Generalised ornstein–uhlenbeck processes and the convergence of lévy integrals

Lecture Notes in Mathematics

Volumn 1857, Issue , 2005, Pages 70-94

  Erickson, K Bruce ^a   Maller, Ross A ^b  

^a UNIVERSITY OF WASHINGTON   (United States)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Compound Poisson process; Lamperti transformation; L vy process; Ornstein Uhlenbeck type process; Stochastic integral; Subordinated perpetuity

Indexed keywords

EID: 84856273300     PISSN: 00758434     EISSN: 16179692     Source Type: Book Series    
DOI: 10.1007/978-3-540-31449-3_6     Document Type: Article

References (43)

1. n in the critical case. Ann. Probab., 25, 478–493.
2. Barndorff-Nielsen, O.E. (Ed.) (2002) Proceedings of the Second MaPhysto Conference, Centre for Math. Phys. & Stochastics Publications, Aarhus, no. 22; http://www.maphysto.dk/
3. Barndorff-Nielsen, O.E. and Shephard, N. (2001) Modelling by Lé vy processes for financial econometrics. In: Lévy processes, Theory and Applications, pp. 283–318
4. O. E. Barndorff-Nielsen, T. Mikosch, S. Resnick, Eds, Birkhäuser, Boston.
5. Bertoin, J. (1996) Lévy Processes. Cambridge University Press.
6. Bertoin, J. (1997) Regularity of the half-line for Lé vy processes. Bull. Sci. Math. 121, 345–354.
7. Brockwell, P.J. (2001) Lé vy driven CARMA processes, Ann. Inst. Statist. Math., 52, 1–18.
8. Brockwell, P.J. (2002) Continuous time ARMA processes, preprint.
9. Brofferio, S. (2003) How a centered random walk on the affine group goes to infinity. Ann. Inst. H. Poincaré, 39, 371–384.
10. Carmona, P., Petit, F. and Marc Yor, (2001) Exponential Functionals of Lé vy Processes. In: Lé vy processes, Theory and Applications, pp. 41–55, O. E. Barndorff-Nielsen, T. Mikosch, S. Resnick, Eds, Birkhäuser, Boston.
11. Chaumont, A. (2003) Personal communication.
12. Dolé ans-Dade, C. (1970) Quelques applications de la formule de changement de variables pour les semimartingales. Z. Wahrsch. Verw. Gebiete 16, 181–194.
13. Doney, R. and Maller, R.A. (2002) Stability and attraction to normality for Lévy processes at zero and at infinity. J. Theoret. Prob., 15 (no 3), 751–792.
14. Dufresne, D. (1990) The distribution of a perpetuity, with application to risk theory and pension funding, Scand. Actuar. J., 9, 39–79.
15. Dzhaparidze, K. and van Zanten, H. (2002) A series expansion of fractional Brownian Motion, CWI report 2002-14.
16. Erickson, K.B. (1973) The strong law of large numbers when the mean is undefined. Trans. Amer. Math. Soc. 185, 371–381.
17. Erickson, K. B. (1981) A limit theorem for renewal sequences with an application to local time. Z. Wahrsch. Verw. Gebiete 57, 535–558.
18. Erickson, K.B. and Maller, R.A. (2003) The strong law for subordinated random walks and Lé vy processes, to appear J. Theoret. Prob.
19. Goldie, C.M. (1991) Implicit renewal theory and the tails of solutions of random equations. Ann. Appl. Probab. 1, 126–166.
20. Goldie, C.M. and Maller, R.A. (2000) Stability of perpetuities. Ann. Probab. 28, 1195–1218.
21. Grincevičius, A. K. (1974) On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines. Theor. Probab. Appl. 19, 163–168.
22. Grincevičius, A. K. (1975a) Limit theorems for products of random linear transformations on the line. Lithuanian Math. J. 15, 568–579.
23. Grincevičius, A. K. (1975b) One limit distribution for a random walk on the line. Lithuanian Math. J. 15, 580–589.
24. Grincevičius, A. K. (1980) Products of random affine transformations. Lithuanian Math. J. 20, 279–282.
25. Grincevičius, A. K. (1981) A random difference equation. Lithuanian Math. J., 21, 302–306.
26. Iksanov, A.M (2003) Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Preprint.
27. Kesten, Harry (1973) Random difference equations and renewal theory for products of random matrices. Acta Math 131, 242–250.
28. Kesten, Harry and Maller, R.A. (1996) Two renewal theorems for random walks tending to infinity, Prob. Theor. Rel. Fields, 106, 1–38.
29. Klüppelberg, C., Lindner, A. and Maller, R.A. (2004) A continuous time GARCH process driven by a Lé vy process: stationarity and second order behaviour, J. Appl. Prob., to appear.
30. Mandelbrot, B.B. and Van Ness, J.W. (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437.
31. Milevsky, M.A. (1999) Martingales, scale functions and stochastic life annuities: a note. Insurance: Mathematics and Economics, 24, 149–154.
32. Norros, I., Valkeila, E. and Virtamo, J. (1999) An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5, 571–587.
33. Paulsen, J. (1993) Risk theory in a stochastic economic environment. Stoch. Proc. and Appl., 46, 327–361.
34. Paulsen, J. and Hove, A. (1999) Markov chain Monte Carlo simulation of the distribution of some perpetuities. Adv. Appl Prob., 31, 112–134.
35. Protter, P. (1990) Stochastic Integration and Differential Equations, A New Approach. Springer-Verlag.
36. Pruitt, W.E. (1981) General one-sided laws of the iterated logarithm. Ann. Probab. 9, 1–48.
37. Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, Cambridge.
38. Sato, K. (2001) Basic results on Lé vy processes. In: Lévy processes, Theory and Applications, pp. 3–38, O. E. Barndorff-Nielsen, T. Mikosch, S. Resnick, Eds, Birkhäuser, Boston.
39. Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob., 11, 750–783.
40. Yor, Marc (1992) Sur certaines fonctionnelles du mouvement brownien ré el, J. Appl Prob. 29, 202–208.
41. Yor, Marc (2001) Interpretations in terms of Brownian and Bessel meanders of the distribution of a subordinated perpetuity. In: Lévy Processes, Theory and Applications, pp. 361–375
42. O. E. Barndorff-Nielsen, T. Mikosch, S. Resnick, Eds, Birkhäuser, Boston.
43. Yor, Marc (2001) Exponential Functionals of Brownian Motion and Related Processes. Springer, New York, Berlin.