# Arbitrage price and American option

I'm studying American Options. If I have $$X=(X_n)$$ an American option, it is not possible to determine a self-financing predictable strategy ($$\alpha, \beta$$) that replicates the option in sense that $$V^{(\alpha,\beta)}_n=X_n$$ for every $$n=0,\ldots,N$$.

This is simply due to the fact $$\tilde{V}^{(\alpha,\beta)}$$ is a $$Q-$$martingale while $$X$$ is a generic adapted process.

Let be: $$\mathcal{A}^-_X=\{(\alpha,\beta) \in \mathcal{A} \mid V_N^{(\alpha,\beta)} \le X\}$$ the family of sub-replicating porfolios for the derivate $$X$$.

Why We can't define a strategy that sub-replicates $$X$$, for every $$n=0,\ldots,N$$?

Thanks.