Prove Corollary 6.7. Corollary 6.7 Let u(x) = log(x) in problem (6.6) and denote by (W*)=0.... the

Prove Corollary 6.7.

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Corollary 6.7 Let u(x) = log(x) in problem (6.6) and denote by (W*)=0.... the corresponding optimal wealth process and by (c)=0.1... the optimal consumption process, characterized by equation (6.23). Then, for any self-financing trading- consumption strategy (0, c) E (xo) such that CT > 0, it holds that W+1| F= for all t = 0, 1,..., T-1, W-ct where (W)=0..... is the wealth process generated by (0,c). In particular, in view of the above corollary, if one considers the simpler problem of maximizing the expected utility from consumption at the terminal date T only (i.e., without intermediate consumption), then an analogous reasoning allows to show that the optimal portfolio process (W)=0.1 is such that, for any portfolio process (W)=0.17, the ratio (W/W)=0.1 is a martingale. In view of this important property, the portfolio process (W)=0.1.7 is also called the numraire portfolio.