Weak solution of SDE, Tanaka equation

A weak solution of $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t \;\;\;\; 0\let\le T, X_0=Z$$ is a triple $(\tilde{X}_t,\tilde{B}_t,\mathcal{H}_t)$ such that the above equation hold and $\tilde{X}_t$ is $\mathcal{H}_t$-adapted and $\tilde{B}_t$ is an $\mathcal{H}_t$-Brownian motion, i.e. $\tilde{B}_t$ is a Brownian motion and $\tilde{B}_t$ is a martingale w.r.t. $\mathcal{H}_t$. In the weak solution of Tanaka equation what would be the filtration $\mathcal{H}_t$ and why would $\tilde{B}_t$ be adapted and a martingale with respect to it?

The filtration $\mathcal{H}_t$ could be the natural filtration of the process $\hat{B}_t$, or any filtration for which $\hat{B}_t$ is a Brownian motion. Since $\tilde{B}_t$ is given by a stochastic integral $$\tilde{B}_t = \int_0^t \mathrm{sgn}(\hat{B}_s)d \hat{B}_s$$ it is adapted to the underlying filtration (as all stochastic integrals of adapted processes are). Since $\mathrm{sgn}(\hat{B}_s)$ is in $L^2$, it follows that $\tilde{B}_t$ is a martingale.