Chaos expansions: applications to a generalized eigenvalue problem for the Malliavin derivative

Fundamental equations with higher order Malliavin operators

“…We present the methodology of chaos expansions on Equation (1), which is the most representative to get familiar with its idea. We also correct the estimate obtained in [9] for the domain of the Skorokhod integral in Theorem 2.8. The first main result of the paper is the proof of the existence and uniqueness of a solution of Equation (2), which will be provided in Theorems 4.1 and 4.5.…”

“…The following theorem estimates the domain and range of the Skorokhod integral in a more precise manner than provided in [9] where a minor error occurred. Proof.…”

“…From the assumption h [ X^S 2p ðRÞ^ðSÞ 21;2q , for some p $ 0, q . p þ 1, it follows that h is integrable in the Skorokhod sense and its integral dðhÞ is of the form (9). Note that the assumption q .…”

“…The domain of the Malliavin derivative is given by The Skorokhod integral, as an extension of the Itô integral for non-adapted processes, can be regarded as the adjoint operator of the Malliavin derivative in ðLÞ 2 -sense. In [9] we have extended the definition of the Skorokhod integral from Hilbert space-valued processes to the class of S 0 -valued generalized processes.…”

“…In [1,3,14,15,19,24] the Malliavin derivative and the Skorokhod integral are defined on a subspace of ðLÞ 2 so that the resulting process after application of these operators always remains in ðLÞ 2 . In [9,10,12] we allowed values in ðSÞ 21 and thus obtained a larger domain for all operators.…”