ON THE MAXIMUM PRINCIPLE FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH A PATH-WISE COST FUNCTIONAL

The maximum principle with a path-wise cost functional is constructed for one-dimensional stochastic differential equations with a symmetric integral with respect to an arbitrary random process with continuous trajectories in the case when non-anticipating control affects the “drift.’’ It is shown that the results obtained are valid in a deterministic formulation of the problem, that is, when the symmetric integral in the equations is taken with respect to a non-random continuous function. Instead of Itö’s calculus, a technique of symmetric integrals with respect to a continuous trajectory of a random process was applied.