Let us consider an adapted sequence $Y_1, Y_2, ...$, and let $c_n \in \mathbb{R}, n \ge1$
Suppose that:
a) $E(Y_{n+1}|\mathcal{F}_n)=Y_n+c_n$
b) $E(Y_{n+1}|\mathcal{F}_n)=Y_n*c_n$
I need to exabith a martingale from a) and b) through a suitable compensation.
I have many doubts about this exercise. Firstly, how can I assume integrability for $Y_n$? Second, I have tried to start from the general proof of compansation of a process to get a martingale but I do not think it is the right path to follow since I get stucked after the first step. How can I reach my conlusion just from this information given?
The hypothesis makes sense only if $Y_n$'s have finite mean (so that conditional expectations are defined). For a) choose a sequence $a_n$ such that $a_{n+1} -a_n=c_n$; for example, $a_1=c_1,a_2=c_1+c_2,\cdots$. Then $\{Y_n-a_n\}$ is a martingale. b) is similar. If $b_n$ is a sequence such that $b_{n+1}=c_nb_n$ then $\{b_nY_n\}$ is a martingale.