 Delta replication from a local-volatility pricing equation

Let a sufficiently smooth $V$ satisfy the pricing equation for a local volatility model, and assume its proposed holdings are stochastically integrable. Hold $h_t^S=V_S(t,S_t)$ stocks and $h_t^B=[V(t,S_t)-S_t{V}_S(t,S_t)]/B_t$ units of the bank account. Their value is $V(t,S_t)$. The Itô formula and the pricing equation giveThus the holdings form a self-financing strategy with terminal value $g(S_T)$, rather than merely a decomposition of today's value. This is a delta hedge. If $V\ge 0$, the replicating strategy has nonnegative wealth and satisfies the standard admissible trading strategy lower-bound condition. The stock and bank wealth amounts are $S_t{V}_S$ and $V-S_t{V}_S$; they differ from the numbers of asset units held.