Numerical determination of hitting time distributions from their Laplace transforms: Simple cases

An important problem in probabilistic modeling consists of the determination of the distribution function of a positive random variable, in particular a hitting time, from its Laplace transform. In this note we use two versions of the method of maximum entropy to invert the Laplace transform using a few of its values on the real line, or equivalently a few fractional moments of the exponential of the variable. To compare the performance of the methods, we consider some simple one dimensional examples in which the Laplace transform can be computed by solving a differential equation exactly, and comparisons can be made.