Itô's formula for noncommutative $C^2$ functions of free Itô processes

Abstract

In a recent paper, the author introduced a rich class $NC^k(\mathbb{R})$ of "noncommutative $C^k$" functions $\mathbb{R} \to \mathbb{C}$ whose operator functional calculus is $k$-times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In the present paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving an Itô formula for noncommutative $C^2$ functions of self-adjoint free Itô processes. To do this, we first extend P. Biane and R. Speicher's theory of free stochastic calculus -- including their free Itô formula for polynomials -- to allow free Itô processes driven by multiple freely independent semicircular Brownian motions. Then, in the self-adjoint case, we reinterpret the objects appearing in the free Itô formula for polynomials in terms of MOIs. This allows us to enlarge the class of functions for which one can formulate and prove a free Itô formula from the space originally considered by Biane and Speicher (Fourier transforms of complex measures with two finite moments) to the strictly larger space $NC^2(\mathbb{R})$. Along the way, we also obtain a useful "traced" Itô formula for arbitrary $C^2$ scalar functions of self-adjoint free Itô processes. Finally, as motivation, we study an Itô formula for $C^2$ scalar functions of $N \times N$ Hermitian matrix Itô processes.