In this paper, using
purely Hilbert space-theoretic methods, an analogue of the Itô integral is
constructed in the symmetric Fock space of a direct integral H of Hilbert
spaces over the real line. The classical Itô integral is the special case when
H = L2[0,∞). An explicit formula is obtained for the projection onto the space
of ‘non-anticipating functionals’, which is then used to prove that simple
non-anticipating functionals are dense in the space of all non-anticipating
functionals. After defining the analogue of the Itô integral, its isometric
nature is established. Finally, the range of this ‘integral’ is identified; this
last result is essentially the Kunita-Watanabe theorem on square-integrable
martingales.
