Prove Proposition 6.6. =0.1....7 Proposition 6.6 Let u(x) = log(x) in problem (6.6) and denote by (W)=0.1

Prove Proposition 6.6.

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=0.1....7 Proposition 6.6 Let u(x) = log(x) in problem (6.6) and denote by (W)=0.1 the corresponding optimal wealth process. Then, the value function satisfies V(W,t) = f(t) log(W) +, where the function f : {0, 1,..., T). for all t = 0, 1,....T. T-8 Ris explicitly given by f(1) = 1-87-1+1 (V)=0.1 is an adapted process defined recursively by r = 0 and (6.22) and 8f(t+1) log f(t) f(t) 0g (10-1)-8 logf(t) 8f (t + 1) +8E[+1], N +Elog +1-1)+1jF n=1 for all t = 0, 1,..., T-1, where (w)=1T denotes the optimal portfolio process. Moreover, the optimal consumption process (c)=0.1.... satisfies W* f(t) = 1-8 1-87-1+1 W. for all t 0, 1,..., T. (6.23) and the optimal portfolio process (w)=1...IT is such that N -rf)+rf 1+1) = 0. (6.24)