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The first passage time problem over a moving boundary for asymptotically stable Lévy processes. (English) Zbl 1354.60049
Let \(\{X_t\}_{t\geq 0}\) be a one-dimensional Lévy process which belongs to the domain of attraction of a stable distribution with stability and skewness parameters \(\alpha\in(0,1)\) and \(\rho\in(0,1)\), respectively. Further, let \(f:[0,\infty)\to\mathbb{R}\) be the so-called moving boundary. In the paper under review, the authors study the asymptotic behavior of the first passage time of \(\{X_t\}_{t\geq0}\) over \(f(t)\), i.e., \[ \mathbb{P}(X_t\leq f(t),\;0\leq t\leq T)\quad\text{as}\quad T\to\infty. \] As the main results they show the following:
(i) If the left tail of the underlying Lévy measure has regularly varying density and \(\limsup_{t\to 0^+}\mathbb{P}(X_t\geq 0)<1\), then for any \(0<\gamma<1/\alpha\), \[ \mathbb{P}(X_t\leq 1-t^\gamma,\;0\leq t\leq T)=T^{-\rho+o(1)}\quad\text{as}\quad T\to\infty. \] (ii) If the right tail of the underlying Lévy measure has regularly varying density, then for any \(0<\gamma<1/\alpha\), \[ \mathbb{P}(X_t\leq 1+t^\gamma,\;0\leq t\leq T)=T^{-\rho+o(1)}\quad\text{as}\quad T\to\infty. \]

60G51 Processes with independent increments; Lévy processes
60F99 Limit theorems in probability theory
60G52 Stable stochastic processes
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[1] Aurzada, F., Kramm, T.: First exit of Brownian motion from a one-sided moving boundary. In: High Dimensional Probability VI: The Banff Volume. Progress in Probability 66, 215-219 (2013) · Zbl 1271.60088
[2] Aurzada, F., Kramm, T., Savov, M.: First passage times of Lévy processes over a one-sided moving boundary. Markov Processes and Related Fields, to appear, http://arxiv.org/abs/1201.1118 · Zbl 1321.60091
[3] Aurzada, F., Simon, T.: Persistence probabilities & exponents. Lévy matters, Springer, to appear, http://arxiv.org/abs/1203.6554 · Zbl 0431.60080
[4] Baltrūnas, A, Some asymptotic results for transient random walks with applications to insurance risk, J. Appl. Probab., 38, 108-121, (2001) · Zbl 0983.60042
[5] Bertoin, J.: Lévy Processes. Cambridge Univ. Press, Cambridge (1996) · Zbl 0861.60003
[6] Bertoin, J; Doney, RA, Some asymptotic results for transient random walks, Adv. Appl. Probab., 28, 207-226, (1996) · Zbl 0854.60069
[7] Bertoin, J., Doney, R.A.: Spitzer’s condition for random walks and Lévy porcesses. Ann. Inst. Henri Poincaré 33, 167-178 (1997) · Zbl 0880.60078
[8] Bingham, NH, Limit theorems in fluctuation theory, Adv. Appl. Probab., 5, 554-569, (1973) · Zbl 0273.60066
[9] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1989) · Zbl 0667.26003
[10] Borovkov, AA, On the asymptotic behavior of the distributions of first-passage times I, Math. Notes, 75, 23-37, (2004) · Zbl 1108.60039
[11] Borovkov, AA, On the asymptotic behavior of the distributions of first-passage times II, Math. Notes, 75, 322-330, (2004) · Zbl 1138.60035
[12] Bray, AJ; Majumdar, SN; Schehr, G, Persistence and first-passage properties in non-equilibrium systems, Adv. Phys., 62, 225-361, (2013)
[13] Breiman, L.: Probability, volume 7 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1992) · Zbl 0753.60001
[14] Denisov, D; Dieker, AB; Shneer, V, Large deviations for random walks under subexponentiality: the big-jump domain, Ann. Probab., 36, 1946-1991, (2008) · Zbl 1155.60019
[15] Denisov, D; Shneer, V, Asymptotics for first-passage times of Lévy processes and random walks, J. Appl. Probab., 50, 64-84, (2013) · Zbl 1264.60031
[16] Denisov, D., Wachtel, V.: Exact asymptotics for the instant of crossing a curve boundary by an asymptotically stable random walk. Preprint, arXiv:1403.5918 · Zbl 1375.60088
[17] Doney, R; Rivero, V, Asymptotic behaviour of first passage time distributions for Lévy processes, Probab. Theory Relat. Fields, 157, 1-45, (2013) · Zbl 1286.60042
[18] Doney, RA, On the asymptotic behaviour of first passage times for transient random walk, Probab. Theory Relat. Fields, 81, 239-246, (1989) · Zbl 0643.60053
[19] Doney, RA; Maller, RA, Moments of passage times for Lévy processes, Ann. Inst. H. Poincaré Probab. Stat., 40, 279-297, (2004) · Zbl 1042.60025
[20] Doney, RA; Maller, RA, Passage times of random walks and Lévy processes across power law boundaries, Probab. Theory Relat. Fields, 133, 57-70, (2005) · Zbl 1082.60037
[21] Esary, JD; Proschan, F; Walkup, DW, Association of random variables, with applications, Ann. Math. Stat., 38, 1466-1474, (1967) · Zbl 0183.21502
[22] Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971) · Zbl 0219.60003
[23] Gärtner, J, Location of wave fronts for the multidimensional KPP equation and Brownian first exit densities, Math. Nachr., 105, 317-351, (1982) · Zbl 0501.60083
[24] Greenwood, PE; Novikov, AA, One-sided boundary crossing for processes with independent increments, Teor. Veroyatnost. i Primenen., 31, 266-277, (1986) · Zbl 0602.60060
[25] Griffin, PS; Maller, RA, Small and large time stability of the time taken for a Lévy process to cross curved boundaries, Ann. Inst. H. Poincaré Probab. Statist., 49, 208-235, (2013) · Zbl 1267.60053
[26] Gut, A, On the moments and limit distributions of some first passage times, Ann. Probab., 2, 277-308, (1974) · Zbl 0278.60031
[27] Jennen, C; Lerche, HR, First exit densities of Brownian motion through one-sided moving boundaries, Z. Wahrsch. Verw. Gebiete, 55, 133-148, (1981) · Zbl 0447.60061
[28] Kwasnicki, M., Malecki, J., Ryznar, M.: Suprema of Lévy processes. Ann. Probab. 41(3B), 2047-2065 (2013) · Zbl 1288.60061
[29] Kyprianou, A.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006) · Zbl 1104.60001
[30] Mogul’skii, AA; Pecherskii, EA, The time of first entry into a region with curved boundary, Sib. Math. Zh., 19, 824-841, (1978) · Zbl 0409.60055
[31] Novikov, A.A.: A Martingale Approach to First Passage Problems and a New Condition for Wald’s Identity. Stochastic Differential Systems (Visegrád, 1980). Lecture Notes in Control and Information Sci., vol. 36. Springer, Berlin (1981)
[32] Novikov, A.A.: The martingale approach in problems on the time of the first crossing of nonlinear boundaries. Trudy Mat. Inst. Steklov., 158, 130-152, 230, 1981. Analytic number theory, mathematical analysis and their applications · Zbl 0491.60038
[33] Novikov, AA, On estimates and asymptotic behavior of nonexit probabilities of Wiener process to a moving boundary, Math. USSR Sbornik, 38, 495-505, (1981) · Zbl 0462.60079
[34] Novikov, AA, The crossing time of a one-sided non-linear boundary by sums of independent random variables, Theory Probab. Appl., 27, 688-702, (1982) · Zbl 0521.60055
[35] Novikov, AA, Martingales, a Tauberian theorem, and strategies for games of chance, Teor. Veroyatnost. i Primenen., 41, 810-826, (1996)
[36] Peškir, G; Shiryaev, AN, On the Brownian first-passage time over a one-sided stochastic boundary, Teor. Veroyatnost. i Primenen., 42, 591-602, (1997) · Zbl 0924.60069
[37] Rogozin, BA, Distribution of the first ladder moment and height, and fluctuations of a random walk, Teor. Verojatnost. i Primenen., 16, 539-613, (1971) · Zbl 0269.60053
[38] Rotar’, V, On the moments of the value and the time of the first passage over a curvilinear boundary, Theory Probab. Appl., 12, 690-691, (1967) · Zbl 0178.19202
[39] Salminen, P, On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary, Adv. Appl. Probab., 20, 411-426, (1988) · Zbl 0647.60087
[40] Samorodnitsky, G., Taqqu, M.S.: Stable non-Gaussian random processes. Stochastic Modeling. Chapman & Hall, New York (1994). Stochastic models with infinite variance · Zbl 0925.60027
[41] Sato, K.: Lévy Processes and Infinitely Divisible Distributions, volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999)
[42] Uchiyama, K, Brownian first exit from and sojourn over one-sided moving boundary and application, Z. Wahrsch. Verw. Gebiete, 54, 75-116, (1980) · Zbl 0431.60080
[43] Vondraček, Z, Asymptotics of first-passage time over a one-sided stochastic boundary, J. Theor. Probab., 13, 279-309, (2000) · Zbl 0970.60091
[44] Zolotarev, V.M.: One-dimensional stable distributions, volume 65 of Translations of Mathematical Monographs. American Mathematical Society (1986)
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