What is an upperbound for the Expectation of the supremum of a submartingale?

Question

Assuming that:

1. $(c_j)_{j\epsilon \mathbb{N}} \in l_2$
2. $B$ standard Brownian.

Why is this upper bound true?

$$\mathbb{E}\left[\sup_{n\in \mathbb{N}}\left|\sum^{n}_{j=0}c_j(B_{j+1}-B_j)\right|^2\right]\leq 4\sum^{\infty}_{j=0}|c_j|^2.$$

The supremum in the expectation makes it difficult to manipulate things. However, taking the supremum out or using Jensen's makes a lower bound rather than an upper bound.

Answer 1

First, by the monotone convergence theorem, it suffices to prove that for all $N\geqslant 1$, $$ \mathbb{E}\left [\max_{1\leqslant n\leqslant N}\left|\sum^{n}_{j=0}c_j(B_{j+1}-B_j)\right|^2\right]\leqslant 4 \sum^{\infty}_{j=0}|c_j|^2 . $$ To this aim, let $D_j:=c_j(B_{j+1}-B_j)$; then $(D_j)_{j\geqslant 0}$ is a martingale differences sequence. Use Doob's inequality to get that $$ \mathbb{E}\left [\max_{1\leqslant n\leqslant N}\left|\sum^{n}_{j=0}c_j(B_{j+1}-B_j)\right|^2\right]\leqslant 4\mathbb{E}\left [ \left|\sum^{N}_{j=0}c_j(B_{j+1}-B_j)\right|^2\right].$$ Then use the uncorrelatedness of $\left(D_j\right)_{j= 0}^n$.