The problem is the following (Problem 3.10 from Karatzas/Shreve):

> 
> Let $Z\_t = \exp \left(\int\_0^t X\_u \, dW\_u-\frac{1}{2}\int\_0^t X^2\_u \, du \right)$. Define $Y\_t=1/Z\_t$. Show the following stochastic differential equation holds
> \begin{align\*}
> dY\_t=Y\_tX\_t^2\,dt-Y\_tX\_t\,dW\_t, \quad Y\_0=1
> \end{align\*}
> where we have assumed $\int\_0^t X\_u^2 \, du<\infty$ almost surely for all $0<t<\infty$, and $W\_t$ is the standard Brownian motion.
> 
> 
> 

My questions are:

1. How can we prove $Y\_t$ is a semi-martingale?
2. For the differential equation, this is my try (which I assume $Y\_t$ is a semi-martingale

\begin{align\*}
Y\_tZ\_t=1 \ \Longrightarrow Z\_t\,dY\_t + Y\_t\,dZ\_t + d\langle Y\_t, Z\_t \rangle = 0
\end{align\*}
Since $Y\_tZ\_t=1$, we have $\langle Y\_t, Z\_t\rangle = 0$. Also, from the textbook, it is shown $dZ\_t= Z\_tX\_t \,dW\_t$
So from the above we have
\begin{align\*}
Y\_tZ\_t \,dY\_t = -Y\_t^2 \, dZ\_t \ \Longleftrightarrow\ dY\_t = -Y\_t^2\,dZ\_t = -Y\_tX\_t\,dW\_t
\end{align\*}
since $dZ\_t = Z\_tX\_t\,dW\_t$ and $Z\_tY\_t=1$. However, it seems the $dt$ term is missing.... I can't really see what is wrong in the calculation though. Does anyone have any comments?