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

Abstract

Density expansions for hypoelliptic diffusions (X1,...,Xd) are revisited. In particular, we are interested in density expansions of the projection (X1T,...,XlT), at time T>0, with l≤d. Global conditions are found which replace the well-known "not-in-cutlocus" condition known from heat-kernel asymptotics. Our small noise expansion allows for a "second order" exponential factor. As application, new light is shed on the Takanobu--Watanabe expansion of Brownian motion and Levy's stochastic area. Further applications include tail and implied volatility asymptotics in some stochastic volatility models, discussed in a compagnion paper.