Annals of Probability

Entrance and exit at infinity for stable jump diffusions

Leif Döring and Andreas E. Kyprianou

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Abstract

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 \begin{equation*}dZ_{t}=\sigma (Z_{t-})\,dX_{t},\end{equation*} driven by stable Lévy processes for $\alpha \in (0,2)$. Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller’s work but exit and entrance from infinite boundaries has long remained open. We show that the presence of jumps implies features not seen in the diffusive setting without drift. Finite time explosion is possible for $\alpha \in (0,1)$, whereas entrance from different kinds of infinity is possible for $\alpha \in [1,2)$. Accordingly, we derive necessary and sufficient conditions on $\sigma $.

Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti–Kiu representation and new Wiener–Hopf factorisations for Lévy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz–Bogdan–Żak transformation, entrance laws for self-similar Markov processes, perpetual integrals of Lévy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt–Nagasawa duality and Getoor’s characterisation of transience and recurrence.

Article information

Source
Ann. Probab., Volume 48, Number 3 (2020), 1220-1265.

Dates
Received: March 2018
Revised: May 2019
First available in Project Euclid: 17 June 2020

Permanent link to this document
https://projecteuclid.org/euclid.aop/1592359227

Digital Object Identifier
doi:10.1214/19-AOP1389

Mathematical Reviews number (MathSciNet)
MR4112713

Zentralblatt MATH identifier
07226359

Subjects
Primary: 60H20: Stochastic integral equations 60G52: Stable processes

Keywords
SDEs entrance explosion Kelvin transform duality stable Lévy processes

Citation

Döring, Leif; Kyprianou, Andreas E. Entrance and exit at infinity for stable jump diffusions. Ann. Probab. 48 (2020), no. 3, 1220--1265. doi:10.1214/19-AOP1389. https://projecteuclid.org/euclid.aop/1592359227


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References

  • [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.
  • [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.
  • [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.
  • [5] Bogdan, K. and Zak, T. (2006). On Kelvin transformation. J. Theoret. Probab. 19 89–120.
  • [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.
  • [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.
  • [8] Chaumont, L. (1996). Conditionings and path decompositions for Lévy processes. Stochastic Process. Appl. 64 39–54.
  • [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.
  • [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.
  • [12] Chaumont, L., Pantí, H. and Rivero, V. (2013). The Lamperti representation of real-valued self-similar Markov processes. Bernoulli 19 2494–2523.
  • [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.
  • [14] Dereich, S., Döring, L. and Kyprianou, A. E. (2017). Real self-similar processes started from the origin. Ann. Probab. 45 1952–2003.
  • [15] Döring, L. and Kyprianou, A. E. (2016). Perpetual integrals for Lévy processes. J. Theoret. Probab. 29 1192–1198.
  • [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.
  • [18] Feller, W. (1954). The general diffusion operator and positivity preserving semi-groups in one dimension. Ann. of Math. (2) 60 417–436.
  • [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.
  • [20] Graversen, S. E. and Vuolle-Apiala, J. (1986). $\alpha$-self-similar Markov processes. Probab. Theory Related Fields 71 149–158.
  • [21] Hunt, G. A. (1957). Markoff processes and potentials. I, II. Illinois J. Math. 1 44–93, 316–369.
  • [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.
  • [24] Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113. Springer, New York.
  • [25] Kiu, S. W. (1980). Semistable Markov processes in ${\mathbf{R}}^{n}$. Stochastic Process. Appl. 10 183–191.
  • [26] Krühner, P. and Schnurr, A. (2018). Time change equations for Lévy-type processes. Stochastic Process. Appl. 128 963–978.
  • [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.
  • [28] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2nd ed. Universitext. Springer, Heidelberg.
  • [29] Kyprianou, A. E. (2016). Deep factorisation of the stable process. Electron. J. Probab. 21 Paper No. 23, 28.
  • [30] Kyprianou, A. E. (2018). Stable Lévy processes, self-similarity and the unit ball. ALEA Lat. Am. J. Probab. Math. Stat. 15 617–690.
  • [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.
  • [32] Kyprianou, A. E., Rivero, V. and Sengül, B. (2017). Conditioning subordinators embedded in Markov processes. Stochastic Process. Appl. 127 1234–1254.
  • [33] Kyprianou, A. E., Rivero, V. M. and Satitkanitkul, W. (2019). Conditioned real self-similar Markov processes. Stochastic Process. Appl. 129 954–977.
  • [34] Kyprianou, A. E. and Vakeroudis, S. (2018). Stable windings at the origin. Stochastic Process. Appl. 128 4309–4325.
  • [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.
  • [36] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205–225.
  • [37] Li, P.-S. (2019). A continuous-state polynomial branching process. Stochastic Process. Appl. 129 2941–2967.
  • [38] Nagasawa, M. (1964). Time reversions of Markov processes. Nagoya Math. J. 24 177–204.
  • [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.
  • [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.
  • [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.
  • [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.
  • [47] Walsh, J. B. (1972). Markov processes and their functionals in duality. Z. Wahrsch. Verw. Gebiete 24 229–246.
  • [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.
  • [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.
  • [51] Zanzotto, P. A. (1997). On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion. Stochastic Process. Appl. 68 209–228.
  • [52] Zanzotto, P. A. (2002). On stochastic differential equations driven by a Cauchy process and other stable Lévy motions. Ann. Probab. 30 802–825.