Fatou's Lemma for a discounted compound poisson process

Question

consider a discounted poisson process $X$ with (i.i.d.) jumps in $L^2(\mathbf{P})$, denoted by $Y$, so

$Y:=e^{-rt}X_t$,

and a sequence of stopping times $T_n$ converging to another stopping time $T$ from below, such that $T_n<T$ on $\left\{T>0\right\}$ and $\lim_{n\rightarrow\infty}T_n=T$. (In other words, foretelling the predictable stopping time $T$).

I have a paper at hand, where they apply Fatou's Lemma to get

$\limsup_{n\rightarrow\infty}\mathbf{E}[Y_{T_n}]\leq \mathbf{E}[Y_{T-}]$,

but why can you apply Fatou's Lemma here?

Answer 1

$Y$ Is uniformely integrable due to the discounting. Consider

$Y^n:=Y\wedge n$.

It is bounded and converges to $Y$ from below. Additionally, for all stopping times $S$, we have $\lim_{n} {E}[Y^n_S]<\infty$ by u.i. Thus, we get

$\limsup_m {E}[Y_{S_m}] = \limsup_m \lim_{n} [Y^n_{S_m}] \leq \lim_{n} {E}[Y^n_{S_m-}] = {E}[Y_{S-}]$

I think, often in definitions, the boundedness ist just used to ensure integrability, which one can get other ways as well.