Pointwise convergence for submartingale

Question

If $(\Omega,\mathscr{F},\mathbb{P})$ is a probability space, $(X_n,\mathscr{F}_n)_{n \in \mathbb{N}}$ is a submartingale such that exists a constant $c>0$ with $\left |{X_{n+1}-X_n}\right |<c$. For $\omega \in \Omega$ such that exists a constant $M \in \mathbb{N}$ with $X_n(\omega)<M$ for all $n$, can I say that $X_n(\omega)$ converges? I am trying to analyze question from the Martingale convergence theorem.

Answer 1

Let $T:=\inf\{n:X_n\ge M\}$ and use the stopped process $X^T_n:=X_{T\wedge n}$, a submartingale. You can check that $E[(X^T_n)^+]\le M^++c$ for all $n$, so the martingale convergence theorem applies to $X^T$. Finally, notice that $\{X_n<M$ for all $n\}\subset\{T=\infty\}$ and that $\lim_nX_n=\lim_nX^T_n$ on $\{T=\infty\}$.