# Uniform Integrability of a Martingale up to time T

I am trying to prove that a certain martingale $$(R_t)_{t\geq 0}$$ is uniformly integrable over a finite time interval $$[0,T]$$.

Now I know that the definition of uniform integrability is that $$\lim_{a\to\infty}\sup_{t\geq 0}\mathbb{E}[|R_t|\mathbb{I}_{|R_t|>a}]=0$$ where $$\mathbb{I}$$ is the indicator function.

I know also that in general showing $$\sup_{t}\mathbb{E}{|R_t|}<\infty$$ is not sufficient to show $$R$$ is uniformly integrable (but $$\sup_t \mathbb{E}|R_t|^2 < \infty$$ is sufficient) but what about in the case of a bounded time interval $$[0,T]$$. Is $$\mathbb{E}[\sup_{t\leq T} |R_t|]<\infty$$ sufficient to show uniform integrability on $$[0,T]$$; i.e. that $$\lim_{a\to\infty}\sup_{t\leq T}\mathbb{E}[|R_t|\mathbb{I}_{|R_t|>a}]=0$$?

It feels intuively like the reason $$\sup_t\mathbb{E}|R_t|<\infty$$ fails as a condition is because in some vague sense: "when $$t$$ is very large $$|R_t|$$ has a much larger chance of being >a" but I don't know how I would go about proving something like this?

• If $R_t \to R_T$ in $L^1$-norm then it's equivalent to the fact that the family $R_t$ converges in probability to $R_T$ and is Uniformly Integrable. This is sometimes referred to as Vitali's theorem unless mistaken en.wikipedia.org/wiki/Vitali_convergence_theorem (I think you can find proofs of this fact on MSE). Your question seems involved to me, so if you can show $L^1$ cv in your case then you don't need to go through the complex argument you are looking for. May 23 at 9:35

If you have a (deterministic) time interval $$[0,T]$$, and a positive martingale $$(R_t)_{0 \le t \le T}$$ in some filtration $$(\mathcal{F}_t)_{0 \le t \le T}$$, then for every $$t \in [0,T]$$, $$R_t = E[R_T|\mathcal{F}_t]$$. This forces uniform integrability, since given any integrable random variable $$Z$$, and any family of sub-$$\sigma$$ fields $$(\mathcal G_i)_{i \in I}$$, the family of conditional expectations $$(E[Z|\mathcal G_i])_{i \in I}$$ is uniformly integrable.