The Optimal Control Problem of Stochastic Differential System with Extended Mixed Delays and Applications

This paper investigates an optimal control problem where the system is described by a stochastic differential equation with extended mixed delays that contain point delay, extended distributed delay, and extended noisy memory. The model is general in that the extended mixed delays of the state variable and control variable are components of all the coefficients, in particular, the diffusion term and the terminal cost. To address the difficulties induced by the extended noisy memory, by stochastic Fubini theorem, we transform the delay variational equation into a Volterra integral equation without delay, and then a kind of backward stochastic Volterra integral equation with Malliavin derivatives is introduced by the developed coefficient decomposition method and the generalized duality principle. Therefore, the stochastic maximum principle and the verification theorem are established. Subsequently, with Clark-Ocone formula, the adjoint equation is expressed as a set of anticipated backward stochastic differential equations. Finally, a nonzero-sum stochastic differential game with extended mixed delays and a linear-quadratic solvable example are discussed, as applications.