The logistic-normal integral and its generalizations

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Abstract

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)). This includes as a particular case the logistic-normal integral, which corresponds to f(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 t can be computed exactly on a grid of equidistant points of width σt in terms of the solutions of the heat equation with initial condition 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.

Original languageEnglish
Pages (from-to)460-469
Number of pages10
JournalJournal of Computational and Applied Mathematics
Volume237
Issue number1
DOIs
StatePublished - 1 Jan 2013

Keywords

  • Fourier series
  • Heat equation
  • Logistic-normal integral

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