Let \((S, D)\) be a couple of price-dividend processes not admitting arbitrage opportunities and let \(\mathbb{P}^{*}\) be a risk neutral probability measure (which exists by Theorem 6.23). As in Exercise 6.2, define the discounted quantities \(\bar{S}_{t}:=S_{t} / r_{f}^{t}\) and \(\bar{D}_{t}:=D_{t} / r_{f}^{t}\), for all \(t=0,1, \ldots, T\). Similarly, for any trading-consumption strategy \((\theta, c)\), define the discounted portfolio value \(\bar{W}_{t}(\theta):=W_{t}(\theta) / r_{f}^{t}\) and the discounted consumption \(\bar{c}_{t}:=c_{t} / r_{f}^{t}\), for all \(t=0,1, \ldots, T\). Prove that, for every \(x \in \mathbb{R}_{+}\), a non-negative adapted process \(c=\left(c_{t}\right)_{t=0,1, \ldots, T}\) belongs to \(\mathscr{C}_{0}^{+}(x)\) if and only if the following two requirements are satisfied:

(i) \(\bar{c}_{T}=x+\sum_{t=1}^{T} \theta_{t}^{\top}\left(\Delta \bar{S}_{t}+\bar{D}_{t}\right)-\sum_{t=0}^{T-1} \bar{c}_{t}\), for some predictable process \(\left(\theta_{t}\right)_{t=0,1, \ldots, T}\)

(ii) \(\sum_{t=0}^{T} \mathbb{E}^{*}\left[\bar{c}_{t}\right]=x\).