State Levy's characterisation of Brownian motion and use it to prove that B_t = ∫_0^t sgn(W_s) dW_s is a standard Brownian motion. Then explain why the filtration generated by B is strictly smaller than that of W.

State Levy's characterisation of Brownian motion and use it to prove that B_t = ∫_0^t sgn(W_s) dW_s is a standard Brownian motion. Then explain why the filtration generated by B is strictly smaller than that of W.

Approach: Compute the quadratic variation of the candidate process and check continuity and the local martingale property, then use Tanaka's formula to identify what information the new process retains.

Levy's characterisation says a continuous local martingale started at zero whose quadratic variation is t is a standard Brownian motion, and B qualifies because [B]_t = ∫_0^t sgn(W_s)^2 ds = t. The integrand is bounded and adapted, so the integral is a continuous square integrable martingale, and the quadratic variation of a stochastic integral is the integral of the squared integrand, which is 1 almost everywhere since the time W spends at zero has Lebesgue measure zero. The filtration is the interesting part: Tanaka's formula gives |W_t| = B_t + L_t^0 and local time is a measurable function of the path of |W|, so B generates the same information as |W| and carries no information about the sign of W. Therefore W solves dW = sgn(W) dB as a weak solution driven by B while remaining unmeasurable with respect to B, which is the standard example separating weak solutions from strong ones.

Follow-up: Why does dX = sgn(X) dB with X_0 = 0 admit no strong solution, and where does the Lipschitz existence theorem fail?

Key concepts: Levy characterisation, quadratic variation, weak solution, Tanaka's formula.