Skip to main content
SpringerLink
Log in

On the transition laws of p-tempered \(\alpha \)-stable OU-processes

  • Original paper
  • Published:
Computational Statistics Aims and scope Submit manuscript

Abstract

We derive an explicit representation for the transition law of a p-tempered \(\alpha \)-stable process of Ornstein–Uhlenbeck-type and use it to develop a methodology for simulation. Our results apply in both the univariate and multivariate cases. Special attention is given to the case where \(p\le \alpha \), which is more complicated and requires developing the new class of so-called incomplete gamma distributions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Barndorff-Nielsen OE, Maejima M, Sato K (2006) Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12(1):1–33

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen OE, Shephard N (2001) Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Ser B 63(2):167–241

    Article  MathSciNet  Google Scholar 

  • Bianchi ML, Rachev ST, Kim YS, Fabozzi FJ (2011) Tempered infinitely divisible distributions and processes. Theory Prob Appl 55(1):2–26

    Article  MathSciNet  Google Scholar 

  • Bianchi ML, Rachev ST, Fabozzi FJ (2017) Tempered stable Ornstein–Uhlenbeck processes: a practical view. Commun Stat Simul Comput 46(1):423–445

    Article  MathSciNet  Google Scholar 

  • Byczkowski T, Nolan JP, Rajput B (1993) Approximation of multidimensional stable densities. J Multivar Anal 46(1):13–31

    Article  MathSciNet  Google Scholar 

  • Cont R, Tankov P (2004) Financial modeling With jump processes. Chapman & Hall, Boca Raton

    MATH  Google Scholar 

  • Grabchak M (2012) On a new class of tempered stable distributions: moments and regular variation. J Appl Prob 49(4):1015–1035

    Article  MathSciNet  Google Scholar 

  • Grabchak M (2016) Tempered stable distributions: stochastic models For multiscale processes. Springer, Cham, Switzerland

    Book  Google Scholar 

  • Grabchak M (2019) Rejection sampling for tempered Lévy processes. Stat Comput 29(3):549–558

    Article  MathSciNet  Google Scholar 

  • Grabchak M (2020) On the simulation for general tempered stable Ornstein–Uhlenbeck processes. J Stat Comput Simul 90(6):1057–1081

    Article  MathSciNet  Google Scholar 

  • Grabchak M, Cao L (2017) SymTS: symmetric tempered stable distributions. Ver. 1.0, R Package. https://cran.r-project.org/web/packages/SymTS/index.html

  • Grabchak M, Samorodnitsky G (2010) Do financial returns have finite or infinite variance? A paradox and an explanation. Quant Finance 10(8):883–893

    Article  MathSciNet  Google Scholar 

  • Johnson ME (1987) Multivariate statistical simulation. Wiley, New York

    Book  Google Scholar 

  • Kawai R, Masuda H (2012) Infinite variation tempered stable Ornstein–Uhlenbeck processes with discrete observations. Commun Stat Simul Comput 41(1):125–139

    Article  MathSciNet  Google Scholar 

  • Kozubowski TJ, Podgórski K (2003) A log-Laplace growth rate model. Math Sci 28(1):49–60

    MathSciNet  MATH  Google Scholar 

  • Qu Y, Dassios A, Zhao H (2019) Exact simulation of gamma-driven Ornstein–Uhlenbeck processes with finite and infinite activity jumps. J Operat Res Soc https://doi.org/10.1080/01605682.2019.1657368

  • Rocha-Arteaga A, Sato K (2019) Topics in infinitely divisible distributions and Lévy processes, Revised edn. Springer, Cham

  • Rosiński J (2007) Tempering stable processes. Stoch Process Appl 117(6):677–707

    Article  MathSciNet  Google Scholar 

  • Rosiński J, Sinclair JL (2010) Generalized tempered stable processes. Banach Cent Publ 90:153–170

    Article  MathSciNet  Google Scholar 

  • Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Stacy EW (1962) A generalization of the gamma distribution. Ann Math Stat 33(3):1187–1192

    Article  MathSciNet  Google Scholar 

  • Tashiro Y (1977) On methods for generating uniform random points on the surface of a sphere. Ann Inst Stat Math 29(1):295–300

    Article  MathSciNet  Google Scholar 

  • Terdik G, Woyczyński WA (2006) Rosiński measures for tempered stable and related Ornstien–Uhlenbeck processes. Prob Math Stat 26(2):213–243

    MATH  Google Scholar 

  • Zhang S, Zhang X (2008) Exact simulation of IG-OU processes. Methodol Comput Appl Prob 10(3):337–355

    Article  MathSciNet  Google Scholar 

  • Zhang S, Zhang X (2009) On the transition law of tempered stable Ornstein–Uhlenbeck processes. J Appl Prob 46(3):721–731

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University Of North Carolina Charlotte, Charlotte, USA

    Michael Grabchak

Authors
  1. Michael Grabchak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Grabchak.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grabchak, M. On the transition laws of p-tempered \(\alpha \)-stable OU-processes. Comput Stat 36, 1415–1436 (2021). https://doi.org/10.1007/s00180-020-01055-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00180-020-01055-2

Keywords

Search

Navigation