= Reverse SLE derivative martingale
{c}
{title2=$M_s$}
Fix $\kappa>0$ and a deterministic terminal time $T$. The reversed driver $U_{T-s}-U_T$ has the law of $\sqrt\kappa B_s$ in its reversed-increment <filtration>. For the <Reverse Loewner flow>, put $Z_s=X_s+iY_s=h_s(z)-\sqrt\kappa B_s$, $q_s=X_s/Y_s$ and $J_s=|h_s'(z)|$. Direct <differentiation> and the <Itô formula> give
$$
dq_s=-\frac{4q_s}{Y_s^2(1+q_s^2)}\,ds-\frac{\sqrt\kappa}{Y_s}\,dB_s,
\qquad d\log J_s=\frac{2(q_s^2-1)}{Y_s^2(1+q_s^2)^2}\,ds.
$$
Consequently
$$
M_s=J_s^2(1+q_s^2)^{4/\kappa},\qquad
\frac{dM_s}{M_s}=-\frac{8q_s}{\sqrt\kappa Y_s(1+q_s^2)}\,dB_s.
$$
The drift cancellation includes the <quadratic variation> correction. Thus $M$ is a positive <local martingale>, hence a <supermartingale>; a true <martingale> assertion needs additional <integrability>. This yields useful <expected value> bounds for the inverse-map <derivative> without assuming the estimate itself is integrable in advance. The exponent $4/\kappa$ excludes the deterministic case $\kappa=0$.
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