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A Random Walk on Rectangles Algorithm

Author

Listed:
  • Madalina Deaconu

    (INRIA Lorraine and Institut Élie Cartan de Nancy (IECN))

  • Antoine Lejay

    (INRIA Lorraine and Institut Élie Cartan de Nancy (IECN))

Abstract

In this article, we introduce an algorithm that simulates efficiently the first exit time and position from a rectangle (or a parallelepiped) for a Brownian motion that starts at any point inside. This method provides an exact way to simulate the first exit time and position from any polygonal domain and then to solve some Dirichlet problems, whatever the dimension. This method can be used as a replacement or complement of the method of the random walk on spheres and can be easily adapted to deal with Neumann boundary conditions or Brownian motion with a constant drift.

Suggested Citation

  • Madalina Deaconu & Antoine Lejay, 2006. "A Random Walk on Rectangles Algorithm," Methodology and Computing in Applied Probability, Springer, vol. 8(1), pages 135-151, March.
    • Handle: RePEc:spr:metcap:v:8:y:2006:i:1:d:10.1007_s11009-006-7292-3
    DOI: 10.1007/s11009-006-7292-3
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    References listed on IDEAS

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    1. Sabelfeld K.K. & Talay D., 1995. "Integral Formulation of the Boundary Value Problems and the Method of Random Walk on Spheres," Monte Carlo Methods and Applications, De Gruyter, vol. 1(1), pages 1-34, December.
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    Cited by:

    1. Deaconu, M. & Herrmann, S. & Maire, S., 2017. "The walk on moving spheres: A new tool for simulating Brownian motion’s exit time from a domain," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 135(C), pages 28-38.
    2. Maire, Sylvain & Nguyen, Giang, 2016. "Stochastic finite differences for elliptic diffusion equations in stratified domains," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 146-165.
    3. Lejay, Antoine & Maire, Sylvain, 2007. "Computing the principal eigenvalue of the Laplace operator by a stochastic method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 73(6), pages 351-363.
    4. Gobet, Emmanuel & Menozzi, Stéphane, 2010. "Stopped diffusion processes: Boundary corrections and overshoot," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 130-162, February.
    5. Karl K. Sabelfeld & Anastasia E. Kireeva, 2022. "Stochastic Simulation Algorithms for Solving Transient Anisotropic Diffusion-recombination Equations and Application to Cathodoluminescence Imaging," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 3029-3048, December.
    6. Sabelfeld, Karl K. & Kireeva, Anastasya, 2020. "Stochastic simulation algorithms for solving a nonlinear system of drift–diffusion-Poisson equations of semiconductors," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 556(C).
    7. Bras, Pierre & Kohatsu-Higa, Arturo, 2023. "Simulation of reflected Brownian motion on two dimensional wedges," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 349-378.

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    2. Deaconu, M. & Herrmann, S. & Maire, S., 2017. "The walk on moving spheres: A new tool for simulating Brownian motion’s exit time from a domain," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 135(C), pages 28-38.
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