  We consider the nonlinear Schr\"odinger equation with dispersion modulated by
a (formal) derivative of a time-dependent function with fractional Sobolev
regularity of class $W^{\alpha,2}$ for some $\alpha\in (0,1)$. Due to the loss
of smoothness in the problem classical numerical methods face severe order
reduction. In this work, we develop and analyze a new randomized exponential
integrator based on a stratified Monte Carlo approximation. The new
discretization technique averages the high oscillations in the solution
allowing for improved convergence rates of order $\alpha+1/2$. In addition, the
new approach allows us to treat a far more general class of modulations than
the available literature. Numerical results underline our theoretical findings
and show the favorable error behavior of our new scheme compared to classical
methods.
