Marginal density expansions for diffusions and stochastic volatility, Part II: Applications - Deuschel,JD; Friz,PK; Jaquier,A; Violante,S (Communications on Pure and Applied Mathematics, 2013)
Density expansions for hypoelliptic diffusions $(X^1,...,X^d)$ are revisited. In particular, we are interested in density expansions of the projection $(X_T^1,...,X_T^l)$, 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; cf. G. Ben Arous (1988). Our small noise expansion allows for a "second order" exponential factor. Applications include tail and implied volatility asymptotics in some correlated stochastic volatility models; in particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009).