A simple method to calculate first-passage time densities with arbitrary initial conditions

Numerous applications all theway frombiology and physics to economics depend on the density of first crossings over a boundary.Motivated by the lack of general purpose analytical tools for computingfirst-passage time densities (FPTDs) for complex problems, we propose a new simple method based on the independent interval approximation (IIA).We generalise previous formulations of the IIA to include arbitrary initial conditions as well as to deal with discrete time and non-smooth continuous time processes.We derive a closed form expression for the FPTD in z and Laplacetransform space to a boundary in one dimension. Two classes of problems are analysed in detail: discrete time symmetric randomwalks (Markovian) and continuous timeGaussian stationary processes (Markovian andnon-Markovian). Our results are in good agreementwith Langevin dynamics simulations.