For Problems 1 and 2 below, we place ourselves within the familiarBinomial Model dis - cussed in detail in class , with Z 1 , Z2, ... independent copies of the Bernoulli variable Z with P[Z = u] = p.; P[Z = d] = Pa (where 0 0, 0 0 is the initial stock -price and K. := _ _1(z,=4) is the number of "up-moves" by day t. The likelihood-ratio and state - price -density processes are then dQ t - K. It : = dP Pd and He := Lt . (1 + R) , respectively, where F. := o(21, ..., Z) is the information accumulated up to day t. Problem # 1: Suppose you start with initial capital w > 0, you are given T days to "play" in this market, and you are told to try and maximize the "expected rate of return" E 1 105 (V) from investment, over all self-financing portfoliosh with P [v* > 0] = 1. How would you go about finding an optimal strategy h* ? What is the corresponding optimal portfolio - value - process V.', t = 0, 1, ...T? (Hint : You may wish to recall ES [VPIF] = - . EP [LTVFIF ]