Regularity of a degenerate parabolic equation appearing in Vecer's unified pricing of Asian options

Vecer derived a degenerate parabolic equation with a boundary condition characterizing the price of Asian options with generally sampled average. It is well understood that there exists a unique probabilistic solution to such a problem but it remained unclear whether the probabilistic solution is a classical solution. We prove that the probabilistic solutions to Vecer's PDE are regular.