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 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.

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