# Show that $B(t)$ is a Brownian motion

Let $$W(t)$$ be a Brownian motion, and define $$B(t) = \int_0^t sign(W(s))dW(s)$$ where $$sign(x) = \begin{cases} 1, & \text{if x \geq 0 } \\ -1 & \text{otherwise} \end{cases}$$

1. Show that $$B(t)$$ is a Brownian motion?
2. Show that $$E(B_tX_t)=0$$

For question 1, I know how to prove $$B(t)$$ has independent increments, but how to prove its increments are normal?

For question 2, using Ito's product rule, $$d(B_tX_t) = B_tdX_t+X_tdB_t+dB_tdX_t$$ Since $$B_t$$ and $$X_t$$ are correlated, how do I deal with $$dB_tdX_t$$ in above equation?

• What is the $X_t$ in part 2? – Math1000 Nov 12 '19 at 3:19