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.


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