  We present an analytic approach to solve a degenerate parabolic problem
associated to the Heston model, which is widely used in mathematical finance to
derive the price of an European option on an risky asset with stochastic
volatility. We give a variational formulation, involving weighted Sobolev
spaces, of the second order degenerate elliptic operator of the parabolic PDE.
We use this approach to prove, under appropriate assumptions on some involved
unknown parameters, the existence and uniqueness of weak solutions to the
parabolic problem on unbounded subdomains of the half-plane.
