A martingale $(X_t)_{t\geqslant 0}$ which is bounded in $L^{1}$ but not uniformly integrable.

Question

Is there any examples of the martingale $X(t)$ which is bounded in $L^{1}$ but not uniformly integrable?

I've just known that $$M_t=\mathrm{e}^{aW_t-a^2t/2}$$ with $a$ not $0$ and $W$ a Brownian Motion is a martingale but not uniformly integrable.

Is there any other examples which are more concrete and more constructive?

Thanks for your help!

Answer 1

There are many examples

1. Show that $(W_t)_{t \geq 0}$ and $(W_t^2-t)_{t \geq 0}$ are not uniformly integrable

2. from Martingale not uniformly integrable

Let $\xi_j$ be i.i.d. with $(\forall j \in \mathbb{N}) \mbox{} \mathbb{P}(\xi_j=0)=\mathbb{P}(\xi_j=2)=1/2$, and define $M_n=\prod_{j=1}^n \xi_j$ for $n\geq 0$. Then $(M_n)$ is a non-negative martingale with $\mathbb{E}(M_n)=1$ for all $n$. But $M_n\to 0$ almost surely, so $(M_n)$ cannot be uniformly integrable.

3. Proof of a martingale not being uniformly integrable

4.From Uniformly Integrable Martingale The $(Y_{n})_{n\in\mathbb{N}}$ is a seq. of positive, independent r.v.'s whose expectation is 1 $\forall n$ satisfying $$\prod_{k=1}^{\infty}\mathbb{E}(\sqrt{Y_{k}})=0$$. Then $\xi_{n}=\prod_{k=1}^{n}Y_{k}$ is a $\mathcal{F_{n}}$-martingale that is not uniformly-integrable.