Summary: The general area of mathematical finance presents some interesting and challenging problems of \textit{stochastic optimal control}, which are typically of two distinct kinds, namely problems of \textit{mean-square minimization} and problems of \textit{utility maximization}. Often these optimal control problems are not especially well suited to direct application of the more ``traditional'' methods of optimal control, such as dynamic programming and the maximum principle. On the other hand, these problems enjoy the very nice properties of being \textit{convex}, with a ``state space'' which is essentially a vector space of \textit{scalar} (rather than vector) valued random variables. These special properties are key to the application of the general method of \textit{conjugate duality} as a tool to characterize and compute optimal trading strategies (i.e. the ``optimal controls'' in a financial context). A stochastic calculus of variations of J-M Bismut and a variational method of R.T. Rockafellar are the most significant implementations of the general principle of conjugate duality for convex problems of optimal control. In this work we shall demonstrate how these apply to stochastic optimal control problems which arise in mathematical finance, paying particular attention to the variational method of Rockafellar.