An identity, namely: $Y_{t}dX_{t}=dY_{t} \Rightarrow d[Y,Y]_{t}=Y^{2}d[X,X]_{t}$ for continuous semimartingale X, is used rather frequently. I understand that $Y_{t}=e^{X_{t}-X_{0}}$ is the solution for initial value $X_{0}$ but cannot see what can be done from here to prove it directly.
2026-03-28 14:56:52.1774709812
Proof that $Y_{t}dX_{t}=dY_{t} \Rightarrow d[Y,Y]_{t}=Y^{2}d[X,X]_{t}$ for continuous semimartingale X?
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