Consider a stock $S,$ whose price at time $n$ is given by $S_n.$ Denote by $\{F_n\}$ the filtration

generated by $\{S_n\}.$ An investor adjusts his holding of the stock dynamically according to his

stratigic function $f.$ More precisely, at time $0,$ he holds $f(S_0)$ shares of the stock. At time $1,$

depending on the stock price $S_1,$ he adjusts his shares of the stock from $f(S_0)$ to $f(S_1).$ At

time $2,$ depending on the stock price $S_2,$ he adjusts his shares of the stock from $f(S_1)$ to $f(S_2),$

At time $3,$dots. Denote by $I_n$ the cumulative gain of this investor on the stock market. Then,

we have

$I_n={\displaystyle \underset{k=1}{\overset{n}{}}}f(S_{k-1})(S_k-S_{k-1}).$

If the price of the stock is an $F_n-$martingale, prove that the gain process $I_n$ is also an $F_{n^{-}}$

martingale. This tells us: one is not able to make money on average, if the underlying asset

does not have an increasing or decreasing trend, even if he$/$she uses a very excellent trading

stategy $f.$