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For positivty, I think it will also depend on the boundary condition. The idea is briefly stated as following. Let say we have a boundary condition $f(S_T,T) = min\{S_T - K,0\}$, $f(b,0) = 0$ and $f(a,t) = 0$. By Feynman-Kac Theorem, the solution of the original PDE can be written as: $f = e^{-rT}\mathbb{E}[min\{S_T-k,0\}]$ , where \$dS_t = rS_tdt + \sigma ...