Why are solutions to the Loewner equation analytic? Consider the Loewner differential equation
$$\frac{\partial g_t(z)}{\partial t} = \frac{2}{g_t(z) - \sqrt{\kappa} B_t}$$
where $B_t$ is a 1-dimensional standard Brownian motion starting at the origin. The solutions $g_t(z)$ should give a family of conformal maps wherever they are defined. I don't see why a solution $g_t(z)$ has to be analytic in $z$. I'm guessing that something like Cauchy-Kowalevski has to be used somehow, but to apply such theorems, we require the function $2/(g - \sqrt{\kappa} B_t)$ to be analytic, which it is not.


The fact that $g_t(z)$ is analytic in $z$ is clear, since the right-hand side is analytic in $g_t(z)$.

(Rohde and Schramm, Basic properties of SLE). 
Goluzin's book presents a self-contained treatment of the deterministic (radial) Loewner equation with piecewise continuous driving term $k(s)$. He actually works with the inverse map and proves its analyticity, along with existence and uniqueness, by following the proof of the Picard existence theorem. I think for the chordal process this would work too: let $g_{0}(z,t)=z$, and for $n\ge 1$
$$g_{n}(z,t) = z+\int_0^t \frac{2}{g_{n-1}(z,s)-k(s)}\,ds$$  Then each $g_n(\cdot, t)$ is holomorphic in $z$ on its domain, and $g_n(\cdot, t)\to g(\cdot, t)$ uniformly on compact subsets of the domain of $g(\cdot,t)$.
