Prove that the projection on the (sigma)-algebra (mathcal{F}_{g_{t}}^{+})of the (mathbf{F}) martingale (left(B_{t}^{2}-t, t geq 0 ight)) is

Prove that the projection on the \(\sigma\)-algebra \(\mathcal{F}_{g_{t}}^{+}\)of the \(\mathbf{F}\) martingale \(\left(B_{t}^{2}-t, t \geq 0\right)\) is \(2\left(t-g_{t}\right)-t\), hence the process

\[\mu_{t}^{2}-(t / 2)=(t / 2)-g_{t}\]

is an \(\left(\mathcal{F}_{g_{t}}^{+}\right)\)-martingale.