= Continuous stochastic logarithm of planar Brownian motion
{title2=$Z_t=Z_0\exp(\int_0^t Z_s^{-1}dZ_s)$}
<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.
Back to article page
