Self-financing conditions for smooth stock and bond holdings
= Self-financing conditions for smooth stock and bond holdings
For $p(x,t)=xg(x,t)+B_th(x,t)$ and deterministic $dB_t=\rho B_tdt$, <Itô formula> shows that a <self-financing portfolio> must have $p_x=g$. This is equivalent to $xg_x+B_th_x=0$. Differentiating it gives $p_{xx}=g_x$, so the remaining gain identity is $xg_t+B_th_t+\sigma^2x^2g_x/2=0$. Conversely these two identities give $dp=g\,dS+h\,dB$. A value PDE alone does not verify arbitrary prescribed holdings.