The Clark–ocone Formula for Vector Valued Random Variables in Abstract Wiener Space

Erratum to " The Clark-Ocone formula for vector valued random variables in abstract Wiener space " , Jour. In this paper we considered the extension of the Clark-Ocone formula for a random variable defined on an abstract Wiener space (W, H, µ) and taking values in a Banach space (denoted there either B or Y). The main result appears in Theorem 3.4. Unfortunately, as first pointed out to us by J. Maas and J. Van Neerven, the dual predictable projection Π Π introduced in Definition 3.1(iii) via the characterization (3.1), does not define a random operator in L 2 (µ; L(H, Y)) as claimed, but rather an element of the larger space L(H, L 2 (µ, Y)). Consequently the right hand side of (3.6) in the main result is ill defined. We have been unable to overcome this difficulty in a meaningful way. We should point out that a Clark-Ocone formula was recently obtained in [3] for random variables on a classical cylindrical Wiener space taking values in a UMD Banach space, in which δ can be explicitly definedà la Itô on adapted processes. Our work, however, was different in spirit and made use of the extended version of δ introduced in [1]. While it is possible to provide an even weaker interpretation of (3.6) in which δ is extended to suitable elements of L(H, L 2 (µ; Y)), the result would have amounted to little more than the collection of classical Clark-Ocone formulae for the scalar random variables {v, The main result, Theorem 3.4, is thus considerably weakened; it remains true a) assuming that Y * * has the Radon Nikodym property (RNP) with respect to µ, and b) for Y-valued random variables v for which one can verify that Π Π∇v ∈ L(H, L 2 (µ; Y)). (The need for the additional RNP condition a) derives from an error, also brought to our attention by J. Maas and J. Van Neerven, in the proof of Proposition 3.14 of [1], (cited here as Lemma 2.3) which has been corrected in [2] under the RNP condition). Section 4 is not affected by the difficulties described above. Erratum: The divergence of Banach space valued random variables on Wiener space " , Prob. Abstract The classical representation of random variables as the Itô integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped …