= Solution
Let $\pi_t$ be the <predictable> <stock> holding during $(t-1,t]$. Cash has constant price, so <self-financing> gives
$$
X_t-X_{t-1}=\pi_t(S_t-S_{t-1}),\qquad
X_t=\mathbb E[\xi\mid\mathcal F_t].
$$
Put $\mathcal G=\mathcal F_{t-1}$. The <tower property of conditional expectation> and $\mathcal F_t$-measurability of $S_t$ give
$$
\operatorname{Cov}(\xi,S_t\mid\mathcal G)
=\operatorname{Cov}(X_t,S_t\mid\mathcal G).
$$
Since $X_{t-1}$, $S_{t-1}$ and $\pi_t$ are $\mathcal G$-measurable, substituting the gains identity yields
$$
\operatorname{Cov}(X_t,S_t\mid\mathcal G)
=\pi_t\operatorname{Var}(S_t\mid\mathcal G).
$$
The denominator is positive on every positive-probability parent atom. If it were zero on such an atom, $S_t$ would be constant there, and the <martingale> property would force that constant to equal $S_{t-1}$, contradicting the nonzero-increment assumption. Hence
$$
\boxed{\pi_t=\frac{\operatorname{Cov}(\xi,S_t\mid\mathcal F_{t-1})}
{\operatorname{Var}(S_t\mid\mathcal F_{t-1})},\qquad1\leq t\leq T.}
$$
This is <conditional covariance hedge ratio>. It uses attainability; a regression coefficient alone would not replicate a general unattainable payoff. After maturity one may liquidate into cash, so the same formula gives zero for later dates wherever its denominator remains nonzero. On a finite sample space a <martingale> cannot have nonzero increments forever; the stated nondegeneracy is naturally a finite-maturity assumption.
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