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QUESTION

Find $u(W,t)$ and $V(W,t)$, where

$$dX=udt+vdW$$

and

\begin{description}

\item[(i)]
$X(t)=W(t)^2$

\item[(ii)]
$X(t)=1+t+\exp(W(t))$

\item[(iii)]
$X(t)=f(t)W(t),\ f$ bounded and continuous.

\end{description}


ANSWER

It\^{o}:

$$dx=\frac{\partial x}{\partial t}dt+\frac{\partial x}{\partial
w}dw+\frac{1}{2}\frac{\partial^2x}{\partial w^2}dt$$

\begin{description}

\item[(i)]
\begin{eqnarray*}
dx&=&0dt+2wdw+\frac{1}{2}2dt\\dx&=&dt+2wdw
\end{eqnarray*}

\item[(ii)]
\begin{eqnarray*}
dx&=&1dt+e^wdw+\frac{1}{2}e^wdt\\dw&=&\left(1+\frac{e^w}{2}\right)dt+e^wdw
\end{eqnarray*}

\item[(iii)]
\begin{eqnarray*}
dx&=&w\frac{df}{dt}dt+fdw+\frac{1}{2}0dt\\ dx&=&wf'dt+fdw
\end{eqnarray*}

\end{description}




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