Wald lemma and Brownian motion

Question

Consider the following version of Wald lemma for Brownian motion.

How can we prove the last inequality: $$E[X_k|\mathcal{F}_{k-1}] \leq M(4^{k-1})+E[\max_{0 \leq r \leq 4^k-4^{k-1}}B_r]-2^{k+1}?$$

Answer 1

First, observe that for every $t \in [0,4^k]$, we have

$$B(t) \le M(4^{k-1})+[\max_{0 \leq s \leq 4^k-4^{k-1}} B_{4^{k-1}+s}-B_{4^{k-1}} ] \,.$$ (Consider separately the cases $t \le 4^{k-1}$ and $t \in [4^{k-1},4^k]$.) In other words, $$M(4^k) \le M(4^{k-1})+ [\max_{0 \leq s \leq 4^k-4^{k-1}} B_{4^{k-1}+s}-B_{4^{k-1}} ] \quad (*) \,.$$

Recall that the Markov property ensures that the process $ \{B_{4^{k-1}+s}-B_{4^{k-1}}\}_{s \ge 0}$ is independent of $\mathcal{F}^+(4^{k-1})$, and has the same law as $ \{B_s\}_{s \ge 0}$. Thus, if $\mathcal{F}_{k-1}$ is (less than ideal) shorthand for $\mathcal{F}^+(4^{k-1})$, taking conditional expectation in $(*)$ yields that

$$E[M(4^k)|\mathcal{F}_{k-1}] \leq M(4^{k-1})+E[\max_{0 \leq r \leq 4^k-4^{k-1}}B_r] \,.$$