  In this paper we consider a variation of the Merton's problem with added
stochastic volatility and finite time horizon. It is known that the
corresponding optimal control problem may be reduced to a linear parabolic
boundary problem under some assumptions on the underlying process and the
utility function. The resulting parabolic PDE is often quite difficult to
solve, even when it is linear. The present paper contributes to the pool of
explicit solutions for stochastic optimal control problems. Our main result is
the exact solution for optimal investment in Heston model.
