  We consider a linear ill-posed equation in the Hilbert space setting under
white noise. Known convergence results for the discrepancy principle are either
restricted to Hilbert-Schmidt operators (and they require a self-similarity
condition for the unknown solution $\hat{x}$, additional to a classical source
condition) or to polynomially ill-posed operators (excluding exponentially
ill-posed problems). In this work we show optimal convergence for a modified
discrepancy principle for both polynomially and exponentially ill-posed
operators (without further restrictions) solely under either H\"older-type or
logarithmic source conditions. In particular, the method includes only a single
simple hyper parameter, which does not need to be adapted to the type of
ill-posedness.
