Theory of Probability and Mathematical Statistics
Inverse first-passage problems of a diffusion with resetting
Mario Abundo
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Abstract: We address some inverse problems for the first-passage place and the first-passage time of a one-dimensional diffusion process with stochastic resetting. This type of diffusion is characterized by the fact that a reset to the position xR can occur according to a homogeneous Poisson process with rate r>0. As regards the inverse first-passage place problem, for random initial position belonging to an interval (0,b) with finite b>0 (and fixed r and xR belonging to (0,b)), let τ be the first time at which the process exits the interval (0,b), and π0 be the probability of exit from the left end of (0,b). Given a probability q, the inverse first-passage place problem consists in finding the density g of the initial position, if it exists, such that π0=q. Concerning the inverse first-passage time problem, for random positive starting point (and fixed r and xR>0, let again τ be the first-passage time of the process through zero. For a given distribution function F(t) on the positive real axis, the inverse first-passage time problem consists in finding the density g of the starting point, if it exists, such that P(τ ≤ t)=F(t), t>0. In addition to the case of random initial position, we also study the case when the initial position and the resetting rate r are fixed, whereas the reset position xR is random. For all types of inverse problems considered, several explicit examples of solutions are reported.
Keywords: Diffusion with resetting, first-passage time, first-passage place
Bibliography: Mario Abundo, The first-passage area of Wiener process with stochastic resetting, Methodol. Comput. Appl. Probab. 25 (2023), no. 4, Paper No. 92, 25. MR 4671697, DOI 10.1007/s11009-023-10069-4
Mario Abundo, The first-passage area of Ornstein-Uhlenbeck process revisited, Stoch. Anal. Appl. 41 (2023), no. 2, 358–376. MR 4549302, DOI 10.1080/07362994.2021.2018335
Mario Abundo, Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process, Control Cybernet. 51 (2022), no. 1, 31–42. MR 4474265, DOI 10.2478/candc-2022-0003
Mario Abundo, An inverse problem for the first-passage place of some diffusion processes with random starting point, Stoch. Anal. Appl. 38 (2020), no. 6, 1122–1133. MR 4165539, DOI 10.1080/07362994.2020.1768867
Mario Abundo, An inverse first-passage problem revisited: the case of fractional Brownian motion, and time-changed Brownian motion, Stoch. Anal. Appl. 37 (2019), no. 5, 708–716. MR 3991056, DOI 10.1080/07362994.2019.1608834
Mario Abundo, An inverse first-passage problem revisited: the case of fractional Brownian motion, and time-changed Brownian motion, Stoch. Anal. Appl. 37 (2019), no. 5, 708–716. MR 3991056, DOI 10.1080/07362994.2019.1608834
Mario Abundo, An overview on inverse first-passage-time problems for one-dimensional diffusion processes, Recent advances in probability and statistics, Lect. Notes Semin. Interdiscip. Mat., vol. 12, Semin. Interdiscip. Mat. (S.I.M.), Potenza, 2015, pp. 1–44. MR 3561465
Mario Abundo, One-dimensional reflected diffusions with two boundaries and an inverse first-hitting problem, Stoch. Anal. Appl. 32 (2014), no. 6, 975–991. MR 3270691, DOI 10.1080/07362994.2014.959595
Mario Abundo, Solving an inverse first-passage-time problem for Wiener process subject to random jumps from a boundary, Stoch. Anal. Appl. 31 (2013), no. 4, 695–707. MR 3175793, DOI 10.1080/07362994.2013.800358
Mario Abundo, Some randomized first-passage problems for one-dimensional diffusion processes, Sci. Math. Jpn. 76 (2013), no. 1, 33–46. MR 3099201
Mario Abundo, The double-barrier inverse first-passage problem for Wiener process with random starting point, Statist. Probab. Lett. 83 (2013), no. 1, 168–176. MR 2998739, DOI 10.1016/j.spl.2012.09.006
Mario Abundo, An inverse first-passage problem for one-dimensional diffusions with random starting point, Statist. Probab. Lett. 82 (2012), no. 1, 7–14. MR 2863016, DOI 10.1016/j.spl.2011.09.005
Mario Abundo, On the first hitting time of a one-dimensional diffusion and a compound Poisson process, Methodol. Comput. Appl. Probab. 12 (2010), no. 3, 473–490. MR 2665271, DOI 10.1007/s11009-008-9115-1
Mario Abundo, On first-passage times for one-dimensional jump-diffusion processes, Probab. Math. Statist. 20 (2000), no. 2, Acta Univ. Wratislav. No. 2256, 399–423. MR 1825652
A. Di Crescenzo, V. Giorno, A. G. Nobile, and L. M. Ricciardi, On the
queue with catastrophes and its continuous approximation, Queueing Syst. 43 (2003), no. 4, 329–347. MR 1976263, DOI 10.1023/A:1023261830362
Arjun K. Gupta and Saralees Nadarajah (eds.), Handbook of beta distribution and its applications, Statistics: Textbooks and Monographs, vol. 174, Marcel Dekker, Inc., New York, 2004. MR 2079703
Ken Jackson, Alexander Kreinin, and Wanhe Zhang, Randomization in the first hitting time problem, Statist. Probab. Lett. 79 (2009), no. 23, 2422–2428. MR 2556323, DOI 10.1016/j.spl.2009.08.016
Fima C. Klebaner, Introduction to stochastic calculus with applications, 2nd ed., Imperial College Press, London, 2005. MR 2160228, DOI 10.1142/p386
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
Petr Lánský and Charles E. Smith, The effect of a random initial value in neural first-passage-time models, Math. Biosci. 93 (1989), no. 2, 191–215. MR 984278, DOI 10.1016/0025-5564(89)90023-0
Mario Lefebvre, Moments of first-passage places for jump-diffusion processes, Sankhya A 83 (2021), no. 1, 245–253. MR 4227212, DOI 10.1007/s13171-019-00181-4
Mario Lefebvre, The inverse first-passage-place problem for Wiener processes, Stoch. Anal. Appl. 40 (2022), no. 1, 96–102. MR 4357216, DOI 10.1080/07362994.2021.1889382
Satya N. Majumdar, Brownian functionals in physics and computer science, The legacy of Albert Einstein, World Sci. Publ., Hackensack, NJ, 2007, pp. 93–129. MR 2330814
A. G. Nobile, L. M. Ricciardi, and L. Sacerdote, Exponential trends of Ornstein-Uhlenbeck first-passage-time densities, J. Appl. Probab. 22 (1985), no. 2, 360–369. MR 789359, DOI 10.1017/s0021900200037827
Henry C. Tuckwell, On the first-exit time problem for temporally homogeneous Markov processes, J. Appl. Probability 13 (1976), no. 1, 39–48. MR 394907, DOI 10.2307/3212663
