# An example of non-uniform-integrable martingale.

[Lemma 4 of JPS07]. Let $\tau: \Omega: \to (0, \infty)$ be a random variable,  such that $\mathbb{P} (\tau> t)>0$ for all $t\in (0, \infty)$. Let $D_t = 1_{\{\tau>t\}}$. Let $\mathcal{D}_t = \sigma(D_s: 0\le s \le t)$ be a natural filtration generated by $D_t$.  Then,

1) $\tau$ is a stopping time w.r.t. $\mathcal{D}_t$.

2)$N_t = \frac{D_t}{\mathbb{P} (\tau>t)}$ is a non-uniform-integrable martingale with $N_\infty = 0$.

Proof. 1) It can be seen from $\{\tau \le t\} = \{D_t = 0\} \in \mathcal{D}_t$.

2)  For all bounded stopping time $T$, one can show a) $N_T \in L^1$; b) $\mathbb{E} [N_T] = N_0 = 1$. Thus,  by Proposition II.3.5 of [RY99], $N_t$ is a martingale. Since $N_\infty = 0 < N_0$, it is not a uniformly integrable martingale by Theorem II.3.1 of [RY99].