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

J. D. Deuschel, P. K. Friz, Antoine Jacquier () and S. Violante

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Abstract: 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 $l \leq d$. 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.

Date: 2011-11, Revised 2013-05
New Economics Papers: this item is included in nep-ets
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