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

Density expansions for hypoelliptic diffusions $(X^1,\mathrm{.}\mathrm{.}\mathrm{.},X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,\mathrm{.}\mathrm{.}\mathrm{.},X_T^l)$, at time $T>0$, with $l\le d$. 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.