Kernel representation formula from complex to real Wiener-Ito integrals and vice versa

TL;DR

This paper establishes explicit formulas linking real and complex Wiener-Ito integrals, enabling conversion between the two representations using recursion and Malliavin calculus techniques.

Contribution

It provides a novel representation formula connecting real and complex Wiener-Ito integrals, with explicit kernel expressions and a finite sum representation.

Findings

Explicit kernel expressions for real and imaginary parts of complex Wiener-Ito integrals.

Representation formula for real Wiener-Ito integrals as sums of complex integrals.

Use of recursion and Malliavin derivatives to establish the connection.

Abstract

We clearly characterize the relation between real and complex Wiener-Ito integrals. Given a complex multiple Wiener-Ito integral, we get explicit expressions for two kernels of its real and imaginary parts. Conversely, consider a two-dimensional real Wiener-Ito integral, we obtain the representation formula by a finite sum of complex Wiener-Ito integrals. The main tools are a recursion technique and Malliavin derivative operators. We build a bridge between real and complex Wiener-Ito integrals.