Use the fact that any continuous, square-integrable martingale with bounded variation paths is constant (cf. Proposition 17.2) to show the following: \(\langle f \cdot Bangle_{t}:=\int_{0}^{t}|f(s)|^{2} d s\) is the unique continuous and increasing process such that \((f \cdot B)^{2}-f^{2} \cdot\langle Bangle\) is a martingale.

Data From Proposition 17.2

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17.2 Proposition. Let (X+, F)to be a martingale with paths which are continuous and of bounded variation on compact intervals. Then Xt = Xo a.s. for all t > 0. Proof. Without loss of generality we may assume that Xo = 0. Denote by V = VAR (X; [0, t]) and T = inf{t>0: V>n} the strong variation of (X+)to and the stopping time when V, exceeds n for the first time. Since t Xt is continuous, so is tVt, cf. Paragraph A.29.f).