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Put ${\mathcal{L}}_{r}V(s)=\frac{1}{2}{\sigma }^2{s}^2{V}^{\prime \prime }(s)+rs{V}^{\prime }(s)$ and define the nonnegative reserve rate $a(s)=rV(s)-{\mathcal{L}}_{r}V(s)$. The obstacle problem gives $V\ge g$ and $a\ge 0$. The American-option superhedge with a funded reserve invests the local surplus in the bond rather than consuming it.

For initial wealth $x\ge V(S_0)$, setand choose the stock and bond holdingsThus $D_t\ge 0$ and $F_t\ge V(S_t)\ge g(S_t)$, pathwise at every time. The Itô formula under the original drift givesAdding these equations yieldsThis is a self-financing strategy, and its nonnegative wealth makes it an admissible trading strategy. Continuity of the stock and local regularity of $V$ ensure local integrability of the holdings. The construction does not require $\mu =r$.

For a classical solution, the usual Itô formula applies directly. The smooth fit solution below is $C^1$ and piecewise $C^2$, with locally absolutely continuous first derivative. The generalized Itô formula applies with its almost-everywhere second derivative; the absence of a derivative jump means no boundary local time of a semimartingale term. This is the usual regularity interpretation of the perpetual American option obstacle equation.

Normalize $B_0=1$ and put ${\tilde{S}}_t=e^{-rt}{S}_t$. Its dynamics are $d{\tilde{S}}_t={\tilde{S}}_t\sigma (t,S_t)d{W}_t$. Bounded volatility makes this stochastic exponential a true martingale on each finite horizon, by the Novikov condition. It also gives a finite second moment: stopping the Itô formula for $S^2$ and applying the Gronwall inequality yields $\mathbb{E}{S}_T^2\le S_0^2{e}^{(2r+L^2)T}$ when $\sigma \le L$.

The discounted payoff $\xi =e^{-rT}(S_T-K{)}^{+}$ is thus square-integrable. Define the nonnegative martingaleThe Brownian martingale representation theorem says that every square-integrable martingale in the Brownian filtration has a representation $M_t=M_0+{\int }_0^t{h}_{u}d{W}_u$ with predictable $h$ and $\mathbb{E}{\int }_0^T{h}_u^{2}du<\infty$. Since $\tilde{S}>0$ and $\sigma >0$, chooseThen discounted gains satisfy $d{M}_t={\pi }_{t}d{\tilde{S}}_t$. Consequently $X_t=B_t{M}_t={\varphi }_t{B}_t+{\pi }_t{S}_t$ is self-financing, nonnegative, and hence an admissible trading strategy. At maturity $X_T=(S_T-K{)}^{+}$, proving claim replication at cost $C(T,K)$.

For minimality, discounted wealth of any admissible self-financing portfolio is a local martingale bounded below, hence a supermartingale by localization and the conditional Fatou lemma. Thus any such replication with initial wealth $x$ obeysTogether with the constructed portfolio, this provesThis is Brownian representation replication in a local volatility market. The given drift $r$ means that the original probability measure already serves as the risk-neutral measure.