5. Monte Carlo Solutions of Differential Equations

  1. James R. Thompson

Published Online: 29 NOV 2011

DOI: 10.1002/9781118109656.ch5

Empirical Model Building: Data, Models, and Reality, Second Edition

Empirical Model Building: Data, Models, and Reality, Second Edition

How to Cite

Thompson, J. R. (2011) Monte Carlo Solutions of Differential Equations, in Empirical Model Building: Data, Models, and Reality, Second Edition, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/9781118109656.ch5

Publication History

  1. Published Online: 29 NOV 2011
  2. Published Print: 24 OCT 2011

ISBN Information

Print ISBN: 9780470467039

Online ISBN: 9781118109656

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Keywords:

  • differential-equation;
  • dirichlet problem;
  • Fokker-Planck equation;
  • gambler's ruin problem;
  • Monte Carlo solution

Summary

This chapter provides a distinction between Monte Carlo and simulation. It shows a closed-form solution for the gambler's ruin problem; it would be ridiculous to use a simulation to solve it. It is by means of an analogy of real-world problems to the general equation that simulation becomes useful. The examples provided in the chapter are given to give the reader a feel as to the practical implementation of simulation-based algorithms as alternatives to the usual numerical approximation techniques. A certain amount of practice quickly brings the user to a point where he or she can write simulation algorithms in days to problems that would require the numerical analyst months to approach. The algorithm for solving the Fokker-Planck problem and Dirichlet problem are not simply analogues of the classical differential-equation formulations of these systems. They are, in fact, descriptions of the axioms that typically give rise to these problems.

Controlled Vocabulary Terms

differential-difference equation