Adapted and casual maximum principle and analytical solution to optimal control for stochastic multiplicative-noise systems with multiple input-delays

The main purpose of the paper is to provide the adapted and casual solution to the stochastic linear quadratic control problem for systems with multiple input-delays (SLQD). We introduce a special stochastic discrete-time maximum principle (SDMP) where instead of the admissible set consisting of the F_k-1-measurable control variable u_k for any k, the admissible set is restricted to compose of the F_k-s-1-measurable with s ≥ 0 u_k for any k. The introduced SDMP plays an important part in obtaining the adapted and casual controller. It should be noted that our derivation avoids the augmented argument, mainly establishes and takes advantage of the link between the optimal state and auxiliary variable.