Apply <Itô formula> to $\phi(x)=x\log x$. Since $\phi''(x)=1/x$,
$$
M_1\log M_1-M_0\log M_0
=\int_0^1(1+\log M_t)\,dM_t
+\frac12\int_0^1\frac{d[M]_t}{M_t}.
$$
Boundedness of $\log M$ makes the stochastic integral a true martingale of mean zero. Taking expectations proves
$$
\boxed{\mathbb E(M_1\log M_1)=\mathbb E(M_0\log M_0)+\frac12\mathbb E\int_0^1\frac{d[M]_t}{M_t}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-202/3/a/solution