Let \(\left(B_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a one-dimensional Brownian motion, \(T<\infty\) and \(\Pi_{n}, n \geqslant 1\), a sequence of partitions of \([0, T]: 0=t_{n, 0}

converge in \(L^{2}(\mathbb{P})\) and find the limit \(L_{T}(\alpha)\). For which \(\alpha\) is \(L_{t}(\alpha), 0 \leqslant t \leqslant T \mathrm{a}\) martingale?