Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 2 / b / ii / Solution

The relation between $B$ and $W$ is the linear equation $d{B}_s=d{W}_s-\beta B_{s}ds$ with $B_0=0$. Multiplication by the deterministic integrating factor $e^{\beta s}$ and the Itô product rule giveHenceThe second equality follows by semimartingale integration by parts. It expresses $B$ as an explicit function of the continuous path of $W$, using only an ordinary time integral. This inversion holds before changing measure; $W$ is already a continuous semimartingale under the original measure.