The Annals of Applied Probability

Hitting time for Bessel processes—walk on moving spheres algorithm (WoMS)

Madalina Deaconu and Samuel Herrmann

Full-text: Open access

Abstract

In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation comes from mathematical finance when dealing with volatility models, but the results can also be used in optimal control problems. The aim here is to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox–Ingersoll–Ross process. The main tools we use are on one side, an adaptation of the method of images to this particular situation and on the other side, the connection that exists between Cox–Ingersoll–Ross processes and Bessel processes.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 6 (2013), 2259-2289.

Dates
First available in Project Euclid: 22 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1382447688

Digital Object Identifier
doi:10.1214/12-AAP900

Mathematical Reviews number (MathSciNet)
MR3127935

Zentralblatt MATH identifier
1298.65018

Subjects
Primary: 65C20: Models, numerical methods [See also 68U20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]

Keywords
Bessel processes Cox–Ingersoll–Ross processes hitting time method of images numerical algorithm

Citation

Deaconu, Madalina; Herrmann, Samuel. Hitting time for Bessel processes—walk on moving spheres algorithm (WoMS). Ann. Appl. Probab. 23 (2013), no. 6, 2259--2289. doi:10.1214/12-AAP900. https://projecteuclid.org/euclid.aoap/1382447688


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References

  • [1] Alili, L. and Patie, P. (2010). Boundary-crossing identities for diffusions having the time-inversion property. J. Theoret. Probab. 23 65–84.
  • [2] Byczkowski, T., Malecki, J. and Ryznar, M. (2011). Hitting times of bessel processes. Available at arXiv:1009.3513.
  • [3] Ciesielski, Z. and Taylor, S. J. (1962). First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103 434–450.
  • [4] Daniels, H. E. (1969). The minimum of a stationary Markov process superimposed on a $U$-shaped trend. J. Appl. Probab. 6 399–408.
  • [5] Devroye, L. (1986). Nonuniform Random Variate Generation. Springer, New York.
  • [6] Ermentrout, G. B. and Terman, D. H. (2010). Mathematical Foundations of Neuroscience. Interdisciplinary Applied Mathematics 35. Springer, New York.
  • [7] Gerstner, W. and Kistler, W. M. (2002). Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge Univ. Press, Cambridge.
  • [8] Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9 313–349.
  • [9] Hamana, Y. and Matsumoto, H. (2011). The probability distribution of the first hitting times of Bessel processes. Available at arXiv:1106.6132.
  • [10] Lànský, P., Sacerdote, L. and Tomassetti, F. (1995). On the comparison of Feller and Ornstein–Uhlenbeck models for neural activity. Biol. Cybernet. 73 457–465.
  • [11] Lerche, H. R. (1986). Boundary Crossing of Brownian Motion. Lecture Notes in Statistics 40. Springer, Berlin.
  • [12] Milstein, G. N. (1997). Weak approximation of a diffusion process in a bounded domain. Stochastics Stochastics Rep. 62 147–200.
  • [13] Muller, M. E. (1956). Some continuous Monte Carlo methods for the Dirichlet problem. Ann. Math. Statist. 27 569–589.
  • [14] Norris, J. R. (1998). Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics 2. Cambridge Univ. Press, Cambridge.
  • [15] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [16] Salminen, P. and Yor, M. (2011). On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes. Period. Math. Hungar. 62 75–101.