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 ; 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) .
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 .
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; cf. G. Ben Arous (1988).
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Applications
As application, new light is shed on the Takanobu--Watanabe expansion of Brownian motion and Levy's stochastic area. Further applications
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correlated
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; in particular, we solve a problem left open by A. Gulisashvili and E.M. Stein (2009)
, discussed in a compagnion paper