Continuous stochastic logarithm of planar Brownian motion ($Z_t=Z_0\exp ({\int }_0^t{Z}_s^{-1}d{Z}_s)$)

Planar Brownian motion started away from zero almost surely avoids zero. On every finite time interval its continuous path therefore has positive minimum modulus, making the logarithmic stochastic integral well-defined. Since complex quadratic covariation satisfies $d[Z,Z]=0$, analytic Itô calculus gives the displayed exponential identity without a drift correction. Its real part is the change in logarithmic radius and its imaginary part is the continuous accumulated argument, including every winding. This is a pathwise logarithmic lift, not a single-valued logarithm on the entire punctured plane.