Let $X(t)$ be a martingale w.r.t. filtration generated by Brownian motion $B(t)$. There is a well-known theorem that states that there is a unique adapted process $H(t)$ such that $$ X(t) = \int_0^t H(s)dB(s).$$ Let's define step process $$M(t)=2 \sum_{j=1}^{\lfloor t \rfloor} 1_{\{ B(j)-B(j-1) > 0 \}} - \lfloor t \rfloor.$$ It's just a simple martingale, generated by fair coin tosses, just written in another way. One can see that $B(t)$ is the only source of uncertainty. What is $H(t)$ for $M(t)$ then? Or am I missing something and M.R.T. is not applicable here?