The exponential space of an \(L^ 2\)-stochastic process with independent increments (Q1104636)

An isometry of the exponential space (Fock space) associated with a Poisson random measure with \(\sigma\)-finite control measure to a specified subspace of \(L^ 2\) is constructed. The same result using multiple Poisson integrals was earlier proved by \textit{D. Surgailis} [Probab. Math. Stat. 3, 217-239 (1982; Zbl 0548.60058)]. The present proof exploits a discrete martingale technique. As a consequence of this result, an isometry of the exponential space associated with an \(L^ 2\)-stochastic process with independent increments to a specified subspace of \(L^ 2\) is also constructed.