Stochastic integral as a spectral integral
by
Volodymyr Tesko
Institute of Mathematics, NAS of Ukraine

It is well known that the Itô stochastic integral of adapted square integrable processes plays a fundamental role in the stochastic calculus. In the case of non-random integrands such integral can be understood (roughly speaking) as an ordinary spectral integral applied to a certain vector from L²-space. The reason for this is that a square integrable martingale (integrator) can be regarded as a resolution of identity applied to above mentioned vector. At the same time, it is considerably more difficult to establish a relation between the Itô and spectral integrals for random integrands, since there is a problem in finding an explicit expression for the corresponding spectral integral. In this context a natural problem arises: to give a suitable definition of "spectral integral" which will generalize the classical Itô integral. In this talk we give a definition and derive some properties of such "spectral integral", and we show that it generalizes the classical Itô stochastic integral of adapted square integrable processes with respect to normal martingales and the Itô integral on a Fock space.

Date received: June 15, 2008