The logistic-normal integral and its generalizations

Abstract: We consider the solutions of the one-dimensional heat equation in an unbounded domain with initial conditions of the form . This includes as a particular case the logistic-normal integral, which corresponds to . 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 can be computed exactly on a grid of equidistant points of width in terms of the solutions of the heat equation with initial condition . 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. [Copyright &y& Elsevier]