Can we define $N_t:=\int_0^t \operatorname{sign}(B_s) \mathrm{d}s$?

Question

Let $B=(B_t)_{t\geq0}$ be a Wiener process in $\mathbb{R}$ and $\operatorname{sign}$ the signum function. Is the process, defiended as stochastic integral or pathwise Lebesgue integral , $$N_t:=\int_0^t \operatorname{sign}(B_s) \mathrm{d}s$$ well-defined? I think as a pathwise Lebesgue integral it should be well defined since it is a measurable. But what about the stochastic integral? Is $\operatorname{sign}(B)$ predictable? What can we say beyond this about the process $N$? The expectation of the process $N$ should be $0$ and it should be of finite variation on compact sets. Are the further interesting properties?