games. The results presented in this paper generalizes/simplifies the relevant ones found in [12] [17]. In addition, a sufficient existence condition of Nash equilibria is proved for nonzero-sum games. Key words. Stochastic controls, variational formula, maximum principle, differential games, minimax principle, sufficient condition, saddle point, Nash equilibrium AMS (MOS) subject classification. 49K, 93E20, 91A15, 91A23 1 Introduction. Let (Ω,F, lF, lP) be a complete filtered probability space, on which a d-dimensional standard Brown-ian motion W (·) is defined with lF = {Ft}t≥0 being its natural filtration, augmented by all lP-nul