Let $Z_t=g_t(z)-U_t$. For $\operatorname{SLE}_4$, $dU_t=2dB_t$, and the <Chordal Loewner equation> gives
$$
dZ_t=\frac2{Z_t}dt-2dB_t,
\qquad d\langle Z\rangle_t=4dt.
$$
The complex <Itô formula> yields
$$
d\log Z_t
=\frac1{Z_t}dZ_t-\frac1{2Z_t^2}d\langle Z\rangle_t
=-\frac2{Z_t}dB_t.
$$
Therefore
$$
\boxed{\log(g_t(z)-U_t)\text{ is a continuous local martingale}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 3 / c / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-203/3/c/i/solution