Problem $2.$

You're an entry level employee managing a financial portfolio. Because

you're a total novice, your boss limits the portfolio to a single stock, and moreover,

each day you're only allowed to make one of three choices: do nothing, purchase

exactly one share, or sell all shares you currently own. You are scheduled to manage

the portfolio for $n$ days, and for each day i the price of the stock is estimated to

be $p_i.$ Assume these estimations are perfect. On any given day i the value of your

portfolio equals

$($#shares you own$)$ amount spent to buy shares