Let be a standard 2-D Brownian motion, and is an adapted process satisfying, for some constants

Let be a martingale of

**[Q.]** Does there exist 1-1 mapping such that

where and are finite variation and martingale terms associated to Doob’s decomposition of ?

In some special cases, the answer is positive. For instance, if for some deterministic function satisfying

then one can check by Ito’s formula

is homeomorphism satisfying the requirement.

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