On the asymptotic behavior of the parameter estimators for some diffusion processes: Application to neuronal models

Ricerche di Matematica

Volumn 58, Issue 1, 2009, Pages 103-127

  Giraudo, Maria Teresa ^a   Mininni, Rosa Maria ^b   Sacerdote, Laura ^a  

^a UNIVERSITY OF TURIN   (Italy)

^b UNIVERSITY OF BARI   (Italy)

Author keywords

Asymptotic properties; Diffusion processes; First passage time; Moment type estimators; Neuronal models; Rate of convergence

Indexed keywords

EID: 67349115963     PISSN: 00355038     EISSN: 18273491     Source Type: Journal    
DOI: 10.1007/s11587-009-0050-4     Document Type: Article

References (34)

1. O.O. Aalen H.K. Gjessing 2005 Survival models based on the Ornstein-Uhlenbeck process Lifetime Data Anal. 10 407 423
2. L. Alili P. Patie J.L. Pedersen 2005 Representation of the first hitting time densiy of an Ornstein- Uhlenbeck process Stoch. Models 21 967 980
3. A.R. Bulsara T.C. Elston C.R. Doering S.B. Lowen K. Lindenberg 1996 Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics Phys. Rev. E 53 4 3958 3969 (Pubitemid 126558901)
4. Casella G., Berger R.L.: Statistical Inference, 2nd edn. Duxbury Advanced Series, USA (2002)
5. J.C. Cox J.E. Ingersoll S.A. Ross 1985 A theory of the term structure of interest rates Econometrica 53 385 407
6. S. Ditlevsen 2007 A result on the FPT of an Ornstein-Uhlenbeck process Statist. Probab. Lett. 77 18 1744 1749
7. S. Ditlevsen O. Ditlevsen 2008 Parameter estimation from observations of first-passage times of the Ornstein-Uhlenbeck process and the Feller process Probab. Eng. Mech. 23 2-3 170 179
8. Ditlevsen, S., Lánský, P.: Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model. Phys. Rev. E 71, Art. No. 011907 (2005)
9. Ditlevsen, S., Lánský, P.: Estimation of the input parameters in the Feller neuronal model. Phys. Rev. E 73, Art. No. 061910 (2006)
10. Ditlevsen, S., Lánský, P.: Parameters of stochastic diffusion processes estimated from observations of first-hitting times: application to the leaky integrate-and-fire neuronal model. Phys. Rev. E 76, Art. No. 041906 (2007)
11. Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G.: Higher transcendental functions. McGrawHill, New York (1953)
12. Feller, W.: Diffusion processes in genetics. In: Neyman, J. (ed.) Proceedings of the second Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp. 227-246 (1951)
13. V. Giorno P. Lánský A.G. Nobile L.M. Ricciardi 1988 Diffusion approximation and first-passage-time problem for a model neuron III. A birth and death proces approach Biol. Cybern. 58 387 404 (Pubitemid 18119828)
14. V. Giorno A.G. Nobile L.M. Ricciardi S. Sato 1989 On the evaluation of first-passage-time probability densities via non singular integral equations Adv. Appl. Probab. 21 20 36
15. M.T. Giraudo L. Sacerdote 1999 An improved technique for the simulation of first passage times for diffusion processes Commun. Stat. Simul. Comput. 28 4 1135 1163
16. M.T. Giraudo L. Sacerdote C. Zucca 2001 A Monte Carlo method for the simulation of first passage times of diffusion processes Methodol. Comput. Appl. Probab. 3 215 231 (Pubitemid 33362075)
17. R.A. Heath C.H. Willcox 1990 A stochastic model for inter-keypress times in a typing task Acta Psychol. (Amst.) 75 1 13 39
18. J. Inoue S. Sato L.M. Ricciardi 1995 On the parameter estimation for diffusion models of single neurons activites Biol. Cybern. 73 209 221
19. Karlin S., Taylor H.M.: A second course in stochastic processes. Academic Press, Inc., London (1981)
20. V. Linetsky 2004 Computing hitting time densities for CIR and OU diffusions: applications to mean-reverting models J. Comput. Finance 7 1 22
21. B. Leblanc O. Scaillet 1998 Path dependent options on yields in the affine term structure Finance Stoch. 2 349 367
22. P. Lánský L. Sacerdote F. Tomassetti 1995 On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity Biol. Cybern. 73 457 465
23. K.W. Morris D. Szynal 1982 On the convergence rate in the central limit theorem of some functions of the average of independent random variables Prob. Math. Stat. 3 1 85 95
24. P. Mullowney S. Iyengar 2008 Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data J. Comput. Neurosci. 24 2 179 194
25. A.G. Nobile L.M. Ricciardi L. Sacerdote 1985 Exponential trends for a class of diffusion processes with steady state distribution J. Appl. Prob. 22 611 618
26. P. Patie 2005 A martingale associated to generalized Ornstein-Uhlenbeck processes and application to finance Stoch. Process. Appl. 115 4 593 607 (Pubitemid 40351934)
27. Ricciardi, L.M.: Diffusion processes and related topics in biology. Lecture Notes in Biomathematics, vol. 14. Springer, Berlin (1977)
28. L.M. Ricciardi L. Sacerdote 1979 The Ornstein-Uhlenbeck process as a model for neuronal activity Biol. Cybern. 35 1 9
29. L.M. Ricciardi S. Sato 1988 First-passage-time density and moments of the Ornstein-Uhlenbeck process J. Appl. Prob. 25 43 57
30. L.M. Ricciardi A. Di Crescenzo V. Giorno A.G. Nobile 1999 An outline of theoretical and algorithm approaches to first passage time problems with applications to biological modeling Math. Jpn. 50 2 247 322
31. Shiryaev A.N.: Probability, 2nd ed. Springer, New York (1996)
32. Shreve S.E.: Stochastic calculus for finance II. Continuous-time models, Springer Finance. Springer, USA (2004)
33. Tuckwell H.C.: Introduction to theoretical neurobiology, vol. 2: Nonlinear and stochastic theories. Cambridge University Press, Cambridge (1988)
34. V. Vaugirard 2004 A canonical first passage time model to pricing nature-linked bonds Econ. Bull. 7 2 1 7