Let
$$
h(y)=g'(g^{-1}(y)).
$$
Differentiating and using the scale equation gives
$$
h'(y)=\frac{g''(x)}{g'(x)}=-2b(x),
\qquad x=g^{-1}(y).
$$
Since $b$ is bounded, $h$ has global <Lipschitz continuity>. Thus
$$
dY_t=h(Y_t)dW_t
$$
has a pathwise unique strong solution by the standard Lipschitz existence-and-uniqueness theorem for a <stochastic differential equation>. Applying the deterministic inverse $g^{-1}$ gives a strong solution $X$, and uniqueness of $Y$ gives pathwise uniqueness of $X$.

Solution

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

 Solution

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