Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Find a polynomial \(\pi(t, x)\) in \(x\) and \(t\), which is of order 4 in the variable \(x\), such that \(\pi\left(t, B_{t}ight)\) is a martingale.

One possibility is to use the exponential Wald identity \(\mathbb{E} \exp \left(\xi B_{\tau}-\frac{1}{2} \xi^{2} \tauight)=1,-1 \leqslant \xi \leqslant 1\), for a suitable stopping time \(\tau\) and a power-series expansion of the left-hand side in \(\xi\).