$5\mathrm{.}4$

An individual faces the following consumption and portfolio choice problem:

$ma{x}_{C_t,{}_t,t}{E}_0[{\displaystyle \underset{t=0}{\overset{T-1}{}}}{}^{t}ln[C_t]+{}^{T}ln[W_T]]$

where each period the individual can choose between a risk$-$free asses paying a time$-$varing return of $R_{ft}$ over the period from $t$ to $t+1$ and a single risky asset. The individual receives no wage income. The risky asset's return over the period from $t$ to $t+1$ is given by

$R_{rt}=\{\begin{array}{ccc}(1+u_t)R_{ft}& \text{w}\text{i}\text{t}\text{h}\text{p}\text{r}\text{o}\text{b}\text{a}\text{b}\text{i}\text{l}\text{i}\text{t}\text{y}& \frac{1}{2}\\ (1+d_t)R_{ft}& \text{w}\text{i}\text{t}\text{h}\text{p}\text{r}\text{o}\text{b}\text{a}\text{b}\text{i}\text{l}\text{i}\text{t}\text{y}& \frac{1}{2}\end{array}$

where $u_t>0$ and ${}_{t}t{}_t^{**}t=0,$dots,$T-1-1.$ Let ${}_{t}be$ the individual's proportion $of$ wealth invested $in$ the risky asset $at$ date $t.$ Solve for the individual's optimal portfolio weight ${}_t^{**}$ for $t=0,$dots,$T-1.$