  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.

Numerical Analysis;Probability