A comparison of diffusions Consider two diffusions given by
$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$
for $j=1,2$ and $t\ge 0$, where $W_\cdot$ is a standard Wiener process/Brownian motion and the $a_j$'s are smooth enough functions such that $0\le a_1\le a_2$.
Does it then follow that $P(|X_1(1)|>x)\le P(|X_2(1)|>x)$ for all real $x$?
At least, does this comparison hold when $a_2$ is a constant?

There are a number of results comparing two diffusions with the same diffusion coefficients but with the drift of one of the two diffusions greater than the drift of the other one -- see e.g. Nakao. However, I have been unable to find a comparison of the desired kind.
 A: The inequality is not true in general — additional assumptions are needed. I think some kind of monotonicity of $a_1$ and $a_2$ should help, but this is merely a guess.
Here is a counterexample. Consider $a_2 = 1$, so that $X_2(1) = W(1)$. Let $a_1(s, x) = 1$ when $|x| < 1$ and $a_1(s, x) = 0$ otherwise. Then $X_2(1) = W(1 \wedge \tau)$, where $\tau$ is the hitting time of $\{-1, 1\}$ for $W$. It is fairly straightforward to see that if $|X_2(1)| \geqslant 1$, then $|X_1(1)| = 1$, while if $|X_2(1)| < 1$, then still $|X_1(1)| = 1$ with positive probability. Thus, the desired inequality fails to hold for $x = 1 - \epsilon$ with $\epsilon > 0$ small enough.
The question asks for smooth $a_1$ and $a_2$, so one should mollify them appropriately, but this is just a technicality, I hope the idea is clear.
A: This is just a comment on a related issue.
You don't have stochastic comparison of processes at a given time but you DO have stochastic comparison of hitting times.
To see this, you can couple the two processes $j=1,2$ AFTER having performed the (respective) time change transformations into standard BM.
As a consequence, the two processes will have the SAME path but with different time parametizations, one FASTER than the other.
This implies stochastic comparison of hitting times.
