Orthogonal Functionals of Independent­Increment Processes

In analogy with the Wiener-Itô theory of multiple integrals and orthogonal polynominals, a set of functionals of general square-integrable martingales is presented which, in the case of independent-increments processes, is orthogonal and complete in the sense that every L^{2} -functional of the independent-increment process can be represented as an infinite sum of these elementary functionals. The functionals are iterated integrals of the basic martingales, similar to the multiple iterated integrals of Itô and can be also thought of as being the analogs of the powers 1,x,x^{2}, \cdots of the usual calculus. The analogy is made even clearer by observing that expanding the Doleans-Dade formula for the exponential of the process in a Taylor-like series leads again to the above elementary functionals. A recursive formula for these functionals in terms of the basic martingale and of lower order functionals is given, and several connections with the theory of reproducing kernel Hilbert spaces associated with independent-increment processes are obtained.