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Entrance and exit at infinity for stable jump diffusions. (English) Zbl 07226359
Summary: In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on \(-\infty\leq a< b\leq\infty\) in terms of their ability to access the boundary (Feller’s test for explosions) and to enter the interior from the boundary. Feller’s technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article, we study exit and entrance from infinity for the most natural generalization, that is, jump diffusions of the form \[ dZ_t=\sigma (Z_{t-})\,dX_t, \]
MSC:
60H20 Stochastic integral equations
60G52 Stable stochastic processes
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[1] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
[2] Bertoin, J. and Savov, M. (2011). Some applications of duality for Lévy processes in a half-line. Bull. Lond. Math. Soc. 43 97-110. · Zbl 1213.60090
[3] Bertoin, J. and Yor, M. (2002). The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17 389-400. · Zbl 1004.60046
[4] Blumenthal, R. M., Getoor, R. K. and Ray, D. B. (1961). On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99 540-554. · Zbl 0118.13005
[5] Bogdan, K. and Zak, T. (2006). On Kelvin transformation. J. Theoret. Probab. 19 89-120. · Zbl 1105.60057
[6] Caballero, M. E. and Chaumont, L. (2006). Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab. 34 1012-1034. · Zbl 1098.60038
[7] Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2011). Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17 34-59. · Zbl 1284.60092
[8] Chaumont, L. (1996). Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 39-54. · Zbl 0879.60072
[9] Chaumont, L. (2013). An introduction to self-similar processes. Lecture notes.
[10] Chaumont, L. and Doney, R. A. (2005). On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 948-961. · Zbl 1109.60039
[11] Chaumont, L., Kyprianou, A., Pardo, J. C. and Rivero, V. (2012). Fluctuation theory and exit systems for positive self-similar Markov processes. Ann. Probab. 40 245-279. · Zbl 1241.60019
[12] Chaumont, L., Pantí, H. and Rivero, V. (2013). The Lamperti representation of real-valued self-similar Markov processes. Bernoulli 19 2494-2523. · Zbl 1284.60077
[13] Chybiryakov, O. (2006). The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups. Stochastic Process. Appl. 116 857-872. · Zbl 1094.60035
[14] Dereich, S., Döring, L. and Kyprianou, A. E. (2017). Real self-similar processes started from the origin. Ann. Probab. 45 1952-2003. · Zbl 1372.60052
[15] Döring, L. and Kyprianou, A. E. (2016). Perpetual integrals for Lévy processes. J. Theoret. Probab. 29 1192-1198. · Zbl 1356.60073
[16] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York.
[17] Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math. (2) 55 468-519. · Zbl 0047.09303
[18] Feller, W. (1954). The general diffusion operator and positivity preserving semi-groups in one dimension. Ann. of Math. (2) 60 417-436. · Zbl 0057.09805
[19] Getoor, R. K. (1966). Continuous additive functionals of a Markov process with applications to processes with independent increments. J. Math. Anal. Appl. 13 132-153. · Zbl 0138.40901
[20] Graversen, S. E. and Vuolle-Apiala, J. (1986). \( \alpha \)-self-similar Markov processes. Probab. Theory Related Fields 71 149-158. · Zbl 0561.60085
[21] Hunt, G. A. (1957). Markoff processes and potentials. I, II. Illinois J. Math. 1 44-93, 316-369. · Zbl 0100.13804
[22] Hunt, G. A. (1958). Markoff processes and potentials. III. Illinois J. Math. 2 151-213.
[23] Kallenberg, O. (1997). Foundations of Modern Probability. Probability and Its Applications (New York). Springer, New York. · Zbl 0892.60001
[24] Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113. Springer, New York. · Zbl 0638.60065
[25] Kiu, S. W. (1980). Semistable Markov processes in \({\mathbf{R}}^n \). Stochastic Process. Appl. 10 183-191. · Zbl 0436.60054
[26] Krühner, P. and Schnurr, A. (2018). Time change equations for Lévy-type processes. Stochastic Process. Appl. 128 963-978. · Zbl 1382.60110
[27] Kuznetsov, A., Kyprianou, A. E., Pardo, J. C. and Watson, A. R. (2014). The hitting time of zero for a stable process. Electron. J. Probab. 19 no. 30, 26. · Zbl 1293.60055
[28] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd ed. Universitext. Springer, Heidelberg. · Zbl 1384.60003
[29] Kyprianou, A. E. (2016). Deep factorisation of the stable process. Electron. J. Probab. 21 Paper No. 23, 28. · Zbl 1338.60130
[30] Kyprianou, A. E. (2018). Stable Lévy processes, self-similarity and the unit ball. ALEA Lat. Am. J. Probab. Math. Stat. 15 617-690. · Zbl 1396.60051
[31] Kyprianou, A. E., Pardo, J. C. and Watson, A. R. (2014). Hitting distributions of \(\alpha \)-stable processes via path censoring and self-similarity. Ann. Probab. 42 398-430. · Zbl 1306.60051
[32] Kyprianou, A. E., Rivero, V. and Sengül, B. (2017). Conditioning subordinators embedded in Markov processes. Stochastic Process. Appl. 127 1234-1254. · Zbl 1373.60085
[33] Kyprianou, A. E., Rivero, V. M. and Satitkanitkul, W. (2019). Conditioned real self-similar Markov processes. Stochastic Process. Appl. 129 954-977. · Zbl 1442.60047
[34] Kyprianou, A. E. and Vakeroudis, S. (2018). Stable windings at the origin. Stochastic Process. Appl. 128 4309-4325. · Zbl 1417.60030
[35] Jacod, J.and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin. · Zbl 1018.60002
[36] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205-225. · Zbl 0274.60052
[37] Li, P.-S. (2019). A continuous-state polynomial branching process. Stochastic Process. Appl. 129 2941-2967. · Zbl 1422.60149
[38] Nagasawa, M. (1964). Time reversions of Markov processes. Nagoya Math. J. 24 177-204. · Zbl 0133.10702
[39] Nagasawa, M. (1976). Note on pasting of two Markov processes. In Séminaire de Probabilités, X (Exposés Supplémentaires, Univ. Strasbourg, Strasbourg, Année Universitaire 1974/1975) 532-535.
[40] Peskir, G. (2015). On boundary behaviour of one-dimensional diffusions: From Brown to Feller and beyond. In William Feller, Selected Papers II 77-93. Springer, Berlin.
[41] Port, S. C. (1967). Hitting times and potentials for recurrent stable processes. J. Anal. Math. 20 371-395. · Zbl 0157.24702
[42] Profeta, C. and Simon, T. (2016). On the harmonic measure of stable processes. In Séminaire de Probabilités XLVIII. Lecture Notes in Math. 2168 325-345. Springer, Cham. · Zbl 1367.60057
[43] Rogozin, B. A. (1972). Distribution of the position of absorption for stable and asymptotically stable random walks on an interval. Teor. Veroyatn. Primen. 17 342-349.
[44] van Casteren, J. A. (1992). On martingales and Feller semigroups. Results Math. 21 274-288. · Zbl 0753.60068
[45] Volkonskii, V. A. (1958). Random substitution of time in strong Markov processes. Teor. Veroyatn. Primen. 3 332-350.
[46] Vuolle-Apiala, J. and Graversen, S. E. (1986). Duality theory for self-similar processes. Ann. Inst. Henri Poincaré Probab. Stat. 22 323-332. · Zbl 0608.60057
[47] Walsh, J. B. (1972). Markov processes and their functionals in duality. Z. Wahrsch. Verw. Gebiete 24 229-246. · Zbl 0248.60054
[48] Werner, F. (2017). Concatenating and pasting of right processes. Available at arXiv:1801.02595.
[49] Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res. 5 67-85. · Zbl 0428.60010
[50] Whitt, W. (2002). Stochastic-Process Limits. Springer Series in Operations Research. Springer, New York. An introduction to stochastic-process limits and their application to queues. · Zbl 0993.60001
[51] Zanzotto, P. A. (1997). On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion. Stochastic Process. Appl. 68 209-228. · Zbl 0911.60037
[52] Zanzotto, P. · Zbl 1017.60058
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