  We provide a simple algorithm for construction of Brownian paths
approximating those of a L\'evy process on a finite time interval. It requires
knowledge of the L\'evy process trajectory on a chosen regular grid and the law
of its endpoint, or the ability to simulate from that. This algorithm is based
on reordering of Brownian increments, and it can be applied in a recursive
manner. We establish an upper bound on the mean squared maximal distance
between the paths and determine a suitable mesh size in various asymptotic
regimes. The analysis proceeds by reduction to the comonotonic coupling of
increments. Applications to model risk and multilevel Monte Carlo are discussed
in detail, and numerical examples are provided.
