(Solving the generalized geometric Brownian motion equation). Let $S(t)$ be a positive stochastic process that satisfies the generalized geometric Brownian motion differential equation (see Example 4.4.8)
$$
d S(t)=\alpha(t) S(t) d t+\sigma(t) S(t) d W(t),
$$
where $\alpha(t)$ and $\sigma(t)$ are processes adapted to the filtration $\mathcal{F}(t), t \geq 0$, associated with the Brownian motion $W(t), t \geq 0$. In this exercise, we show that $S(t)$ must be given by formula (4.4.26) (i.e., that formula provides the only solution to the stochastic differential equation (4.10.2)). In the process, we provide a method for solving this equation.
(i) Using (4.10.2) and the Itô-Doeblin formula, compute $d \log S(t)$. Simplify so that you have a formula for $d \log S(t)$ that does not involve $S(t)$.
(ii) Integrate the formula you obtained in (i), and then exponentiate the answer to obtain (4.4.26).