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Abstrakty
For a certain class of one-dimensional diffusions X(t); we study the distribution of maxtЄ[0;T] X(t) and the distribution of the first instant at whichX(t) attains the maximum by reducingX(t) to Brownian motion. Moreover, for T fixed or random, we study the asymptotics of threshold crossing probability, i.e. the rate of decay of P(maxsЄ[0;T ] X(s) > z ) as z goes to infinity. Some examples are also reported.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Strony
107--120
Opis fizyczny
Bibliogr. 23 poz.
Twórcy
autor
- Dipartimento di Matematica, Università Tor Vergata, via della Ricerca Scientifica, 00133 Roma, Italy
Bibliografia
- [1] M. Abundo, Diffusion approximation of discrete Markov chains with binomial-like transition probabilities, in: Cybernetics and Systems ’96, R. Trappl (Ed.), Austrian Society for Cybernetics Studies, Vienna 1996, pp. 508-513.
- [2] M. Abundo, On some properties of one-dimensional diffusion processes on an interval, Probab. Math. Statist. 17 (2) (1997), pp. 277-310.
- [3] M. Abundo, On the first-passage time of diffusion processes over a one-sided stochastic boundary, Stochastic Anal. Appl. 21 (1) (2003), pp. 1-23.
- [4] M. Abundo, Stopping a stochastic integral process as close as possible to the ultimate value of a functional, Sci. Math. Jpn. 60 (3) (2004), pp. 475-479.
- [5] M. Abundo, Limit at zero of the first-passage time density and the inverse problem for one-dimensional diffusions, Stochastic Anal. Appl. 24 (2006), pp. 1119-1145.
- [6] J. M. Azaïs and C. Delmas, Asymptotic expansions for the distribution of the maximum of a Gaussian random field, Extremes 2 (2002), pp. 181-212.
- [7] S. M. Berman, An asymptotic formula for the distribution of the maximum of Gaussian process with stationary increments, J. Appl. Probab. 22 (1985), pp. 454-460.
- [8] L. Breiman, On some limit theorems similar to the arc-sin law, Theory Probab. Appl. 10 (1965), pp. 323-331.
- [9] R. Carbone, Binomial approximation of Brownian motion and its maximum, Statist. Probab. Lett. 69 (2004), pp. 271-285.
- [10] C. Cierco-Ayrolles, A. Croquette and C. Delmas, Computing the distribution of the maximum of Gaussian random processes, Methodol. Comput. Appl. Probab. 5 (2003), pp. 427-438.
- [11] K. Dębicki and T. Rolski, A note on transient Gaussian fluid models, Queueing Syst. 41 (2002), pp. 321-342.
- [12] K. Dębicki, B. Zwart and S. Borst, The supremum of a Gaussian process over a random interval, Statist. Probab. Lett. 68 (2004), pp. 221-234.
- [13] C. Delmas, An asymptotic expansion for the distribution of the maximum of a class of Gaussian fields, C.R. Acad. Sci. Paris Sér. I Math. 327 (1998), pp. 393-397.
- [14] C. Delmas, Distribution du maximum d’un champ aléatoire et applications statistiques, Ph.D. Thesis, University Toulouse III, France, 2001.
- [15] I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer, New York 1972.
- [16] R. Z. Has’minskij, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980.
- [17] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland 1981.
- [18] V. Lanska, P. Lansky and C. E. Smiths, Synaptic transmission in a diffusion model for neural activity, J. Theor. Biol. 166 (1994), pp. 393-406.
- [19] P. Lévy, Processus stochastiques et mouvement brownien, Gauthier-Villars, Paris 1965.
- [20] V. I. Piterbarg and V. Prisyazhnyuk, Asymptotic behavior of the probability of a large excursion for a nonstationary Gaussian process, Teor. Veroyatnost. Mat. Statist. 18 (1978), pp. 121-133.
- [21] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin-Heidelberg 1991.
- [22] M. A. Urosov, On a property of the moment at which Brownian motion attains its maximum and some optimal stopping problems, Theory Probab. Appl. 49 (1) (2005), pp. 169-176.
- [23] B. Zwart, S. Borst and K. Dębicki, Subexponential asymptotics of hybrid fluid and ruin models, Ann. Appl. Probab. 15 (1A) (2005), pp. 500-517.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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