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A Monte Carlo Method for the Simulation of First Passage Times of Diffusion Processes

  • Maria Teresa Giraudo
  • Laura Sacerdote
  • Cristina Zucca
Article

Abstract

A reliable Monte Carlo method for the evaluation of first passage times of diffusion processes through boundaries is proposed. A nested algorithm that simulates the first passage time of a suitable tied-down process is introduced to account for undetected crossings that may occur inside each discretization interval of the stochastic differential equation associated to the diffusion. A detailed analysis of the performances of the algorithm is then carried on both via analytical proofs and by means of some numerical examples. The advantages of the new method with respect to a previously proposed numerical-simulative method for the evaluation of first passage times are discussed. Analytical results on the distribution of tied-down diffusion processes are proved in order to provide a theoretical justification of the Monte Carlo method.

diffusion processes tied down processes first exit time Monte Carlo methods 

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References

  1. A. Buonocore, A. G. Nobile, and L. M. Ricciardi, “A new integral equation for the evaluation of FPT probability densities,” Adv. Appl. Prob. vol. 19 pp. 784-800, 1987.Google Scholar
  2. L. Favella, M. T. Reineri, L. M. Ricciardi, and L. Sacerdote, “First passage time problems and some related computational problems,” Cybernetics and Systems vol. 13 pp. 95-128, 1982.Google Scholar
  3. W. Feller, “Diffusion processes in one dimension,” Trans. Amer. Math. Soc. vol. 77 pp. 1-31, 1954.Google Scholar
  4. A. Friedman, Stochastic Differential Equations and Applications, Academic Press, 1976.Google Scholar
  5. V. Giorno, A. G. Nobile, L. M. Ricciardi, and L. Sacerdote, “Some remarks on the Rayleigh process,” J. Appl. Prob. vol. 23 pp. 398-408, 1986.Google Scholar
  6. M. T. Giraudo and L. Sacerdote, “An improved technique for the simulation of first passage times for diffusion processes,” Communications in Statistics, Simulation and Computation vol. 28no. 4 pp. 1135-1163, 1999.Google Scholar
  7. J. Honerkamp, Stochastic Dynamical Systems: Concepts, Numerical Methods, Data Analysis, VCH, 1994.Google Scholar
  8. S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, 1981.Google Scholar
  9. P.E. Kloeden and E. Platen, The Numerical Solution of Stochastic Differential Equations, Springer Verlag, 1992.Google Scholar
  10. L. M. Ricciardi, A. Di Crescenzo, V. Giorno, and A. G. Nobile, “An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling,” Mathematic Japonica vol. 50no. 2, pp. 247-322, 1999.Google Scholar
  11. L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, Vol. 2, Wiley Series in Probability and Mathematical Statistics, Wiley, 1987.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Maria Teresa Giraudo
    • 1
  • Laura Sacerdote
    • 1
  • Cristina Zucca
    • 1
  1. 1.Department of MathematicsTorino UniversityTorino(Italy)

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