Stochastic Maximum Principle for Square-Integrable Optimal Control of Linearly Growing Stochastic Differential Systems Subject to a Quadratically Growing Cost Functional

With Ekeland’s variational principle, we prove a general stochastic maximum principle (SMP) for square-integrable optimal control of linearly growing stochastic differential systems subject to a quadratically growing cost functional. The diffusion coefficient is allowed to depend on the control variable, and the admissible control range is allowed to be nonconvex. We relax the existing assumptions of finite moments of arbitrary order on optimal control (see [S. G. Peng, SIAM J. Control Optim., 28 (1990), pp. 966–979]), and of control-bounded coefficients (see [J. M. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Appl. Math. (N. Y.) 43, Springer, 1999, pp. 114 and 118]), so that the typical linear quadratic optimal stochastic control problem is included to satisfy all the assumptions of our SMP.