Let $W$ be a standard one dimensional Brownian motion, and $\mathcal F\_t$ its natural filtration. Consider the SDE


$$dX\_t = \mu\_X (t, \omega) \, dt + \sigma\_X (t, \omega) \, dW\_t$$


$$dY\_t = \mu\_Y (t, \omega) \, dt + \sigma\_Y (t, \omega) \, dW\_t$$


$$X\_0 = x\_0, Y\_0 = y\_0 \text{ a.s.}$$


where $\mu\_X, \mu\_Y, \sigma\_X, \sigma\_Y \geq 0$ are progressively measurable with respect to $\mathcal F\_t$, and $x\_0, y\_0$ are constants.


Assume the existence of a solution to the above two equations up to a determinstic time $T$.


**Question:** Suppose $\sigma\_X \neq \sigma\_Y$ on a subset of $\Omega \times [0, T]$ of positive measure. Then is it true that


$$\mathbb P(Y\_T > X\_T) > 0?$$



A comparison principle for SDE
Don't think so. Take both to satisfy something like $dZ = \sigma |Z| dW$, start one from -1, and one from 1. regardless of the exact parameters, one stays positive and one stays negative.


