Suppose $f, g: \mathbb R \to \mathbb R$ are continuous, non negative functions with $f \leq g$.


Fix some $T > 0$, and denote by $X^f$ the stochastic integral $\int\_{[0, T]} f(s) \ dW\_s$, where $W\_s$ is a standard Brownian motion. Similarly write $X^g$ for the corresponding integral of $g$.


Write $F^f$ for the cumulative distribution function of $|X^f|$, that is $F^f (x) := P(|X^f| < x)$. Similarly write $F^g$ for the corresponding cumulative distribution function of $|X^g|$.



> 
> **Question:** Is it true that $F^f \geq F^g$?
> 
> 
> 



A monotonicity formula for the stochastic integral with respect to Brownian motion
$X^f$ and $X^g$ are Gaussian random variables with zero mean and variances $$\mathbb{E}(|X^f|^2) = \int\_0^T|f(t)|^2dt \leq \int\_0^T|g(t)|^2dt = \mathbb{E}(|X^g|^2).$$


Thus, we know the probability density functions of both random variables. Comparison is up to the reader.


