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First, planar Brownian motion started away from zero almost surely never hits zero. To see this without confusing it with neighborhood recurrence, stop in an annulus with inner radius $ϵ$ and fixed outer radius $R>|Z_0|$. The preceding logarithmic calculation gives probability $\log (R/|Z_0|)/\log (R/ϵ)$ of reaching the inner circle first. A hit of zero before the outer exit would require all these inner hits, so its probability is zero as $ϵ\downarrow 0$. On every finite path interval the continuous path is bounded; a countable union over integer outer radii and integer times proves the all-time assertion.

Consequently the integrand $1/Z$ is locally square integrable: on every compact time interval the nonvanishing continuous path has a positive minimum modulus. Write $Z=X+iY$ with independent standard real coordinates. Complex quadratic covariation satisfies $d[Z,Z]=dt-dt+2id[X,Y]=0$, so analytic Itô's lemma has no second-derivative drift. For $L=\int Z^{-1}dZ$ one likewise has $d[L,L]=0$, and $d[e^{-L},Z]=0$. The product rule therefore givesIts initial value is $Z_0$, proving, simultaneously for all times by continuity,Thus $\Re L_t=\log (|Z_t|/|Z_0|)$ and $\Im L_t$ is the continuous accumulated change of complex argument of the path about zero, measured in radians. It is a winding angle, without reduction modulo $2\pi$. A single-valued logarithm on the whole punctured plane is unnecessary; the path has a continuous stochastic logarithm of planar Brownian motion.