# 5. Monte Carlo Solutions of Differential Equations

Published Online: 29 NOV 2011

DOI: 10.1002/9781118109656.ch5

Copyright © 2011 John Wiley & Sons, Inc. All rights reserved.

Book Title

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

Additional Information

#### 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

- Published Online: 29 NOV 2011
- Published Print: 24 OCT 2011

#### Book Series:

#### ISBN Information

Print ISBN: 9780470467039

Online ISBN: 9781118109656

- Summary
- Chapter
- References

### 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