= Solution
Assume condition (1). Given an adapted cash-flow process $X_1,\ldots,X_T$, construct the holdings backwards. Set $H_{T+1}=0$. Once $H_{t+1}$ is known, the random variable
$$
X_t+H_{t+1}\cdot P_t
$$
is $\mathcal F_t$-measurable. Condition (1) supplies an $\mathcal F_{t-1}$-measurable $H_t$ satisfying
$$
H_t\cdot(P_t+\delta_t)
=X_t+H_{t+1}\cdot P_t.
$$
Hence $\xi_t^H=X_t$ for every $t\leq T$, and setting later holdings to zero proves condition (2).
Conversely, let $X_T$ be any $\mathcal F_T$-measurable random variable and apply condition (2) to the adapted process with cash flow $X_T$ at $T$ and zero cash flow earlier. Since $H_{T+1}=0$,
$$
X_T=\xi_T^H=H_T\cdot(P_T+\delta_T),
$$
where $H_T$ is $\mathcal F_{T-1}$-measurable. This is condition (1). The conditions are therefore equivalent and describe <market completeness>.
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