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].

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