I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM does the trick, which filtration can one take?), a meaningful way to define stochastic integrals of the form

$\int_{a}^{b} X_s \, d B_s$, $-\infty \leq a < b \leq \infty$ and possibly a connection to continuous (semi-)martingales on $\mathbb{R}$ (is there a martingale representation theorem?).

One example is the semimartingale decomposition of Brownian local time with respect to the space variable (see Perkins, 1981). For $x,y \in \mathbb{R}$ with $x < y$ one has that

$$L_t^y - L_t^x = 2 \int_{x}^{y} \sqrt{L_t^u} \, d B_u + \text{ finite variaton part}$$

How is this integral defined in case $x<0$?

I've encountered such integrals in several articles by established authors, dating back to the eighties, and they happily apply the standard theory (Burkholder-Davis-Gundy, Ito-isometry etc.) to it, so I'm quite sure that I'm missing something here.

Any help is appreciated!