SLE interior-point martingale ($M_t$)

For $\kappa >0$, $\rho >0$, let $J_t=|g_t^{\prime }(z)|$, $S_t=\sin {\theta }_t$ and ${{\rm Y}}_t=Y_t/J_t$. Thenis a positive continuous local martingale before the Loewner swallowing time. Indeed the Itô formula gives $d\log J_t=-2(X_t^2-Y_t^2)|Z_t{|}^{-4}dt$, $d\log {{\rm Y}}_t=-4Y_t^2|Z_t{|}^{-4}dt$, andCombining the exponents cancels the drift of $M_t$, including the quadratic-variation correction, leaving $d{M}_t/M_t=-\rho X_t/(\sqrt{\kappa }|Z_t{|}^2)d{B}_t$.