  We propose a novel framework to investigate lead-lag relationships between
two financial assets. Our framework bridges a gap between continuous-time
modeling based on Brownian motion and the existing wavelet methods for lead-lag
analysis based on discrete-time models and enables us to analyze the
multi-scale structure of lead-lag effects. We also present a statistical
methodology for the scale-by-scale analysis of lead-lag effects in the proposed
framework and develop an asymptotic theory applicable to a situation including
stochastic volatilities and irregular sampling. Finally, we report several
numerical experiments to demonstrate how our framework works in practice.
