Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2009 / A / 2 / c / Solution

The solution is the continuous Doléans-Dade exponential:Here $[X]$ is the continuous quadratic variation starting from zero, denoted by the angle-bracket notation in the source. If $X=M+A$ with $A$ continuous of finite variation, then $[X]=[M]$.

Set $U=X-X_0-\frac{1}{2}[X]$. Its quadratic variation is still $[X]$. Applying Itô's formula to $e^U$ givesThe initial value is $S_0=1$. This verification applies to the full semimartingale, including its finite-variation part.