# zbMATH — the first resource for mathematics

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 $$0<\gamma<1/\alpha$$, $\mathbb{P}(X_t\leq 1+t^\gamma,\;0\leq t\leq T)=T^{-\rho+o(1)}\quad\text{as}\quad T\to\infty.$

##### MSC:
 60G51 Processes with independent increments; Lévy processes 60F99 Limit theorems in probability theory 60G52 Stable stochastic processes
Full Text:
##### References:
 [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.