Show that (beta_{t}=int_{0}^{t} operatorname{sgn}left(B_{s} ight) d B_{s}) is a (mathrm{BM}^{1}). Use Lvy's characterization of a (mathrm{BM}^{1}), Theorem

Show that \(\beta_{t}=\int_{0}^{t} \operatorname{sgn}\left(B_{s}\right) d B_{s}\) is a \(\mathrm{BM}^{1}\).

Use Lévy's characterization of a \(\mathrm{BM}^{1}\), Theorem 9.13 or 19.5.

Data From Theorem 9.13

Data From Theorem 19.5

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9.13 Theorem (Lvy 1948). Let (X, X = 0, be a martingale with continuous sample paths such that (X-t, tots a martingale. Then (X) is a Brownian motion. Proof (Gikhman, Skorokhod 1972). We have to show that X-Xss and X-Xs-N(0, t-s) for all 0 < s