Define the <scale function of a one-dimensional diffusion>
$$
g(x)=\int_0^x\exp\left(-2\int_0^yb(z)dz\right)dy.
$$
Then $g'>0$, so $g$ is strictly increasing, and
$$
\frac12g''+bg'=0.
$$
By <Itô formula>,
$$
dY_t=d(g(X_t))=g'(X_t)dW_t,
$$
so $Y=g(X)$ is a <local martingale>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 5 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-202/5/b/solution