Can a process be shifted by some $cn$ and it retains martingale? Martingale linearity?
Particularly the problem in my case is that:
My process is defined by $M_n=Y_n-cn$, $c>0$, where $Y_n=f(X_1,...,X_n)$ is known as sub-martingale process.
My $F_n = \sigma(X_1,...,X_n)$.
By linearity of conditional expectation:
$$\mathbb{E}[Y_n|F_n]-\mathbb{E}[cn|F_n]$$
Clearly this also doesn't violate measurability.
I can estimate this down:
$$\geq Y_{n-1}-\mathbb{E}[cn|F_n]$$
But what then?
I've checked:
I'm not sure whether I can say that $cn$ depends or doesn't depend on $F_n$, since I don't have information about $X_i$ except for $\mathbb{E}[X_1]=0$, they are i.i.d., bounded
I'm not sure if I'm allowed to do something like "let $cn=\text{ some r.v. I know }$".
It's possible to say: $cn\mathbb{E}[1|F_n]$, but I'm not sure if this is any simpler.