= Solution
{c}
The diffusion associated with the displayed pricing equation is $d\widetilde S_s=0.04\widetilde S_sds+0.2\widetilde S_s dW_s$. For $Y_s=e^{-0.05s}V(s,\widetilde S_s)$, <Itô's formula> gives
$$
dY_s=e^{-0.05s}\bigl(V_t+0.04\widetilde S_sV_S+0.02\widetilde S_s^2V_{SS}-0.05V\bigr)ds
+e^{-0.05s}0.2\widetilde S_sV_S\,dW_s.
$$
The drift vanishes by the PDE. Under the usual integrability conditions for the payoff and solution, localization and conditional expectation make $Y$ a true martingale; for polynomial-growth data the lognormal moment bounds provide the needed integrability. Applying the terminal condition yields the <Feynman-Kac formula>
$$
\boxed{V(t,S)=e^{-0.05(T-t)}\mathbb E[\Psi(\widetilde S_T)\mid\widetilde S_t=S].}
$$
The printed diffusion contains an additional undefined factor $\sigma$ multiplying $0.2\widetilde S_s dW_s$. If literally retained, its second-order generator coefficient would be $0.02\sigma^2S^2$, so the requested representation would correspond to that different PDE unless $\sigma^2=1$. The formula above uses the volatility $0.2$ specified by the stock model and by part (c).
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