$MN-\langle M,N \rangle$ is uniformly integrable when M, N are $H^2$?

Question

This proof below really seems to have little to do with uniform integrability, what does the DCT application give exactly?

Any alternative proofs would be good too

Answer 1

The calculation in the proof shows that

$$\begin{align*} X_t := |M_t N_t - \langle M,N \rangle_t| &\leq \sup_{s \geq 0} |M_s| \cdot \sup_{s \geq 0} |N_s| + \sqrt{\langle M \rangle_{\infty}} \sqrt{\langle N \rangle_{\infty}} =: X \in L^1. \end{align*}$$

Consequently,

$$\int_{X_t \geq R} X_t \, d\mathbb{P} \leq \int_{X \geq R} X \, d\mathbb{P}.$$

Since this implies

$$\sup_{t \geq 0} \int_{X_t \geq R} X_t \, d\mathbb{P} \leq \int_{X \geq R} X \, d\mathbb{P},$$

the claim follows by letting $R \to \infty$ (using dominated convergence at the right-hand side).