= Solution
Write $T=t_0$, $B_t=e^{-\rho(T-t)}$, and let $V_t=f(S_t,t)$ with $f(x,t)=xg(x,t)+B_th(x,t)$. The <portfolio> holds $g(S_t,t)$ shares of <stock> and $h(S_t,t)$ units of the <zero-coupon bond>. It is <self-financing> when all changes of holdings are funded within it, which is the gain identity
$$
dV_t=g(S_t,t)\,dS_t+h(S_t,t)\,dB_t.
$$
Under the physical <probability measure>, write $dS_t=\mu S_tdt+\sigma S_tdW_t$, where $\sigma>0$ and $dB_t=\rho B_tdt$. The <Itô formula> gives
$$
df(S_t,t)=\left(f_t+\mu S_tf_x+\frac12\sigma^2S_t^2f_{xx}\right)dt
+\sigma S_tf_x\,dW_t.
$$
Matching the <Brownian motion> coefficients forces $f_x=g$. Since $f_x=g+xg_x+B_th_x$, this is precisely
$$
xg_x+B_th_x=0.
$$
Differentiating $f_x=g$ gives $f_{xx}=g_x$. Since $f_t=xg_t+\rho B_th+B_th_t$, matching the <drift> coefficients now gives
$$
xg_t+B_th_t+\frac12\sigma^2x^2g_x=0.
$$
These are the <self-financing conditions for smooth stock and bond holdings>. Conversely, substituting both identities into the <Itô formula> gives the gain identity, so they are sufficient as well. Smoothness and the positive <lognormal distribution> density of $S_t$ extend the identities along the price process to every $x>0$, with boundary times obtained by continuity.
The <portfolio with constant stock value in bond units> must have $xg(x,t)=\theta B_t$, so $g(x,t)=\theta B_t/x$. The first condition then gives $h_x=\theta/x$, hence
$$
h(x,t)=\theta\log(x/S_0)+c(t).
$$
The second condition reduces to $c'(t)=\theta(\sigma^2/2-\rho)$. Initial <portfolio wealth> requires $h(S_0,0)=w_0/B_0-\theta=w_0e^{\rho T}-\theta$. Therefore the holdings are
$$
\boxed{g(x,t)=\frac{\theta e^{-\rho(T-t)}}x,\qquad
h(x,t)=w_0e^{\rho T}-\theta+\theta\log(x/S_0)+\theta(\sigma^2/2-\rho)t.}
$$
The corresponding <portfolio wealth> is
$$
\boxed{V_t=e^{-\rho(T-t)}\left[w_0e^{\rho T}+\theta\log(S_t/S_0)+\theta(\sigma^2/2-\rho)t\right].}
$$
Under the <risk-neutral measure>, $\log(S_t/S_0)=(\rho-\sigma^2/2)t+\sigma W_t^Q$, so its value in <zero-coupon bond> units is $V_t/B_t=w_0e^{\rho T}+\theta\sigma W_t^Q$. This also verifies the <self-financing> gain equation after using the bond as <numéraire>.
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