Marginal density expansions for diffusions and stochastic volatility I: Theoretical foundations

Abstract

Density expansions for hypoelliptic diffusions (X1,..,Xd) are revisited. We are particularly interested in density expansions of the projection ( XT1,..,XTl) at time T > 0 with l ≤ d. Global conditions are found that replace the well-known "not-in-cut-locus" condition known from heat kernel asymptotics. Our small-noise expansion allows for a "second order" exponential factor. As an application, new light is shed on the Takanobu-Watanabe expansion of Brownian motion and Lévy's stochastic area. Further applications include tail and implied volatility asymptotics in some stochastic volatility models, discussed in the companion paper. © 2013 Wiley Periodicals, Inc.