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Exponential functionals of Markov additive processes. (English) Zbl 07206374
Summary: We provide necessary and sufficient conditions for convergence of exponential integrals of Markov additive processes. By contrast with the classical Lévy case studied by K. B. Erickson and R. A. Maller [J. Theor. Probab. 18, No. 2, 359–375 (2005; Zbl 1075.60044)] we have to distinguish between almost sure convergence and convergence in probability. Our proofs rely on recent results on perpetuities in a Markovian environment by G. Alsmeyer and F. Buckmann [J. Difference Equ. Appl. 23, No. 4, 699–740 (2017; Zbl 1371.60174)].
MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J74 Jump processes on discrete state spaces
60G51 Processes with independent increments; Lévy processes
60J25 Continuous-time Markov processes on general state spaces
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[1] L. Alili and D. Woodford. On the finiteness and tails of perpetuities under a Lamperti-Kiu MAP. 2019. Preprint. Available on arXiv:1811.10286v2.
[2] G. Alsmeyer and F. Buckmann. Stability of perpetuities in Markovian environment. J. Difference Equ. Appl., 23:699-740, 2017. · Zbl 1371.60174
[3] G. Alsmeyer and F. Buckmann. Fluctuation theory for Markov random walks. J. Theoret. Probab., 31:2266-2342, 2019. · Zbl 1434.60255
[4] S. Asmussen. Applied Probability and Queues. Springer, 2nd edition, 2003. · Zbl 1029.60001
[5] S. Asmussen and O. Kella. A multidimensional martingale for Markov additive processes and its applications. Adv. Appl. Probab., 32:376-393, 2000. · Zbl 0961.60081
[6] A. Behme. Exponential functionals of Lévy processes with jumps. ALEA, 12:375-397, 2015. · Zbl 1321.60092
[7] A. Behme and A. Lindner. On exponential functionals of Lévy processes. J. Theor. Probab., 28:681-720, 2015. · Zbl 1323.60063
[8] A. Behme, A. Lindner, and R. Maller. Stationary solutions of the stochastic differential equation \(d{V}_t={V}_{t-}d{U}_t+d{L}_t\) with Lévy noise. Stoch. Proc. Appl., 121:91-108, 2011. · Zbl 1209.60033
[9] J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surveys, 2:191-212, 2005. · Zbl 1189.60096
[10] E. Çinlar. Markov additive processes I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 24:85-93, 1972. · Zbl 0236.60047
[11] E. Çinlar. Markov additive processes II. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 24:95-121, 1972. · Zbl 0236.60048
[12] L. Chaumont, H. Panti, and V. Rivero. The Lamperti representation of real-valued self-similar Markov processes. Bernoulli, 19:2494-2523, 2013. · Zbl 1284.60077
[13] S. Dereich, L. Döring, and A. E. Kyprianou. Real self-similar processes started from the origin. Ann. Probab., 45:1952-2003, 2017. · Zbl 1372.60052
[14] K. B. Erickson and R. A. Maller. Drift to infinity and the strong law for subordinated random walks and Lévy processes. J. Theoret. Probab., 18:359-375, 2005. · Zbl 1075.60044
[15] K. B. Erickson and R. A. Maller. Generalised Ornstein-Uhlenbeck processes and the convergence of Lévy integrals. In M. Emery, M. Ledoux, and M. Yor, editors, Séminaire de Probabilités XXXVIII, Lecture Notes in Mathematics, volume 1857, pages 70-94. Springer, Berlin, 2005. · Zbl 1066.60053
[16] C. M. Goldie and R. Maller. Stability of perpetuities. Ann. Probab., 28:1195-1218, 2000. · Zbl 1023.60037
[17] B. Grigelionis. Additive Markov processes. Lith. Math. J., 18:340-342, 1978. · Zbl 0409.60079
[18] J. D. Hamilton. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrika, 57:357-384, 1989. · Zbl 0685.62092
[19] O. Kella and S. Ramasubramanian. Asymptotic irrelevance of initial conditions for Skorohod reflection mapping on the nonnegative orthant. Mathematics of Operations Research, 37:301-312, 2012. · Zbl 1248.90029
[20] H. Kesten. The limit points of a normalized random walk. Ann. Math. Statist., 41:1173-1205, 1970. · Zbl 0233.60062
[21] C. Klüppelberg, A. Lindner, and R. Maller. A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour. J. Appl. Probab., 41:601-622, 2004. · Zbl 1068.62093
[22] P. Klusik and Z. Palmowski. A note on Wiener-Hopf factorization for Markov additive processes. J. Theoret. Probab., 27.1:202-219, 2014. · Zbl 1305.60034
[23] A. Kuznetsov, J. C. Pardo, and M. Savov. Distributional properties of exponential functionals of Lévy processes. Electron. J. Probab., 17:1-35, 2012. · Zbl 1246.60073
[24] A. E. Kyprianou, V. Rivero, and W. Satitkanitkul. Conditioned real self-similar Markov processes. Stoch. Proc. Appl., 129:954-977, 2019. · Zbl 1442.60047
[25] A. Lindner and R. Maller. Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes. Stoch. Proc. Appl., 115:1701-1722, 2005. · Zbl 1080.60056
[26] J. C. Pardo, P. Patie, and M. Savov. A Wiener-Hopf type factorization of the exponential functional of Lévy processes. J. London Math. Soc., 86:930-956, 2012. · Zbl 1272.60027
[27] J. C. Pardo, V. Rivero, and K. van Schaik. On the density of exponential functionals of Lévy processes. Bernoulli, 19:1938-1964, 2013. · Zbl 1305.60035
[28] J. Paulsen. Risk theory in a stochastic economic environment. Stoch. Proc. Appl., 46:327-361, 1993. · Zbl 0777.62098
[29] P. E. Protter. Stochastic Integration and Differential Equations. Springer, Berlin, 2nd edition, 2004. · Zbl 1041.60005
[30] P. Salminen and L. Vostrikova. On exponential functionals of processes with independent increments. Theory Probab. Appl., 63(2):267-291, 2018. · Zbl 1414.60033
[31] P. Salminen and L. Vostrikova. On moments of integral exponential functionals of additive processes. Statist. Probab. Lett., 146: 139-146, 2019. · Zbl 1407.60115
[32] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge, 2nd edition, 2013. · Zbl 1287.60003
[33] R. Stephenson. On the exponential functional of Markov additive processes, and applications to multi-type self-similar fragmentation processes and trees. ALEA, Lat. Am. J. Probab. Math. Stat., 15:1257-1292, 2018. · Zbl 1414.60022
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