We obtain a maximum principle for stochastic control problem of general
controlled stochastic differential systems driven by fractional Brownian
motions (of Hurst parameter H>1/2). This maximum principle specifies a system
of equations that the optimal control must satisfy (necessary condition for the
optimal control). This system of equations consists of a backward stochastic
differential equation driven by both fractional Brownian motion and the
corresponding underlying standard Brownian motion. In addition to this backward
equation, the maximum principle also involves the Malliavin derivatives. Our
approach is to use conditioning and Malliavin calculus. To arrive at our
maximum principle we need to develop some new results of stochastic analysis of
the controlled systems driven by fractional Brownian motions via fractional
calculus. Our approach of conditioning and Malliavin calculus is also applied
to classical system driven by standard Brownian motion while the controller has
only partial information. As a straightforward consequence, the classical
maximum principle is also deduced in this more natural and simpler way.Comment: 44 page
