Is a martingale with bounded variance therefore bounded in $L^2$?

Question

If a martingale $W_n$ has bounded variance, does this mean that $W_n$ is automatically bounded in $L^2$? I feel like this ought to be obvious but I don't see how to prove it and I haven't been able to find any references.

Thank you!

Answer 1

Using the martingale property,

$$ \begin{align} \text{E}(W_n) &= \text{E}[\text{E}(W_n | W_{n - 1})] \\ &= \text{E}(W_{n - 1}) \\ &= ... \\ &= \text{E}(W_1) , \end{align} $$

and so you have a uniform bound on both $\text{Var}(W_n)$ and $\text{E}(W_n)^2$.