Let $X_n\sim \text{Bernoulli}(\frac{1}{2})$ with state space $\{-1,1\}$ i.i.d for all $n$. Set $S_n=X_0+X_1+...+X_n$.
Suppose that $X_0=2$ a.s. then $\{S_n\}$ has independent increment $\text{Bernoulli}(p,n)$.
I know that $$\int_0^3S_ndS_n=S_0(S_1-S_0)+S_1(S_2-S_1)+S_2(S_3-S_2).$$
Financially speaking, $S_n$ is the quantity of asset at time $n$ or at time $n+1$ ? I'm a bit confuse...
My attempts
Let $n$ denote the days.
At $n=0$ I buy $X_0=2$ assets that have price $S_0$. Now, suppose that the price change at midnight and until midnight I can't buy or sale any asset.
At $n=1$ at midnight the price of the asset is now $S_1$. So, I have now $S_0$ asset at $S_1$ dollars. My profit is therefore $$S_1S_0-S_0S_0=S_0(S_1-S_0)=\int_0^1 S_ndS_n,$$
Now I buy $X_1$ asset at price $S_1$, and thus you have $S_0+X_1=S_1$ assets that have price $S_1$ dollars. I can't buy nor sale any asset until midnight.
At $n=2$ at midnight, the price of the assets is now $S_2$, and thus, I have $S_1$ assets at $S_2$ dollars. Therefore, my profit since the day $1$ is $$-X_1S_1+S_2S_1.$$
Question : But it should be $$-S_1S_1+S_2S_1=(S_2-S_1)S_1=\int_1^2 S_ndS_n,$$ but I really don't understand how to get that.
At the very end I get $$-S_0S_0+S_0S_1-X_1S_1+S_2S_1$$ instead of $$S_0S_0+S_0S_1-S_1S_1+S_2S_1,$$ and I really don't understand how to get this previous formula. Could someone help ?
$S_n$ is the value of the asset on day $n$. On day $0$, the asset has value $S_0=2$. That midnight, the price changes by $X_1$ to $S_0+X_1=S_1$, so $S_1$ is the price on day $1$.
Since we are doing $\int_0^n S_n\,dS_n$, we are also assuming that $S_n$ is the amount of the asset you own after buying shares on day $n$. You start by buying $S_0=2$ units on day $0$, then during the day on day $1$ you buy $X_1$ shares, so you own $S_0+X_1=S_1$ of the asset.
After buying $X_1$ assets on day $1$, you have $S_0+X_1=S_1$ assets at price $S_1$.
On day $2$, after the price change at midnight, you have $S_1$ assets at price $S_2$.
Therefore, your overnight profit is $$ -S_1\times S_1+S_1\times S_2=S_1(S_2-S_1). $$