Precise asymptotics of certain Wiener functionals (Q1178826)

An asymptotic expansion is given for a class of Wiener functionals as the variance \(s\) of the underlying Wiener measure tends to zero. The asymptotic expansion is obtained via a stochastic analogue of Taylors formula. In order to prove this formula, the authors develop a time- dependent version of the Malliavin calculus based upon the heat (as opposed to the Ornstein-Uhlenbeck) operator \(\Delta-\partial/\partial t\), and introduce an associated regularity criterion, which they term \(P\)- uniform regularity. Their asymptotic expansion is derived for the class of \(P\)-uniformly regular Wiener functionals \(F\). Assuming the usual condition that (their analogue) of the Malliavin covariance matrix \(A\) corresponding to \(F\) has \((\hbox{det} A)^{-1}\in L^ p\), for all \(p\), they obtain an asymptotic expansion for the density (with respect to Lebesgue measure) of \(F\) as \(s\) tends to zero. In the final section, they show that the solution map of a general stochastic integral equation with history-dependent coefficient functions is \(P\)-uniformly regular.