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Marginal Density Expansions for Diffusions and Stochastic Volatility I: Theoretical Foundations
Author(s) -
Deuschel J. D.,
Friz P. K.,
Jacquier A.,
Violante S.
Publication year - 2014
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21478
Subject(s) - mathematics , stochastic volatility , brownian motion , heat kernel , geometric brownian motion , volatility (finance) , exponential function , stochastic discount factor , mathematical analysis , statistical physics , diffusion process , econometrics , statistics , economics , physics , economy , capital asset pricing model , service (business)
Density expansions for hypoelliptic diffusions ( X 1 ,…, X d ) are revisited. We are particularly interested in density expansions of the projection(X T 1 , ⋯ , X T l )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 [12][Deuschel, J., ].© 2013 Wiley Periodicals, Inc.