Advances in Applied Probability

Joint densities of first hitting times of a diffusion process through two time-dependent boundaries

Laura Sacerdote, Ottavia Telve, and Cristina Zucca

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Abstract

Consider a one-dimensional diffusion process on the diffusion interval I originated in x0I. Let a(t) and b(t) be two continuous functions of t, t > t0, with bounded derivatives, a(t) < b(t), and a(t), b(t) ∈ I, for all t > t0. We study the joint distribution of the two random variables Ta and Tb, the first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of Ta and Tb in terms of P(Ta < t, Ta < Tb) and P(Tb < t, Ta > Tb), and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed.

Article information

Source
Adv. in Appl. Probab., Volume 46, Number 1 (2014), 186-202.

Dates
First available in Project Euclid: 1 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.aap/1396360109

Digital Object Identifier
doi:10.1239/aap/1396360109

Mathematical Reviews number (MathSciNet)
MR3189054

Zentralblatt MATH identifier
1304.60086

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 65R20: Integral equations

Keywords
First hitting time diffusion process Brownian motion Ornstein-Uhlenbeck process copula

Citation

Sacerdote, Laura; Telve, Ottavia; Zucca, Cristina. Joint densities of first hitting times of a diffusion process through two time-dependent boundaries. Adv. in Appl. Probab. 46 (2014), no. 1, 186--202. doi:10.1239/aap/1396360109. https://projecteuclid.org/euclid.aap/1396360109


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References

  • Abate, J. and Whitt, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7, 36–43.
  • Albano, G. and Giorno, V. (2006). A stochastic model in tumor growth. J. Theoret. Biol. 242, 329–336.
  • Alili, L., Patie, P. and Pedersen, J. L. (2005). Representations of the first hitting time density of an Ornstein–Uhlenbeck process. Stoch. Models 21, 967–980.
  • Benedetto, E., Sacerdote, L. and Zucca, C. (2013). A first passage problem for a bivariate diffusion process: numerical solution with an application to neuroscience when the process is Gauss-Markov. J. Comput. Appl. Math. 242, 41–52.
  • Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion–-Facts and Formulae, 2nd edn. Birkhäuser, Basel.
  • Buonocore, A., Nobile, A. G. and Ricciardi, L. M. (1987). A new integral equation for the evaluation of first-passage-time probability densities. Adv. Appl. Prob. 19, 784–800.
  • Buonocore, A., Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1990). On the two-boundary first-crossing-time problem for diffusion processes. J. Appl. Prob. 27, 102–114.
  • Capocelli, R. M. and Ricciardi, L. M. (1976). On the transformation of diffusion processes into the Feller process. Math. Biosci. 29, 219–234.
  • Davydov D. and Linetsky, V. (2003). Pricing options on scalar diffusions: an eigenfunction expansion approach. Operat. Res. 51, 185–209.
  • Di Crescenzo, A. G., Giorno, V., Nobile, A. G. and Ricciardi, L. M. (1995). On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes. J. Appl. Prob. 32, 316–336.
  • Galleani, L., Sacerdote, L., Tavella, P. and Zucca, C. (2003). A mathematical model for the atomic clock error. Metrologia 40, S257–S264.
  • Giorno, V., Nobile, A. G. and Ricciardi, L. M. (2011). On the densities of certain bounded diffusion processes. Ric. Mat. 60, 89–124.
  • Giorno, V., Nobile, A. G., Ricciardi, L. M. and Sacerdote, L. (1986). Some remarks on the Rayleigh process. J. Appl. Prob. 23, 398–408.
  • Giraudo, M. T. and Sacerdote, L. (1999). An improved technique for the simulation of first passage times for diffusion processes. Commun. Statist. Simul. Comput. 28, 1135–1163.
  • Giraudo, M. T., Sacerdote, L. and Zucca, C. (2001). A Monte Carlo method for the simulation of first passage times of diffusion processes. Meth. Comp. Appl. Prob. 3, 215–231.
  • Itô, K. and McKean, H. P., Jr. (1974). Diffusion Processes and Their Sample Paths. Springer, Berlin.
  • Lapidus, L. and Pinder, G. F. (1999). Numerical Solution of Partial Differential Equations in Science and Engineering. John Wiley, New York.
  • Linetsky, V. (2004). Computing hitting time densities for CIR and OU diffusions: applications to mean-reverting models. J. Comput. Finance 7, 1–22.
  • Linz, P. (1985). Analytical and Numerical Methods for Volterra Equations (SIAM Studies Appl. Math. 7). SIAM, Philadelphia, PA.
  • Nelsen, R. B. (1999). An Introduction to Copulas (Lecture Notes Statist. 139). Springer, New York.
  • Novikov, A., Frishling, V. and Kordzakhia, N. (1999). Approximations of boundary crossing probabilities for a Brownian motion. J. Appl. Prob. 36, 1019–1030.
  • Orsingher, E. and Beghin, L. (2006). Probabilità e Modelli Aleatori. Aracne, Roma.
  • Panfilo, G., Tavella, P. and Zucca, C. (2004). How long does a clock error remain inside two threshold barriers? An evaluation by means of stochastic processes. In Proc. Europ. Freq. Time Forum, Guilford, pp. 110–115.
  • Peskir, G. (2002). Limit at zero of the Brownian first-passage density. Prob. Theory Relat. Fields 124, 100–111.
  • Ricciardi, L. M. (1976). On the transformation of diffusion processes into the Wiener process. J. Math. Anal. Appl. 54, 185-199.
  • Ricciardi, L. M. (1977). Diffusion Processes and Related Topics in Biology (Lecture Notes Biomath. 14). Springer, Berlin.
  • Ricciardi, L. M. and Sacerdote, L. (1987). On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary. J. Appl. Prob. 24, 355–369.
  • Ricciardi, L. M. and Sato, S. (1990). Diffusion processes and first-passage-time problems. In Lectures in Applied Mathematics and Informatics, Manchester University Press, pp. 206–285.
  • Ricciardi, L. M., Sacerdote, L. and Sato, S. (1984). On an integral equation for first-passage-time probability densities. J. Appl. Prob. 21, 302–314.
  • Ricciardi, L. M., Di Crescenzo, A., Giorno, V. and Nobile, A. G. (1999). An outline of theoretical and algorithmic approaches to first passage time problems with applications to biological modeling. Math. Japon. 50, 247–322.
  • 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 Math. 2058), pp. 99–148.
  • Smith, G. D. (1978). Numerical Solution of Partial Differential Equations. Finite Difference Methods, 2nd edn. Oxford University Press.
  • Smith, P. L. (2000). Stochastic dynamic models of response time and accuracy: a foundational primer. J. Math. Psych. 44, 408–463.
  • Van Loan, C. (1992). Computational Frameworks for the Fast Fourier Transform. SIAM, Philadelphia, PA.
  • Zucca, C. and Sacerdote, L. (2009). On the inverse first-passage-time problem for a Wiener process. Ann. Appl. Prob. 19, 1319–1346.