Global Convergence of Successive Approximations for Non-convex Stochastic Optimal Control Problems

This paper focuses on finding approximate solutions to the stochastic optimal control problems where the state trajectory is subject to controlled stochastic differential equations permitting controls in the diffusion coefficients. An algorithm based on the method of successive approximations is described for finding a set of small measure, in which the control is varied finitely so as to reduce the value of the functional and, as the control domains are not necessarily convex, the second-order adjoint processes are introduced in each minimization step of the Hamiltonian. Under certain convexity conditions, we prove that the values of the cost functional descend to the global minimum as the number of iterations tends to infinity. In particular, a convergence rate for a class of linear-quadratic systems is available.