Consider the ito integral of the sign of the Brownian motion $W_s$ from $0$ to $t$: $$\int_0^t \operatorname{sign}(W_s)\,dW_s$$ This appears for instance in the Tanaka formula. I think this is a Brownian motion, by Levy's characterization, since this is a continuous Martingale, according to the theory of stochastic integration, whose quadratic variation is $t$. However, I don't quite understand it intuitively.

For instance, it seems that the above ito integral is always positive, at least bounded form below. Say, $W_s$ stays positive, then the above ito integral is $W_t$, which is positive, since the sign of $W_s$ is 1. If $W_s$ stays negative, then the ito integral is $|W_s|$, which is also positive, since the sign of $W_s$ is $-1$. So it seems the above ito integral is positive or bounded from below. In fact, the above ito integral appears in the semi-Martingale representation of $|W_t|$, which is given by Tanaka formula: $$|W_t|=\int_0^t \operatorname{sign}(W_s)\,dW_s +L_t$$ where $L_t$ is the Brownian local time.

So, my question is how can the ito integral be a Brownian motion, if it seems that the integral is positive. Maybe, my intuition has something wrong. Can someone explain to me what goes wrong here?

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