Good morning,
I have the following problem. I have a stochastic process $(D_t)_{t\geq T}$ (the starting point $T$ is not relevant) adapted to a filtration $(\mathcal{F}_t)_{t\geq T}$ and such that
$\mathbb{E}[D(t+1)|\mathcal{F}_t]=D(t)\left(1-\frac{1}{t^2}\right)+\frac{t+1}{t^2}$.
I know that $\mathbb{E}[|D(t)|]$ is finite for all $t$. I need to find a sub/super-martingale defined in terms of $D(t)$ which allows me to conclude that $D(t)/\log(t)\rightarrow X$ almost surely. Do you have any idea? Thanks in advance.