I am trying to solve a system of non-linear index-1 DAEs in which the derivatives of the state variables, $x(t)$ are corrupted by additive noise, $w(t)$ (whose co-variance matrix is known).
$\dot x(t) = f(x(t),t,u(t)) + w(t)$
Is there a known (open source) solver/library/package/toolbox in MATLAB/python/Julia (with or without adaptive time-stepping capability) that can solve the system?