Let $a_1,a_2,\dots$ be random variables that are adapted to the filtration {$\mathcal{F}_{t}$ } such that $$\mathbb{E}[a_i \mid \mathcal{F}_{i-1}] \leq K$$ for all $i$. Then for any stopping time $\tau$, we have that
$$ \mathbb{E}\left[\sum\limits_{i=1}^{\tau}a_i\right] \leq \mathbb{E}[\tau] K. $$ See for instance the answer to this question.
Now suppose that the $a_i$ satisfy a growth bound: $$\mathbb{E}[a_{i} \mid \mathcal{F}_{i-1}] \leq a_{i-1} + K.$$ Intuitively, I want to reason that $a_i \approx i K$, and therefore $$\mathbb{E}\left[\sum\limits_{i=1}^{\tau}a_i\right] \leq \mathbb{E}[\tau^2] K.$$ Is there a way to formally establish this inequality, or is there some reason why it would not hold?