= Solution
Normalize $B_0=1$ and put $\widetilde S_t=e^{-rt}S_t$. Its dynamics are $d\widetilde S_t=\widetilde S_t\sigma(t,S_t)\,dW_t$. 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 $\mathbb E S_T^2\leq S_0^2e^{(2r+L^2)T}$ when $\sigma\leq L$.
The discounted payoff $\xi=e^{-rT}(S_T-K)^+$ is thus square-integrable. Define the nonnegative <martingale>
$$
M_t=\mathbb E[\xi\mid\mathcal F_t],\qquad M_0=C(T,K).
$$
The <Brownian martingale representation theorem> says that every square-integrable martingale in the Brownian filtration has a representation $M_t=M_0+\int_0^th_u\,dW_u$ with predictable $h$ and $\mathbb E\int_0^Th_u^2du<\infty$. Since $\widetilde S>0$ and $\sigma>0$, choose
$$
\boxed{\pi_t=\frac{h_t}{\widetilde S_t\sigma(t,S_t)},\qquad
\phi_t=M_t-\pi_t\widetilde S_t.}
$$
Then discounted gains satisfy $dM_t=\pi_t\,d\widetilde S_t$. Consequently $X_t=B_tM_t=\phi_tB_t+\pi_tS_t$ 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 $x$ obeys
$$
x\geq\mathbb E[e^{-rT}X_T]=C(T,K).
$$
Together with the constructed portfolio, this proves
$$
\boxed{\text{minimal admissible replication cost}=C(T,K).}
$$
This 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>.
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