Delta replication from a local-volatility pricing equation ($h^S=V_S,h^B=(V-S{V}_S)/B$)

If $V$ satisfies the pricing equation for a local volatility model, 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 pricing equation for a local volatility model and Itô formula giveThus this is a self-financing strategy, with terminal value $g(S_T)$. It is a delta hedge, and nonnegative $V$ makes its wealth an admissible trading strategy. The stock and bank wealth amounts are $S_t{V}_S$ and $V-S_t{V}_S$, rather than the asset-unit holdings.