= SLE interior-point martingale
{c}
{title2=$M_t$}
For $\kappa>0$, $\rho>0$, let $J_t=|g_t'(z)|$, $S_t=\sin\theta_t$ and $\Upsilon_t=Y_t/J_t$. Then
$$
M_t=J_t^{(8-\kappa+\rho)\rho/(4\kappa)}\Upsilon_t^{\rho(\rho+8)/(8\kappa)}S_t^{-\rho/\kappa}
$$
is a positive continuous <local martingale> before the <Loewner swallowing time>. Indeed the <Itô formula> gives $d\log J_t=-2(X_t^2-Y_t^2)|Z_t|^{-4}dt$, $d\log\Upsilon_t=-4Y_t^2|Z_t|^{-4}dt$, and
$$
d\log S_t=\frac{(\kappa/2-4)X_t^2-\kappa Y_t^2/2}{|Z_t|^4}\,dt+\sqrt\kappa\frac{X_t}{|Z_t|^2}\,dB_t.
$$
Combining the exponents cancels the drift of $M_t$, including the quadratic-variation correction, leaving $dM_t/M_t=-\rho X_t/(\sqrt\kappa|Z_t|^2)\,dB_t$.
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