The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations

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 Hölder 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.

Keywords: local martingales, local stochastic solutions, degenerate Cauchy problems, Feynman-Kac formula, necessary and sufficient condition for uniqueness, comparison principle

References in corpus (1)

• Bubbles, convexity and the Black--Scholes equation

Cited by in corpus (2)

• Partial Information in a Mean-Variance Portfolio Selection Game
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