Hilbert transform of G-Brownian local time

Abstract

Let B be a G-Brownian motion with quadratic process 〈B〉 under the G-expectation. In this paper, we consider the integrals [Formula: see text] We show that the integral diverges and the convergence [Formula: see text] exists in 𝕃2 for all a ∈ ℝ, t > 0. This shows that [Formula: see text] coincides with the Hilbert transform of the local time [Formula: see text] of G-Brownian motion B for every t. The functional is a natural extension to classical cases. As a natural result we get a sublinear version of Yamada's formula [Formula: see text] where the integral is the Itô integral under the G-expectation.