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ABSTRACT
In 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.
MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgments
The authors are extremely grateful to the referees for their valuable comments and suggestions on improvement of the first version of the present paper.
Funding
This work is supported by National Natural Science Foundation of China (Grant No. 11461061).