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Some explicit identities associated with positive self-similar Markov processes. (English) Zbl 1170.60017
Some special classes of Lévy processes are considered with no Gaussian component whose Lévy measure is of the type $$\pi(dx)=e^{\gamma x}\nu(e^x-1)dx$$, where $$\nu$$ is the density of the stable Lévy measure and $$\gamma$$ is a positive parameter which depends on its characteristics. These processes were introduced by M. E. Caballero and L. Chaumont [J. Appl. Probab. 43, 967–983 (2006; Zbl 1133.60316)] as the underlying Lévy processes in the Lamperti representation of conditioned stable Lévy processes. The law of these Lévy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points are computed explicitly.

##### MSC:
 60G18 Self-similar stochastic processes 60G51 Processes with independent increments; Lévy processes 60G52 Stable stochastic processes 60G40 Stopping times; optimal stopping problems; gambling theory
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##### References:
 [1] V. Bernyk, R.C. Dalang, G. Peskir, The law of the supremum of a stable Lévy process with no negative jumps, Ann. Probab. (2008) (in press) · Zbl 1185.60051 [2] Bertoin, J., Lévy processes, (1996), Cambridge University Press Cambridge · Zbl 0861.60003 [3] Bertoin, J.; Caballero, M.E., Entrance from 0+ for increasing semi-stable Markov processes, Bernoulli, 8, 2, 195-205, (2002) · Zbl 1002.60032 [4] Bertoin, J.; Yor, M., The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes, Potential anal., 17, 4, 389-400, (2002) · Zbl 1004.60046 [5] Bertoin, J.; Yor, M., On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes, Ann. fac. sci. Toulouse VI ser. math., 11, 1, 33-45, (2002) · Zbl 1031.60038 [6] Bertoin, J.; Yor, M., Exponential functionals of Lévy processes, Probab. surv., 2, 191-212, (2005) · Zbl 1189.60096 [7] Blumenthal, R.; Getoor, R.K.; Ray, D.B., On the distribution of first hits for the symmetric stable processes, Trans. amer. math. soc., 99, 540-554, (1961) · Zbl 0118.13005 [8] Boyarchenko, S.I.; Levendorskii, S.Z., Non-Gaussian merton – black – scholes theory, (2002), World Scientific Singapore · Zbl 0997.91031 [9] Caballero, M.E.; Chaumont, L., Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes, Ann. probab., 34, 1012-1034, (2006) · Zbl 1098.60038 [10] Caballero, M.E.; Chaumont, L., Conditioned stable Lévy processes and the Lamperti representation, J. appl. probab., 43, 967-983, (2006) · Zbl 1133.60316 [11] Carr, P.; Geman, H.; Madan, D.B.; Yor, M., The fine structure of asset returns: an empirical investigation, J. business, 75, 305-332, (2002) [12] Chaumont, L., Conditionings and path decompositions for Lévy processes, Stochastic process. appl., 64, 39-54, (1996) · Zbl 0879.60072 [13] Chaumont, L.; Pardo, J.C., The lower envelope of positive self-similar Markov processes, Electron J. probab., 11, 1321-1341, (2006) · Zbl 1127.60034 [14] Cont, R.; Tankov, P., Financial modeling with jump processes, (2004), Chapman and Hall/CRC Boca Raton, FL · Zbl 1052.91043 [15] Doney, R.A., Fluctuation theory for Lévy processes, () · Zbl 0982.60048 [16] Doney, R.A.; Kyprianou, A.E., Overshoots and undershoots of Lévy processes, Ann. appl. probab., 16, 1, 91-106, (2006) · Zbl 1101.60029 [17] Getoor, R.K., Continuous additive functionals of a Markov process with applications to processes with independent increments, J. math. anal. appl., 13, 132-153, (1966) · Zbl 0138.40901 [18] Kou, S.G.; Wang, H., First passage times of a jump diffusion process, Adv. in appl. probab., 35, 504-531, (2003) · Zbl 1037.60073 [19] Kyprianou, A.E., Introductory lectures of fluctuations of Lévy processes with applications, (2006), Springer · Zbl 1104.60001 [20] Lamperti, J.W., Semi-stable Markov processes, Z. wahrscheinlichkeitstheor. verwandte geb., 22, 205-225, (1972) · Zbl 0274.60052 [21] A. Lewis, E. Mordecki, Wiener-Hopf factorization for Lévy processes having negative jumps with rational transforms. (2005) (preprint) [22] Méjane, O., Upper bound of a volume exponent for directed polymers in a random environment, Ann. inst. H. Poincaré probab. statist., 40, 3, 299-308, (2004) · Zbl 1041.60079 [23] Monrad, D.; Silverstein, M.L., Stable processes: sample function growth at a local minimum, Z. wahrscheinlichkeitstheor. verwandte geb., 49, 2, 177-210, (1979) · Zbl 0431.60041 [24] Pistorius, M.R., On maxima and ladder processes for a dense class of Lévy processes, J. appl. probab., 43, 208-220, (2006) · Zbl 1102.60044 [25] V. Rivero, Recouvrements aléatoires et processus de Markov auto-similaires. Thèse de doctorat de l’université Paris VI, (2004) [26] Rivero, V., Recurrent extensions of self-similar Markov processes and cramér’s condition, Bernoulli, 11, 3, 471-509, (2005) · Zbl 1077.60055 [27] Rogozin, B.A., The distribution of the first hit for stable and asymptotically stable random walks on an interval, Theory probab. appl., 17, 332-338, (1972) · Zbl 0272.60050 [28] Sato, K.I., Lévy processes and infinitely divisible distributions, (1999), Cambridge University Press Cambridge · Zbl 0973.60001 [29] Schoutens, W., Lévy processes in finance. pricing finance derivatives, (2003), Wiley New York [30] Zolotarev, V.M., ()
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