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Marginal density expansions for diffusions and stochastic volatility

Please always quote using this URN:urn:nbn:de:0296-matheon-11534
  • Density expansions for hypoelliptic diffusions (X1^,...,X^d� ) are revisited. In particular, we are interested in density expansions of the projection (X^1_T,...,X^l_T) 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; cf. G. Ben Arous (88). 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).

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Metadaten
Author:Jean-Dominique Deuschel, Peter Friz, Antoine Jacquier, Sean Violante
URN:urn:nbn:de:0296-matheon-11534
Referee:Peter Karl Friz
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2012/07/17
Release Date:2012/07/17
Tag:Laplace method on Wiener space; generalized density expansions in small noise and small time; large strike and small time asymptotics for implied volatility; stochastic volatility; sub-Riemannian geometry with drift
Institute:Technische Universität Berlin
MSC-Classification:53-XX DIFFERENTIAL GEOMETRY (For differential topology, see 57Rxx. For foundational questions of differentiable manifolds, see 58Axx) / 53Bxx Local differential geometry / 53B21 Methods of Riemannian geometry
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Hxx Stochastic analysis [See also 58J65] / 60H07 Stochastic calculus of variations and the Malliavin calculus
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Hxx Stochastic analysis [See also 58J65] / 60H10 Stochastic ordinary differential equations [See also 34F05]
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Hxx Stochastic analysis [See also 58J65] / 60H30 Applications of stochastic analysis (to PDE, etc.)
60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Hxx Stochastic analysis [See also 58J65] / 60H35 Computational methods for stochastic equations [See also 65C30]
Preprint Number:971
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