Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Universality of the asymptotics of the one-sided exit problem for integrated processes

Frank Aurzada and Steffen Dereich

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Abstract

We consider the one-sided exit problem – also called one-sided barrier problem – for ($\alpha$-fractionally) integrated random walks and Lévy processes.

Our main result is that there exists a positive, non-increasing function $\alpha\mapsto\theta(\alpha)$ such that the probability that any $\alpha$-fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{-\theta(\alpha)+\mathrm{o}(1)}$ for large $T$. We also investigate when the fixed level can be replaced by a different barrier satisfying certain growth conditions (moving boundary).

This, in particular, extends Sinai’s result on the survival exponent $\theta(1)=1/4$ for the integrated simple random walk to general random walks with some finite exponential moment.

Résumé

Nous considérons le problème unilatéral de sortie – ou problème unilatéral de barrière – pour des intégrales ($\alpha$-fractionnelles) de marches aléatoires et de processus de Lévy.

Notre résultat principal est l’existence d’une fonction positive, décroissante $\alpha\mapsto\theta(\alpha)$ telle que la probabilité qu’une intégrale d’un processus de Lévy $\alpha$-fractionnel quelconque (ou marche aléatoire) avec certains moments exponentiels finis reste en dessous d’un niveau fixe jusqu’à un temps $T$ se comporte comme $T^{-\theta(\alpha)+\mathrm{o}(1)}$ pour $T$ grand. Nous analysons aussi la possibilité de remplacer le niveau fixe par une barrière différente qui satisfait certaines conditions de croissance (marge mouvante).

Cela, en particulier, étend le résultat de Sinai sur l’exposant de survie d’une marche aléatoire simple intégrée à des marches aléatoires générales de moment exponentiel fini.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 236-251.

Dates
First available in Project Euclid: 29 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1359470133

Digital Object Identifier
doi:10.1214/11-AIHP427

Mathematical Reviews number (MathSciNet)
MR3060155

Zentralblatt MATH identifier
1285.60042

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes 60J65: Brownian motion [See also 58J65] 60G15: Gaussian processes 60G18: Self-similar processes

Keywords
Integrated Brownian motion Integrated Lévy process Integrated random walk Lower tail probability Moving boundary One-sided barrier problem One-sided exit problem Persistence probabilities Survival exponent

Citation

Aurzada, Frank; Dereich, Steffen. Universality of the asymptotics of the one-sided exit problem for integrated processes. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 236--251. doi:10.1214/11-AIHP427. https://projecteuclid.org/euclid.aihp/1359470133


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References

  • [1] R. F. Bass, N. Eisenbaum and Z. Shi. The most visited sites of symmetric stable processes. Probab. Theory Related Fields 116 (2000) 391–404.
  • [2] G. Baxter and M. D. Donsker. On the distribution of the supremum functional for processes with stationary independent increments. Trans. Amer. Math. Soc. 85 (1957) 73–87.
  • [3] J. Bertoin. Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge, 1996.
  • [4] J. Bertoin. The inviscid Burgers equation with Brownian initial velocity. Comm. Math. Phys. 193 (1998) 397–406.
  • [5] N. H. Bingham. Maxima of sums of random variables and suprema of stable processes. Z. Wahrsch. Verw. Gebiete 26 (1973) 273–296.
  • [6] F. Caravenna and J.-D. Deuschel. Pinning and wetting transition for $(1+1)$-dimensional fields with Laplacian interaction. Ann. Probab. 36 (2008) 2388–2433.
  • [7] R. A. Doney. Spitzer’s condition and ladder variables in random walks. Probab. Theory Related Fields 101 (1995) 577–580.
  • [8] A. Dembo, B. Poonen, Q.-M. Shao and O. Zeitouni. Random polynomials having few or no real zeros. J. Amer. Math. Soc. 15 (2002) 857–892 (electronic).
  • [9] J. D. Esary, F. Proschan and D. W. Walkup. Association of random variables, with applications. Ann. Math. Statist. 38 (1967) 1466–1474.
  • [10] W. Feller. An Introduction to Probability Theory and Its Applications II, 2nd edition. Wiley, New York, 1971.
  • [11] M. Goldman. On the first passage of the integrated Wiener process. Ann. Math. Statist. 42 (1971) 2150–2155.
  • [12] P. Groeneboom, G. Jongbloed and J. A. Wellner. Integrated Brownian motion, conditioned to be positive. Ann. Probab. 27 (1999) 1283–1303.
  • [13] Y. Isozaki and S. Watanabe. An asymptotic formula for the Kolmogorov diffusion and a refinement of Sinai’s estimates for the integral of Brownian motion. Proc. Japan Acad. Ser. A Math. Sci. 70 (1994) 271–276.
  • [14] J. Komlós, P. Major and G. Tusnády. An approximation of partial sums of independent ${\mathrm{RV}}$’s and the sample ${\mathrm{DF}}$. I. Z. Wahrsch. Verw. Gebiete 32 (1975) 111–131.
  • [15] A. Lachal. Sur le premier instant de passage de l’intégrale du mouvement Brownien. Ann. Inst. Henri Poincaré Probab. Stat. 27 (1991) 385–405.
  • [16] A. Lachal. Sur les excursions de l’intégrale du mouvement Brownien. C. R. Acad. Sci. Paris Sér. I Math. 314 (1992) 1053–1056.
  • [17] W. V. Li and Q.-M. Shao. A normal comparison inequality and its applications. Probab. Theory Related Fields 122 (2002) 494–508.
  • [18] W. V. Li and Q.-M. Shao. Lower tail probabilities for Gaussian processes. Ann. Probab. 32 (2004) 216–242.
  • [19] W. V. Li and Q.-M. Shao. Recent developments on lower tail probabilities for Gaussian processes. Cosmos 1 (2005) 95–106.
  • [20] M. Lifshits. Gaussian Random Functions. Mathematics and Its Applications. Kluwer Academic, Dordrecht, 1995.
  • [21] S. N. Majumdar. Persistence in nonequilibrium systems. Current Sci. 77 (1999) 370–375.
  • [22] H. P. McKean, Jr. A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227–235.
  • [23] G. M. Molchan. Maximum of a fractional Brownian motion: Probabilities of small values. Comm. Math. Phys. 205 (1999) 97–111.
  • [24] G. M. Molchan. On the maximum of fractional Brownian motion. Teor. Veroyatn. Primen. 44 (1999) 111–115.
  • [25] G. M. Molchan. Unilateral small deviations of processes related to the fractional Brownian motion. Stochastic Process. Appl. 118 (2008) 2085–2097.
  • [26] I. Monroe. On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 (1972) 1293–1311.
  • [27] A. A. Novikov. Estimates for and asymptotic behavior of the probabilities of a Wiener process not crossing a moving boundary. Mat. Sb. 110 (1979) 539–550 (in Russian). English translation in: Sb. Math. 38 (1981) 495–505.
  • [28] J. Obłój. The Skorokhod embedding problem and its offspring. Probab. Surv. 1 (2004) 321–390 (electronic).
  • [29] T. Simon. The lower tail problem for homogeneous functionals of stable processes with no negative jumps. ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007) 165–179 (electronic).
  • [30] T. Simon. On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data. J. Stat. Phys. 131 (2008) 733–747.
  • [31] Y. G. Sinai. Distribution of some functionals of the integral of a random walk. Teoret. Mat. Fiz. 90 (1992) 323–353.
  • [32] D. Slepian. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 (1962) 463–501.
  • [33] J. M. Steele. Stochastic Calculus and Financial Applications. Applications of Mathematics 45. Springer, New York, 2001.
  • [34] K. Uchiyama. Brownian first exit from and sojourn over one sided moving boundary and application. Z. Wahrsch. Verw. Gebiete 54 (1980) 75–116.
  • [35] V. Vysotsky. Clustering in a stochastic model of one-dimensional gas. Ann. Appl. Probab. 18 (2008) 1026–1058.
  • [36] V. Vysotsky. On the probability that integrated random walks stay positive. Stochastic Process. Appl. 120 (2010) 1178–1193.