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On the numerical solution of Brownian motion processes

Published online by Cambridge University Press:  14 July 2016

R. S. Anderssen ,
F. R. De Hoog  and
R. Weiss
R. Weiss
Affiliation:
Computer Centre, The Australian National University
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Abstract

A new class of finite difference methods based on the concept of product integration is proposed for the numerical solution of the systems of weakly singular first kind Volterra equations which arise in the study of Brownian motion processes.

Keywords

BROWNIAN MOTION PROCESSESSINGULAR VOLTERRA INTEGRAL EQUATIONSNUMERICAL SOLUTIONFINITE DIFFERENCE METHODSPRODUCT INTEGRATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 2 , June 1973 , pp. 409 - 418
Copyright
Copyright © Applied Probability Trust 1973 

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References

Daniels, H. E. (1969) The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.Google Scholar
De Hoog, F. and Weiss, R. (1972a) On the solution of Volterra integral equations of the first kind. Numer. Math. To appear.Google Scholar
De Hoog, F. and Weiss, R. (1972b) High order methods for Volterra integral equations of the first kind. SIAM J. Numer. Anal. To appear.CrossRefGoogle Scholar
Durbin, J. (1971) Boundary crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Prob. 8, 431–453.Google Scholar
Fortet, R. (1943) Les functions aléatoires du type de Markov associées a certaines équations linéaires aux derivées partielles du type parabolique. J. Math. Pures Appl. 22, 177–243.Google Scholar
Isaacson, E. and Keller, H. B. (1966) Analysis of Numerical Methods. John Wiley and Sons, New York.Google Scholar
Kowalewski, G. (1930) Integralgleichungen. Walter de Gruyter and Co., Berlin and Leipzig.Google Scholar
Smith, C. S. (1972) A note on boundary probabilities for Brownian motion. Submitted for publication.Google Scholar
Weiss, R. (1972) Product integration for the generalized Abel equation. Math. Comp. 26, 177–190.CrossRefGoogle Scholar
Weiss, R. and Anderssen, R. S. (1972) A product integration method for a class of singular first kind Volterra equations. Num. Math. 18, 442–456.Google Scholar
Young, A. (1954) The application of approximate product integration to the numerical solution of integral equations. Proc. Roy. Soc. A 224, 561–573.Google Scholar