The following pictures comes from "Probability with Martingales" which denotes a stochastic integral(discrete):
$$Y:=H\cdot X$$
Here $H$ is previsible.According to the gambling strategy ,$H=0$ in the white balls and $H=1$ in the black balls.
I wonder why $H_n(X_n-X_{n-1})$ is the profits you get at time $n$.
Here is my understanding:
$X_n$ is the price of a stock.At time $n$,if you didn't buy before you can decide whether or not to buy one share ,if you already had one share ,you can decide whether or not to hold it.(right?)
According to he textbook ,at time $t_4$(in the picture),the profits is $X_4-X_3$.
But at time $t_3$,we decide not to buy one share because the value of $t_0,t_1,t_2$ is higher than $a$.
At time $t_4$ we decide to buy one share because the value of $t_3$ is lower than $a$.
At time $t_5$ we decide to hold the stocks because the value of $t_4$ is lower than $b$
So at $t_4$ we don't make profits and at $t_5$ make profits $X_5-X_4$
So the profit at time $n$ is $H_{n-1}(X_n-X_{n-1}) $
contradict with the conclusion in the textbook and I can not get the upcrossing lemma now.Could you tell me where is wrong ?