Gronwall lemma with conditional expectation

Question

The discrete Gronwall's inequality states that if $x_n$ and and $u_n$ are non-negative sequences such that
$$ x_{n+1}\le a+\sum_{k=0}^n u_k x_k$$ then $$x_n\le a\prod_{k=0}^{n-1} (1+u_k)$$ (It can be shown using a simple induction as in https://math.stackexchange.com/questions/325565/gronwalls-lemma-discrete-version)

For the analysis of a random process, I came with an inequality that is similar to the one above but with a conditional expectation instead of an almost-sure inequality.

More precisely, let $X_n$ and $U_n$ be two discrete time stochastic processes adapted to a filtration $F_n$ such that $X_n>0$, $U_n\in(0,1)$, and $$ E[X_{n+1} | F_n] \le a - \sum_{k=0}^n U_k X_k $$ Let $T$ be a stopping time that is almost surely bounded. I am wondering if the following inequality holds: $$E[X_T]\le a E[\prod_{k=0}^{T-1}(1-U_k)]$$

Is it a known result? I have the impression that the induction used in the deterministic case does not work. If it helps, I can assume that $U_k$ is $F_{k-1}$ measurable for all $k$.

Answer 1

I just realized that the assertion that I was trying to prove is false because of the minus sign. In this case, it is not true even for deterministic sequences.

If I replace the minus with a plus : $$E[X_{n+1}|F_n]\le a+\sum_{k=0}^nU_kX_k$$ then the proof used in https://arxiv.org/pdf/1601.07503v1.pdf can be adapted to show that $$ E\left[\frac{X_T}{\prod_{k=0}^{T-1}(1+U_k)}\right] \le a $$ (the idea of the proof is to set $M_{n+1}=X_{n+1}-E[X_{n+1}|F_n]$ which is a martingale difference sequence. Then following https://arxiv.org/pdf/1601.07503v1.pdf, one obtain Equation (9) of https://arxiv.org/pdf/1601.07503v1.pdf with a $L_n$ that is martingale difference sequence). I will try to provide more details later.