Robust stochastic maximum principle: Complete proof and discussions

This paper develops a version of Robust Stochastic Maximum Principle (RSMP) applied to the Minimax Mayer Problem formulated for stochastic differential equations with the control-dependent diffusion term. The parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the Lebesque integral over a parametric set of the standard stochastic Hamiltonians corresponding to a fixed value of the uncertain parameter. The paper deals with a cost function given at finite horizon and containing the mathematical expectation of a terminal term. A terminal condition, covered by a vector function, is also considered. The optimal control strategies, adapted for available information, for the wide class of uncertain systems given by an stochastic differential equation with unknown parameters from a given compact set, are constructed. This problem belongs to the class of minimax stochastic optimization problems. The proof is based on the recent results obtained for Minimax Mayer Problem with a finite uncertainty set [14,43-45] as well as on the variation results of [53] derived for Stochastic Maximum Principle for nonlinear stochastic systems under complete information. The corresponding discussion of the obtain results concludes this study.