Theory of Probability and Mathematical Statistics
Asymptotic results for certain first-passage times and areas of renewal processes
Claudio Macci and Barbara Pacchiarotti
Link
Abstract: We consider the process {x-N(t): t≥0}, where x∈R+ and {N(t): t≥0} is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of (τ(x),A(x)) where τ(x) is the first-passage time of {x-N(t): t≥0} to reach zero or a negative value, and A(x)=∫0τ(x-N(t))dt is the corresponding first-passage (positive) area swept out by the process {x-N(t): t≥0}. We remark that we can define the sequence {(τ(n),A(n)): n≥1} by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as x→∞ in the fashion of large (and moderate) deviations.
Keywords: Large deviations, moderate deviations, joint distribution, integrated random walk
Bibliography: Mario Abundo, On the first-passage area of a one-dimensional jump-diffusion process, Methodol. Comput. Appl. Probab. 15 (2013), no. 1, 85–103. MR 3030212, DOI 10.1007/s11009-011-9223-1
Mario Abundo and Danilo Del Vescovo, On the joint distribution of first-passage time and first-passage area of drifted Brownian motion, Methodol. Comput. Appl. Probab. 19 (2017), no. 3, 985–996. MR 3683981, DOI 10.1007/s11009-017-9546-7
Mario Abundo and Sara Furia, Joint distribution of first-passage time and first-passage area of certain Lévy processes, Methodol. Comput. Appl. Probab. 21 (2019), no. 4, 1283–1302. MR 4029242, DOI 10.1007/s11009-018-9677-5
Marek Arendarczyk, Krzysztof Dȩbicki, and Michel Mandjes, On the tail asymptotics of the area swept under the Brownian storage graph, Bernoulli 20 (2014), no. 2, 395–415. MR 3178504, DOI 10.3150/12-BEJ491
D.S. Bates, Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options, The Review of Financial Studies 9 (1996), no. 1, 69–107.
J. Blanchet and M. Mandjes, Asymptotics of the area under the graph of a Lévy-driven workload process, Oper. Res. Lett. 41 (2013), no. 6, 730–736. MR 3131857, DOI 10.1016/j.orl.2013.10.004
Stephen A. Book, Large deviation probabilities for weighted sums, Ann. Math. Statist. 43 (1972), 1221–1234. MR 331486, DOI 10.1214/aoms/1177692474
A. A. Borovkov, O. J. Boxma, and Z. Palmowski, On the integral of the workload process of the single server queue, J. Appl. Probab. 40 (2003), no. 1, 200–225. MR 1953775, DOI 10.1017/s0021900200022348
Narasinga R. Chaganty, Large deviations for joint distributions and statistical applications, Sankhyā Ser. A 59 (1997), no. 2, 147–166. MR 1665683
Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, 2nd ed., Applications of Mathematics (New York), vol. 38, Springer-Verlag, New York, 1998. MR 1619036, DOI 10.1007/978-1-4612-5320-4
Denis Denisov, Elena Perfilev, and Vitali Wachtel, Tail asymptotics for the area under the excursion of a random walk with heavy-tailed increments, J. Appl. Probab. 58 (2021), no. 1, 217–237. MR 4222426, DOI 10.1017/jpr.2020.85
Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal, Fundamentals of convex analysis, Grundlehren Text Editions, Springer-Verlag, Berlin, 2001. Abridged version of Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and II [ibid.; MR1295240 (95m:90002)]. MR 1865628, DOI 10.1007/978-3-642-56468-0
P. Jorion, On jump processes in the foreign exchange and stock markets, The Review of Financial Studies 1 (1988), no. 4, 427–445.
Michael J. Kearney, Satya N. Majumdar, and Richard J. Martin, The first-passage area for drifted Brownian motion and the moments of the Airy distribution, J. Phys. A 40 (2007), no. 36, F863–F869. MR 2396208, DOI 10.1088/1751-8113/40/36/F03
Michael J. Kearney, Andrew J. Pye, and Richard J. Martin, On correlations between certain random variables associated with first passage Brownian motion, J. Phys. A 47 (2014), no. 22, 225002, 11. MR 3208169, DOI 10.1088/1751-8113/47/22/225002
Frank B. Knight, The moments of the area under reflected Brownian bridge conditional on its local time at zero, J. Appl. Math. Stochastic Anal. 13 (2000), no. 2, 99–124. MR 1768498, DOI 10.1155/S1048953300000137
S. G. Kou and Hui Wang, First passage times of a jump diffusion process, Adv. in Appl. Probab. 35 (2003), no. 2, 504–531. MR 1970485, DOI 10.1239/aap/1051201658
Elena Perfilev and Vitali Wachtel, Local asymptotics for the area under the random walk excursion, Adv. in Appl. Probab. 50 (2018), no. 2, 600–620. MR 3832886, DOI 10.1017/apr.2018.26
Mihael Perman and Jon A. Wellner, On the distribution of Brownian areas, Ann. Appl. Probab. 6 (1996), no. 4, 1091–1111. MR 1422979, DOI 10.1214/aoap/1035463325
R. Rudzkis, L. Saulis, and V. Statuljavičus, Large deviations for sums of independent random variables, Litovsk. Mat. Sb. 19 (1979), no. 1, 169–179, 232 (Russian, with English and Lithuanian summaries). MR 537824
