Abstract: When a local supermartingale is bounded from below, is it a proper supermartingale?
Question: In remark 4.2 (p.16) of the lecture notes by Martin Hairer, the conditions when $$ [0,\infty)\ni \quad t\mapsto F(X_t)-\int^t_0G(X_s)ds $$ is supermartingale is concerned, where $F,G$ is continuous functions, $X$ is a càdlàg Markov process. Then he claims that the condition can be weakened to just requiring it to be a local supermartingale, when $F$ is a continuous function bounded from below.
↓ Could you please check if my proof draft below is correct? ↓
My Draft: It suffices to consider non-negative local supermartingale $Y_t$. Let $\{\tau_n\}_{n=1}^\infty$ be a reducing sequence, almost surely increasing, with $\lim_{n\to\infty}\tau_n=\infty$ a.s. Choose arbitrary $0\le s<t$. To establish $\operatorname{E}[Y_t|\mathcal{F}_s]\le Y_s$ a.s., first we notice that for any $n\in\mathbb{N}$, we have $\operatorname{E}[Y_{t\land\tau_n}|\mathcal{F}_s]\le Y_{s\land\tau_n}$ a.s. In particular,
$$ \operatorname{E}[Y_{t\land\tau_n}]\le\operatorname{E}[Y_{s\land\tau_n}]\le\operatorname{E}[Y_0]<\infty,\qquad0\le s\le t. $$
From these two inequalities, using Falou's lemma, we obtain $$ \operatorname{E}[Y_t]=\operatorname{E}\left[\liminf_{n\to\infty}Y_{t\land\tau_n}\right]\le\operatorname{E}[Y_0]<\infty,\qquad0\le t, $$ $$ \operatorname{E}[Y_t|\mathcal{F}_s]=\operatorname{E}\left[\liminf_{n\to\infty}Y_{t\land\tau_n}\,\middle|\,\mathcal{F}_s\right]\le\liminf_{n\to\infty}Y_{s\land\tau_n}=Y_s,\qquad0\le s\le t. $$
From the first inequality and the non-negativity of $Y$, we conclude $\operatorname{E}[|Y_t|]<\infty$ for any $0\le t$. Combining this with the second inequality, we conclude that $\{Y_t\}$ is actually a supermartingale.
Is my argument correct so far? Any help is appreciated. Thanks in advance.
P.S. This achieves generalization of the result of this MSE post.
