= Solution
Put $\mathcal L_rV(s)=\frac12\sigma^2s^2V''(s)+rsV'(s)$ and define the nonnegative reserve rate $a(s)=rV(s)-\mathcal L_rV(s)$. The <obstacle problem> gives $V\geq g$ and $a\geq0$. The <American-option superhedge with a funded reserve> invests the local surplus in the bond rather than consuming it.
For initial wealth $x\geq V(S_0)$, set
$$
D_t=e^{rt}\left(x-V(S_0)+\int_0^te^{-ru}a(S_u)\,du\right),\qquad F_t=V(S_t)+D_t,
$$
and choose the <stock> and bond holdings
$$
\boxed{\pi_t=V'(S_t),\qquad \phi_t=\frac{F_t-\pi_tS_t}{B_t}.}
$$
Thus $D_t\geq0$ and $F_t\geq V(S_t)\geq g(S_t)$, pathwise at every time. The <Itô formula> under the original drift gives
$$
dV(S_t)=V'(S_t)\,dS_t+\tfrac12\sigma^2S_t^2V''(S_t)\,dt,
\qquad dD_t=(rD_t+a(S_t))\,dt.
$$
Adding these equations yields
$$
\boxed{dF_t=\pi_t\,dS_t+r(F_t-\pi_tS_t)\,dt
=\pi_t\,dS_t+\phi_t\,dB_t.}
$$
This is a <self-financing strategy>, and its nonnegative wealth makes it an <admissible trading strategy>. Continuity of the <stock> and local regularity of $V$ ensure local integrability of the holdings. The construction does not require $\mu=r$.
For a classical solution, the usual <Itô formula> applies directly. The <smooth fit> solution below is $C^1$ and piecewise $C^2$, with locally absolutely continuous first derivative. The generalized <Itô formula> applies with its almost-everywhere second derivative; the absence of a derivative jump means no boundary <local time of a semimartingale> term. This is the usual regularity interpretation of the perpetual <American option> obstacle equation.
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