Let $B$ be a standard 2-D Brownian motion, and $\sigma: \Omega\times \mathbb R^{+} \mapsto \mathbb R^{2 \times 2}$ is an $\mathcal F_{t}$ adapted process satisfying, for some constants $0<\lambda<\Lambda$ $$\lambda |\xi|^{2} \le \xi' \sigma \xi \le \Lambda |\xi|^{2}, \quad \forall \xi \in \mathbb R^{2}, \quad a.s..$$ Let $M$ be a martingale of $$M_{t} = \int_{0}^{t} \sigma_{s} d B_{s}.$$
[Q.] Does there exist 1-1 mapping $f:\mathbb R^{2} \mapsto \mathbb R^{2}$ such that $$f(B_{t}) = A_{t} + M_{t}$$ where $A$ and $M$ are finite variation and martingale terms associated to Doob's decomposition of $f(B_{t})$?
In some special cases, the answer is positive. For instance, if $\sigma_{s} = \hat \sigma(B_{s})$ for some deterministic function $\hat\sigma: \mathbb R^{2} \mapsto \mathbb R^{2\times 2}$ satisfying $$\partial_{y} \hat \sigma_{i1}(x,y) = \partial_{x} \hat\sigma_{i2}(x,y) := \hat \sigma_{i}(x,y)>0, \quad i = 1, 2,$$ then one can check by Ito's formula $$f(x,y) = \int_{0}^{y}\int_{0}^{x} \hat \sigma(r,s) dr ds$$ is a homeomorphism satisfying the requirement.
