The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations

TL;DR

This paper proves the existence and uniqueness of classical solutions to a degenerate parabolic equation from option pricing, relaxing regularity conditions and providing new probabilistic insights.

Contribution

It extends the regularity requirements for stochastic solutions in degenerate parabolic equations and offers new probabilistic proofs for martingality criteria.

Findings

Classical solutions exist under weaker Hölder continuity conditions.

Unicity established without linear growth assumptions.

Characterization of non-smooth solutions as limits of smooth approximations.

Abstract

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally H\"older continuous with exponent $\delta\in (0, 1]$, the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement $\delta\ge 1/2$. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.