Growth bounds for iterated integrals of L2-Itô random processes

Fliess operators with deterministic inputs have been studied since the late 1970s and are well understood. When the inputs are stochastic processes the theory is less developed. There have been several interesting approaches for Wiener process inputs. But applications such as the interconnection of systems is not well-posed in this context. This paper has two specific goals. The first goal is to show that a certain sum of iterated stochastic integrals of general L₂-Ito random processes can be represented in a compact form, which amounts to a variation of the well-known Hu-Meyer formula. This latter formula allows one to write an iterated Stratonovich integral in terms of a finite sum of iterated Ito integrals. The second goal is to derive an L₂ upper bound for these iterated integrals.