A characterization of operators on functionals of discrete-time normal martingales

ABSTRACTIn this article, we aim at characterizing operators acting on functionals of discrete-time normal martingales. Let be a discrete-time normal martingale that has the chaotic representation property. We first introduce a transform, called 2D-Fock transform, for operators from the testing functional space to the generalized functional space of M. Then we characterize continuous linear operators from to via their 2D-Fock transforms. Our characterization theorems show that there exists a one-to-one correspondence between continuous linear operators from to and functions on Γ × Γ that only satisfy some type of growth condition, where Γ designates the finite power set of . Finally, we give some applications of our characterization theorems.