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Selfsimilar Additive Processes and Stationary Ornstein–Uhlenbeck Type Processes

  • Alfonso Rocha-Arteaga
  • Ken-iti Sato
Chapter
Part of the SpringerBriefs in Probability and Mathematical Statistics book series (SBPMS )

Abstract

Selfsimilar processes on \(\mathbb {R}^{d}\) are those stochastic processes whose finite-dimensional distributions are such that any change of time scale has the same effect as some change of spatial scale. Under the condition of stochastic continuity and non-zero, the relation of the two scale changes is expressed by a positive number c called exponent. A Lévy process is selfsimilar if and only if it is strictly stable.

References

  1. 17.
    Carr, P., Geman, H., Madan, D. B, & Yor, M. (2005). Pricing options on realized variance. Finance and Stochastics, 9, 453–475.Google Scholar
  2. 23.
    Eberlein, E., & Madan, D. B. (2009). Sato processes and the valuation of structured products. Quantitative Finance, 9, 1, 27–42.Google Scholar
  3. 38.
    Jeanblanc, M., Pitman, J., & Yor, M. (2002). Self-similar processes with independent increments associated with Lévy and Bessel processes. Stochastic Processes and their Applications, 100, 223–231.Google Scholar
  4. 48.
    Kokholm, T., & Nicolato, E. (2010). Sato processes in default modelling. Applied Mathematical Finance, 17(5), 377–397.Google Scholar
  5. 51.
    Kyprianou, A. E. (2014). Fluctuations of Lévy processes with applications (2nd ed.). Berlin: Springer.Google Scholar
  6. 53.
    Lamperti, J. (1962). Semi-stable stochastic processes. Transactions of the American Mathematical Society, 104, 62–78.MathSciNetCrossRefGoogle Scholar
  7. 60.
    Maejima, M., & Sato, K. (2003). Semi-Lévy processes, semi-selfsimilar additive processes, and semi-stationary Ornstein–Uhlenbeck type processes. Journal of Mathematics of Kyoto University, 43, 609–639.Google Scholar
  8. 62.
    Maejima, M., Sato, K., & Watanabe, T. (2000). Distributions of selfsimilar and semi-selfsimilar processes with independent increments. Statistics & Probability Letters, 47, 395–401.Google Scholar
  9. 63.
    Maejima, M., Suzuki, K., & Tamura, Y. (1999). Some multivariate infinitely divisible distributions and their projections. Probability and Mathematical Statistics, 19, 421–428.Google Scholar
  10. 88.
    Sato, K. (1990). Distributions of class L and self-similar processes with independent increments. In T. Hida, H. H. Kuo, J. Potthoff & L. Streit (Eds.), White noise analysis. Mathematics and applications (pp. 360–373). Singapore: World Scientific.Google Scholar
  11. 89.
    Sato, K. (1991). Self-similar processes with independent increments. Probability Theory and Related Fields, 89, 285–300.MathSciNetCrossRefGoogle Scholar
  12. 92.
    Sato, K. (1998). Multivariate distributions with selfdecomposable projections. Journal of the Korean Mathematical Society, 35, 783–791.Google Scholar
  13. 93.
    Sato, K. (1999). Lévy processes and infinitely divisible distributions. Cambridge: Cambridge University Press.Google Scholar
  14. 96.
    Sato, K. (2004). Stochastic integrals in additive processes and application to semi-Lévy processes. Osaka Journal of Mathematics, 41, 211–236.Google Scholar
  15. 106.
    Sato, K., & Yamamuro, K. (1998). On selfsimilar and semi-selfsimilar processes with independent increments. Journal of the Korean Mathematical Society, 35, 207–224.Google Scholar
  16. 107.
    Sato, K., & Yamamuro, K. (2000). Recurrence-transience for self-similar additive processes associated with stable distributions. Acta Applicandae Mathematica, 63, 375–384.Google Scholar
  17. 132.
    Watanabe, T. (1996). Sample function behavior of increasing processes of class L. Probability Theory and Related Fields, 104, 349–374.MathSciNetCrossRefGoogle Scholar
  18. 137.
    Watanabe, T., & Yamamuro, K. (2010). Limsup behaviors of multi-dimensional selfsimilar processes with independent increments. ALEA Latin American Journal of Probability and Mathematical Statistics, 7, 79–116.Google Scholar
  19. 141.
    Yamamuro, K. (2000a). Transience conditions for self-similar additive processes. Journal of the Mathematical Society of Japan, 52, 343–362.MathSciNetCrossRefGoogle Scholar
  20. 142.
    Yamamuro, K. (2000b). On recurrence for self-similar additive processes. Kodai Mathematical Journal, 23, 234–241.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alfonso Rocha-Arteaga
    • 1
  • Ken-iti Sato
    • 2
  1. 1.Facultad de Ciencias Físico-MatemáticasUniversidad Autónoma de SinaloaCuliacánMexico
  2. 2.Hachiman-yama 1101-5-103Tenpaku-kuJapan

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