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I have a two-asset Black-Scholes model:

$dB_t = B_t r dt$

$dS_t = S_t (\mu dt + \sigma dW_t)$

I introduce a European claim $\xi = \max(K,S_T)$ with maturity $T$, for some fixed $K$. I have calculated what the no-arbitrage price of this claim should be at each time $t<T$ by computing expectations under the equivalent martingale measure, which is a function of $S_t$, $t$, and the fixed parameters in the model. and I am now asked to find a replicating portfolio in the original 2 asset market for this claim.

I know that if $V(t,S)$ is a solution to the Black-Scholes PDE $\frac{\partial V}{\partial t} + rV \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} = rV$ subject to the terminal condition $V(T,S)=\max(K,S)$, then $V(t,S_t)$ is a no-arbitrage time-t price for $\xi$, and that the trading strategy given by taking initial wealth to be $V(0,S_0)$ and the time-t holding in the stock to be $\frac{\partial V}{\partial S}(t,S_t)$ is a replicating strategy for the claim.

My question is: If I view the pricing function I originally found (by computing expectations) as a function $\xi(t,S_t)$, is it necessarily true taking initial wealth to be $\xi(0,S_0)$ and taking time-t holding in the stock to be $\frac{\partial \xi}{\partial S}(t,S_t)$ will give a replicating portfolio? If this is the case, is there some way of proving this fact without using the PDE approach?

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