= Solution
Let $\theta_t$ and $\eta_t$ be the <predictable> holdings of the <stock> and the <bank account>. With no <consumption>, the <self-financing portfolio> has $X=\theta S+\eta B$ and $dX=\theta dS+\eta dB$. The <Itô product rule>, together with the dynamics from part (a), gives
$$
\begin{aligned}
d(X_tY_t)
&=\bigl(Y_t\theta_tS_t\mu_t+Y_t\eta_tB_tr_t-r_tX_tY_t-Y_t\theta_tS_t\sigma_t\lambda_t\bigr)dt\\
&\quad+Y_t(\theta_tS_t\sigma_t-X_t\lambda_t)\,dW_t\\
&=Y_t(\theta_tS_t\sigma_t-X_t\lambda_t)\,dW_t.
\end{aligned}
$$
The <drift> vanishes because $X=\theta S+\eta B$ and $\sigma\lambda=\mu-r$. Thus $XY$ is a <local martingale>. It is nonnegative by the assumed nonnegative wealth and strict positivity of $Y$, and hence
$$
\boxed{XY\ \text{is a supermartingale}.}
$$
For example, this last fact follows directly from <Conditional Fatou lemma> applied to a <localizing sequence>; the initial capital is assumed finite. As usual, holdings must be integrable against the asset <semimartingales> for the <self-financing portfolio> equation to be defined.
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