Brownian representation replication in a local volatility market (${\pi }_t=h_t/({\tilde{S}}_t\sigma (t,S_t))$)

For a square-integrable discounted claim martingale $dM=hdW$ and discounted stock $d\tilde{S}=\tilde{S}\sigma dW$, choose stock holdings $h/(\tilde{S}\sigma )$. The remaining wealth lies in the bond. A nonnegative claim makes this replication admissible; discounted admissible wealth is a supermartingale, establishing the minimal initial cost.