# Reference request: Stochastic integration and martingale theory on the whole real line

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!

• Do you really need to define $B_s$ on the entire line or just $\mathrm{d}B_s$? Because the increments $\mathrm{d}B_s$ are easily defined and that's enough to integrate against. Aug 9, 2014 at 20:53
• @PabloLessa I think so, usually you want a (semi-)martingale as integrator. The mere definition doesn't seem to be a problem, take the one of standard BM and change the index set from $\mathbb{R}_{+}$ to $\mathbb{R}$. In the articles of the big guys (Marc Yor etc.), you can read statements like "there exists a Brownian motion $(B_x)_{x \in \mathbb{R}}$ with respect to a prob. measure $P$ and a filtration $\{\mathcal{E}_x, x \in \mathbb{R}\}$" inside of their theorems and then you see stochastic integrals of the form I wrote above, so there must be a proper way to define everything. Aug 9, 2014 at 21:25