The quadratic covariation of two continuous \(L^{2}\) martingales \(M, N \in \mathcal{M}_{T}^{2, c}\) is defined as in the discrete case by the polarization formula (15.8).

Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\) and \(f, g \in \mathcal{L}_{T}^{2}\). Show that \(\langle f \bullet B, g \bullet Bangle_{t}=\int_{0}^{t} f(s) g(s) d s\) for all \(t \in[0, T]\).

Data From Formula (15.8)

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(M, N) := ((M + N) - (M-N)) for all M, N e M. (15.8)