Let \(X=M+A\) be the Doob-Meyer decomposition of a strictly positive continuous sub-martingale. Let \(Y\) be the solution of

\[d Y_{t}=Y_{t} \frac{1}{X_{t}} d M_{t}, Y_{0}=X_{0}\]

and let \(Z\) be the solution of \(d Z_{t}=-Z_{t} \frac{1}{X_{t}} d A_{t}, Z_{0}=1\). Prove that \(U=Y / Z\) satisfies \(d U_{t}=U_{t} \frac{1}{X_{t}} d X_{t}\) and deduce that \(U=X\).