Distribution-valued weak solutions to a parabolic problem arising in financial mathematics

We study distribution-valued solutions to a parabolic problem that arises from a model of the Black-Scholes equation in option pricing. We give a minor generalization of known existence and uniqueness results for solutions in bounded domains $Omega subset mathbb{R}^{n+1}$ to give existence of solutions for certain classes of distributions $fin mathcal{D}'(Omega)$. We also study growth conditions for smooth solutions of certain parabolic equations on $mathbb{R}^nimes (0,T)$ that have initial values in the space of distributions.