Prove Proposition 6.18. Proposition 6.18 Let c = (c)=0,1T and c = (c)=0.1...T be two consumption processes

Prove Proposition 6.18.

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Proposition 6.18 Let c = (c)=0,1T and c = (c)=0.1...T be two consumption processes financed by the strategies 0 = (8)=0.1 and 8 = (0)=0.1...T. respectively, and such that P(c = c) = 1 for all t = 0, 1,..., T. Then the following hold: (i) if there are no arbitrage opportunities of the second kind, then W(0) = W(0); (ii) if there are no arbitrage opportunities of the first kind, then W,(0) = W,(0) with probability one, for all t = 1,..., T. For any consumption process c = (c)=0.1... + (x), the initial wealth x represents the cost at t = 0 of financing the consumption stream. More generally, for a self-financing trading-consumption strategy (0, c) = (xo), the quantity W,(e) represents the cost (or market value) at time t of the future consumption stream (Cs), for t {0, 1,..., T), i.e., the amount of wealth at time t that allows to finance the future consumption stream. In this sense, the Law of One Price (Proposition 6.18) implies that if two consumption processes can be financed by trading in the available securities and deliver identical payment streams, then they are financed by the same wealth. The absence of arbitrage opportunities also implies that the wealth associated to a self-financing trading-consumption strategy must always be greater (or equal) than the current consumption, as shown in the following lemma (see Exercises 6.16 and 6.17 for the proofs).