## Bernoulli

• Bernoulli
• Volume 23, Number 4B (2017), 3744-3771.

### Simulation of hitting times for Bessel processes with non-integer dimension

#### Abstract

In this paper, we complete and improve the study of the simulation of the hitting times of some given boundaries for Bessel processes. These problems are of great interest in many application fields as finance and neurosciences. In a previous work (Ann. Appl. Probab. 23 (2013) 2259–2289), the authors introduced a new method for the simulation of hitting times for Bessel processes with integer dimension. The method, called walk on moving spheres algorithm (WoMS), was based mainly on the explicit formula for the distribution of the hitting time and on the connection between the Bessel process and the Euclidean norm of the Brownian motion. This method does not apply anymore for a non-integer dimension. In this paper we consider the simulation of the hitting time of Bessel processes with non-integer dimension $\delta\geq1$ and provide a new algorithm by using the additivity property of the laws of squared Bessel processes. We split each simulation step of the algorithm in two parts: one is using the integer dimension case and the other one considers hitting time of a Bessel process starting from zero.

#### Article information

Source
Bernoulli, Volume 23, Number 4B (2017), 3744-3771.

Dates
Revised: May 2016
First available in Project Euclid: 23 May 2017

https://projecteuclid.org/euclid.bj/1495505108

Digital Object Identifier
doi:10.3150/16-BEJ866

Mathematical Reviews number (MathSciNet)
MR3654822

Zentralblatt MATH identifier
06778302

#### Citation

Deaconu, Madalina; Herrmann, Samuel. Simulation of hitting times for Bessel processes with non-integer dimension. Bernoulli 23 (2017), no. 4B, 3744--3771. doi:10.3150/16-BEJ866. https://projecteuclid.org/euclid.bj/1495505108

#### 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] Buonocore, A., Giorno, V., Nobile, A.G. and Ricciardi, L.M. (2002). A neuronal modeling paradigm in the presence of refractoriness. Biosystems 67 35–43.
• [3] Buonocore, A., Nobile, A.G. and Ricciardi, L.M. (1987). A new integral equation for the evaluation of first-passage-time probability densities. Adv. in Appl. Probab. 19 784–800.
• [4] Burkitt, A.N. (2006). A review of the integrate-and-fire neuron model. I. Homogeneous synaptic input. Biol. Cybernet. 95 1–19.
• [5] Byczkowski, T., Małecki, J. and Ryznar, M. (2013). Hitting times of Bessel processes. Potential Anal. 38 753–786.
• [6] 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.
• [7] Daniels, H.E. (1969). The minimum of a stationary Markov process superimposed on a $U$-shaped trend. J. Appl. Probab. 6 399–408.
• [8] Daniels, H.E. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli 2 133–143.
• [9] Deaconu, M. and Herrmann, S. (2013). Hitting time for Bessel processes – walk on moving spheres algorithm (WoMS). Ann. Appl. Probab. 23 2259–2289.
• [10] Devroye, L. (1986). Nonuniform Random Variate Generation. New York: Springer.
• [11] Durbin, J. (1985). The first-passage density of a continuous Gaussian process to a general boundary. J. Appl. Probab. 22 99–122.
• [12] Durbin, J. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Probab. 29 291–304.
• [13] Ferebee, B. (1982). The tangent approximation to one-sided Brownian exit densities. Z. Wahrsch. Verw. Gebiete 61 309–326.
• [14] Ferebee, B. (1983). An asymptotic expansion for one-sided Brownian exit densities. Z. Wahrsch. Verw. Gebiete 63 1–15.
• [15] Gerstner, W. and Kistler, W.M. (2002). Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge: Cambridge Univ. Press.
• [16] Giorno, V., Nobile, A.G., Ricciardi, L.M. and Sato, S. (1989). On the evaluation of first-passage-time probability densities via nonsingular integral equations. Adv. in Appl. Probab. 21 20–36.
• [17] Gobet, E. (1998). Schéma d’Euler continu pour des diffusions tuées et options barrière. C. R. Acad. Sci. Paris Sér. I Math. 326 1411–1414.
• [18] Gobet, E. (2000). Weak approximation of killed diffusion using Euler schemes. Stochastic Process. Appl. 87 167–197.
• [19] Gobet, E. (2001). Euler schemes and half-space approximation for the simulation of diffusion in a domain. ESAIM Probab. Stat. 5 261–297 (electronic).
• [20] Gobet, E. and Menozzi, S. (2010). Stopped diffusion processes: Boundary corrections and overshoot. Stochastic Process. Appl. 120 130–162.
• [21] Göing-Jaeschke, A. and Yor, M. (2003). A survey and some generalizations of Bessel processes. Bernoulli 9 313–349.
• [22] Hamana, Y. and Matsumoto, H. (2013). The probability distributions of the first hitting times of Bessel processes. Trans. Amer. Math. Soc. 365 5237–5257.
• [23] Herrmann, S. and Tanré, E. (2016). The first-passage time of the Brownian motion to a curved boundary: An algorithmic approach. SIAM J. Sci. Comput. 38 A196–A215.
• [24] Ichiba, T. and Kardaras, C. (2011). Efficient estimation of one-dimensional diffusion first passage time densities via Monte Carlo simulation. J. Appl. Probab. 48 699–712.
• [25] Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets. Springer Finance. Springer: London.
• [26] Lánský, P., Sacerdote, L. and Tomassetti, F. (1995). On the comparison of Feller and Ornstein–Uhlenbeck models for neural activity. Biol. Cybernet. 73 457–465.
• [27] Lerche, H.R. (1986). Boundary Crossing of Brownian Motion: Its Relation to the Law of the Iterated Logarithm and to Sequential Analysis. Lecture Notes in Statistics 40. Berlin: Springer.
• [28] Norris, J.R. (1998). Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics 2. Cambridge: Cambridge Univ. Press.
• [29] Patie, P. and Winter, C. (2008). First exit time probability for multidimensional diffusions: A PDE-based approach. J. Comput. Appl. Math. 222 42–53.
• [30] Pötzelberger, K. and Wang, L. (2001). Boundary crossing probability for Brownian motion. J. Appl. Probab. 38 152–164.
• [31] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Berlin: Springer.
• [32] Ricciardi, L.M., Sacerdote, L. and Sato, S. (1984). On an integral equation for first-passage-time probability densities. J. Appl. Probab. 21 302–314.
• [33] Sacerdote, L. and Giraudo, M.T. (2013). Stochastic integrate and fire models: A review on mathematical methods and their applications. In Stochastic Biomathematical Models. Lecture Notes in Math. 2058 99–148. Heidelberg: Springer.
• [34] Sacerdote, L. and Tomassetti, F. (1996). On evaluations and asymptotic approximations of first-passage-time probabilities. Adv. in Appl. Probab. 28 270–284.
• [35] 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.
• [36] Shiga, T. and Watanabe, S. (1973). Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete 27 37–46.
• [37] Strassen, V. (1967). Almost sure behavior of sums of independent random variables and martingales. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) 315–343. Berkeley, CA: Univ. California Press.
• [38] Tuckwell, H.C. (1989). Stochastic Processes in the Neurosciences. CBMS-NSF Regional Conference Series in Applied Mathematics 56. Philadelphia, PA: SIAM.
• [39] Wang, L. and Pötzelberger, K. (2007). Crossing probabilities for diffusion processes with piecewise continuous boundaries. Methodol. Comput. Appl. Probab. 9 21–40.