Let $0<d<1<u$ and $\{Y_k\}$ be iid. random variables with $P(Y_k=u)=p=1-P(Y_k=d)$. Let $A_0>0$ and $A_n=A_0\sum_{k=1}^nY_k$, $F_n=\sigma\{Y_1,\dots,Y_n\}$. What are the weakest conditions for $A_n$ to be a (Sub-)martingale.
First I tried to check the 3 conditions for a submartingal:
i) $E(|A_n|)=E(A_0\prod_{k=1}^nY_k)=A_0E(Y_1)^n$ which Should be less than $\infty$. So the first condition would be $E(Y_1)<\infty$.
ii) $A_n$ is adapted seems trivial since it is just the product.
iii) $E(A_{n+1}|F_n)=A_0A_nE(A_{n+1}|F_n)$ should be $\geq A_n$, so I find the condition $A_0E(Y_{n+1}|F_n)\geq1$
In order for $A_n$ to be a martingale I find $E(Y_{n+1}|F_n)=1/A_0$. However, I cannot see how this conditions can just hold for all $n$. I am still pretty insecure in prob. theory.
There is a follow up question, too, which asks if there is a stopping time $T$ with $E(T)<\infty$ such that $E(A_T)>A_0$. I would appreciate hints for that, too. Thank you very much in advance.
Also, I have no idea where I have to use the assumption $P(Y_k=u)=p=1-P(Y_k=d)$.