Orthogonal functionals of the Poisson process

• Authors: H. Ogura
• Venue: IEEE Transactions on Information Theory
• Year: 2006

In analogy to the orthogonal functionals of the Brownian-motion process developed by Wiener, ltô, and others, a theory of the orthogonal functionals of the Poisson process is presented making use of the concept of multivariate orthogonal polynomials. Following a brief discussion of Charlier polynomials of a single variable, multivariate Charlier polynomials are introduced. An explicit representation as well as an orthogonality property are given. A multiple stochastic integral of a multivariate function with respect to the Poisson process, called the multiple Poisson-Wiener integral, is defined using the multivariate Charlier polynomials. A multiple Poisson-Wiener integral, which gives a polynomial functional of the Poisson process, is orthogonal to any other of different degree. Several explicit forms are given for the sake of application. It is shown that any nonlinear functional of the Poisson process with finite variance can be developed in terms of these orthogonal functionals, corresponding to the Cameron-Martin theorem in the case of the Brownian-motion process. Finally, some possible applications to nonlinear problems associated with the Poisson process are briefly discussed.