  The excursion set approach is a framework for estimating how the number
density of nonlinear structures in the cosmic web depends on the expansion
history of the universe and the nature of gravity. A key part of the approach
is the estimation of the first crossing distribution of a suitably chosen
barrier by random walks having correlated steps: The shape of the barrier is
determined by the physics of nonlinear collapse, and the correlations between
steps by the nature of the initial density fluctuation field. We describe
analytic and numerical methods for calculating such first up-crossing
distributions. While the exact solution can be written formally as an infinite
series, we show how to approximate it efficiently using the Stratonovich
approximation. We demonstrate its accuracy using Monte-Carlo realizations of
the walks, which we generate using a novel Cholesky-decomposition based
algorithm, which is significantly faster than the algorithm that is currently
in the literature.
