# Q. Existence of 1-1 mapping

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 ${0<\lambda<\Lambda}$

$\displaystyle \lambda |\xi|^{2} \le \xi' \sigma \xi \le \Lambda |\xi|^{2}, \quad \forall \xi \in \mathbb R^{2}, \quad a.s..$

Let ${M}$ be a martingale of

$\displaystyle M_{t} = \int_{0}^{t} \sigma_{s} d B_{s}.$

[Q.] Does there exist 1-1 mapping ${f:\mathbb R^{2} \mapsto \mathbb R^{2}}$ such that

$\displaystyle f(B_{t}) = A_{t} + M_{t}$

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

$\displaystyle \partial_{y} \hat \sigma_{i1}(x,y) = \partial_{x} \hat\sigma_{i2}(x,y) := \hat \sigma_{i}(x,y)>0, \quad i = 1, 2,$

then one can check by Ito’s formula

$\displaystyle f(x,y) = \int_{0}^{y}\int_{0}^{x} \hat \sigma(r,s) dr ds$

is homeomorphism satisfying the requirement.