Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 202 / 5 / b / Solution

Before the lifetime, $X>0$, so apply the Itô formula to $g(x)=\sqrt{x}$ after stopping inside compact subintervals of $(0,\infty )$. The derivatives are $g^{\prime }(x)=1/(2\sqrt{x})$ and $g^{\prime \prime }(x)=-1/(4x^{3/2})$. Since the quadratic variation of $X$ satisfies $d[X{]}_t=X_{t}dt$, we getThe negative drift is the second-order correction in the Itô formula. This is a Lamperti transform up to a constant scale: the square root transformation makes the noise coefficient constant. Its drift is singular at zero, which is why this stochastic differential equation is stated only before the boundary lifetime.