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\textbf{Question}
Assume that an asset $S$ has growth rate $\mu$, volatility $\sigma$
and pays a continuous dividend yield $q$ and that it evolves according
to the stochastic differential equation
$$\frac{dS}{S} = (\mu - q)dt + \sigma dX$$
where $dX$ is a Wiener process with the properties that
\begin{eqnarray*}
\textrm{\Large{$\varepsilon$}} (dX) & = & 0\\
\textrm{\Large{$\varepsilon$}} (dX^2) & = & dt\\
\lim_{dt\to 0} dX^2 = dt
\end{eqnarray*}
Give a heuristic derivation of It$\underline{\textrm{o}}$'s lemma for
a sufficiently differentiable function $V(S,t)$ which depends on both
$S$ and $t$.
Suppose that an option is written on this asset with the properties
that at expiry it is equal to the asset, and prior to its expiry it
pays out a known sum $K(S,t)dt$ during each time interval $(t,
t+dt)$. By constructing an instantaneously risk-free portfolio and
considering cash flows, show that it value $V$ must satisfy the
problem
$$\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2
\frac{\partial^2 V}{\partial S^2} + (r-q)S \frac{\partial V}{\partial
S} -rV = -K(S,t)$$
$$t