 Reverse SLE derivative martingale

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 $Z_s=X_s+i{Y}_s=h_s(z)-\sqrt{\kappa }{B}_s$, $q_s=X_s/Y_s$ and $J_s=|h_s^{\prime }(z)|$. Direct differentiation and the Itô formula giveConsequentlyThe 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 $\kappa =0$.