The logistic-normal integral and its generalizations

We consider the solutions of the one-dimensional heat equation in an unbounded domain with initial conditions of the form f(x)/(1+exp(σx))f(x)/(1+exp(σx)). This includes as a particular case the logistic-normal integral, which corresponds to f(x)=1f(x)=1. Such initial conditions appear in stochastic calculus problems, and the numerical simulation of short-rate interest rate models and credit models with log-normally distributed short rates and hazard rates respectively. We show that the solutions at time tt can be computed exactly on a grid of equidistant points of width σtσt in terms of the solutions of the heat equation with initial condition f(x)f(x). The exact results on the grid can be used as nodes for a precise interpolation. Series representation of the solutions can be obtained by an application of the Poisson summation formula.