Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Estimation for the discretely observed telegraph process

Authors: S. M. Iacus and N. Yoshida
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 37-47
MSC (2000): Primary 60K99; Secondary 62M99
Published electronically: August 4, 2009
MathSciNet review: 2446847
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The telegraph process $ \{X(t), t>0\}$ is supposed to be observed at $ n+1$ equidistant time points $ t_i=i\Delta_n$, $ i=0,1,\dots, n$. The unknown value of $ \lambda$, the underlying rate of the Poisson process, is a parameter to be estimated. The asymptotic framework considered is the following: $ \Delta_n \to 0$, $ n\Delta_n=T \to \infty$ as $ n \to \infty$. We show that previously proposed moment type estimators are consistent and asymptotically normal but not efficient. We study further an approximated moment type estimator which is still not efficient but comes in explicit form. For this estimator the additional assumption $ n\Delta_n^3 \to 0$ is required in order to obtain asymptotic normality. Finally, we propose a new estimator which is consistent, asymptotically normal and asymptotically efficient under no additional hypotheses.

References [Enhancements On Off] (What's this?)

  • 1. F. Black and M. S. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973), 637-654.
  • 2. A. De Gregorio and S. M. Iacus, Parametric estimation for the standard and the geometric telegraph process observed at discrete times, Unimi Research Papers,
  • 3. A. Di Crescenzo and B. Martinucci, On the effect of random alternating perturbations on hazard rates, Scientiae Mathematicae Japonicae 64 (2006), no. 2, 381-394. MR 2254153
  • 4. A. Di Crescenz and F. Pellerey, On prices' evolutions based on geometric telegrapher's process, Applied Stochastic Models in Business and Industry 18 (2002), 171-184. MR 1907356 (2003d:60130)
  • 5. G. B. Di Masi, Y. M. Kabanov, and W. J. Runggaldier, Mean-variance hedging of options on stocks with Markov volatilities, Theory of Probability and its Applications 39 (1994), 172-182. MR 1348196 (96h:90020)
  • 6. D. Florens-Zmirou, Approximate discrete time schemes for statistics of diffusion processes, Statistics 20 (1989), 547-557. MR 1047222 (91e:62215)
  • 7. S. K. Fong and S. Kanno, Properties of the telegrapher's random process with or without a trap, Stochastic Processes and their Applications 53 (1994), 147-173. MR 1290711 (95g:60089)
  • 8. S. Goldstein, On diffusion by discontinuous movements and the telegraph equation, The Quarterly Journal of Mechanics and Applied Mathematics 4 (1951), 129-156. MR 0047963 (13:960b)
  • 9. E. E. Holmes, Is diffusion too simple? Comparisons with a telegraph model of dispersal, American Naturalist 142 (1993), 779-796.
  • 10. E. E. Holmes, M. A. Lewis, J. E. Banks, and R. R. Veit, Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology 75 (1994), no. 1, 17-29.
  • 11. S. M. Iacus, Statistical analysis of the inhomogeneous telegrapher's process, Statistics and Probability Letters 55 (2001), no. 1, 83-88. MR 1860195 (2002g:60161)
  • 12. M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain Journal of Mathematics 4 (1974), 497-509. MR 0510166 (58:23185)
  • 13. Yu. A. Kutoyants, Statistical Inference for Spatial Poisson Processes, Lecture Notes in Statistics, Springer-Verlag, 1998. MR 1644620 (99k:62149)
  • 14. C. Mazza and D. Rullière, A link between wave governed random motions and ruin processes, Insurance: Mathematics and Economics 35 (2004), 205-222. MR 2095886 (2005h:60125)
  • 15. R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science 4 (1973), no. 1, 141-183. MR 0496534 (58:15058)
  • 16. E. Orsingher, Hyperbolic equations arising in random models, Stochastic Processes and their Applications 21 (1985), 49-66. MR 834990 (87f:35145)
  • 17. E. Orsingher, Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws, Stochastic Processes and their Applications 34 (1990), 49-66. MR 1039562 (91g:60086)
  • 18. M. Pinsky, Lectures on Random Evolution, World Scientific, River Edge, New Jersey, 1991. MR 1143780 (93b:60160)
  • 19. N. Ratanov, A jump telegraph model for option pricing, Quantitative Finance 7 (2007), no. 5, 575-583. MR 2358921 (2008m:60167)
  • 20. N. Ratanov, Quantile hedging for telegraph markets and its applications to a pricing of equity-linked life insurance contracts, Borradores de Investigatión, no. 62, apr. 2005,
  • 21. W. Stadje and S. Zacks, Telegraph processes with random velocities, Journal of Applied Probability 41 (2004), 665-678. MR 2074815 (2005g:60116)
  • 22. Y. Yao, Estimation of noisy telegraph process: Nonlinear filtering versus nonlinear smoothing, IEEE Transactions on Information Theory 31 (1985), no. 3, 444-446. MR 794444 (86f:94007)
  • 23. N. Yoshida, Estimation for diffusion processes from discrete observation, Journal of Multivariate Analysis 41 (1992), no. 2, 220-242. MR 1172898 (93g:62113)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60K99, 62M99

Retrieve articles in all journals with MSC (2000): 60K99, 62M99

Additional Information

S. M. Iacus
Affiliation: Department of Economics, Business and Statistics, University of Milan, Via Conservatorio 7, 20122 Milan, Italy

N. Yoshida
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguroku, Tokyo 153-8914, Japan

Keywords: Telegraph process, discretely observed process, inference for stochastic processes
Received by editor(s): December 28, 2006
Published electronically: August 4, 2009
Additional Notes: The work of the first author was supported by JSPS (Japan Society for the Promotion of Science) Program FY2006, grant ID No. S06174. He is also thankful to the Graduate School of Mathematical Sciences, University of Tokyo as host research institute for the JSPS Program
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society