A Quadratic Brownian Martingale

$B_t^2$ itself grows in expectation, so subtract a deterministic compensator to make it flat.

Check that $E[B_t^2\mid {\mathcal{F}}_s]=B_s^2+(t-s)$ for $s<t$.

1. 1. Compute the conditional expectation

   For $s<t$, write $B_t=B_s+(B_t-B_s)$ and expand $B_t^2$; using independence of the increment, $E[B_t^2\mid {\mathcal{F}}_s]=B_s^2+(t-s)$.

2. 2. Remove the deterministic drift

   Subtracting $t$ compensates for the growing conditional mean, leaving a martingale.

   $$M_t=B_t^2-t,E[M_t\mid {\mathcal{F}}_s]=M_s$$

Answer: B_t^2 - t