## Journal of Applied Probability

### Stationary-increment student and variance-gamma processes

#### Abstract

A continuous-time model with stationary increments for asset price {Pt} is an extension of the symmetric subordinator model of Heyde (1999), and allows for skewness of returns. In the setting of independent variance-gamma-distributed returns the model resembles closely that of Madan, Carr, and Chang (1998). A simple choice of parameters renders {e-rtPt} a familiar martingale. We then specify the activity time process, {Tt}, for which {Tt - t} is asymptotically self-similar and {τt}, with τt = Tt - Tt-1, is gamma distributed. This results in a skew variance-gamma distribution for each log price increment (return) Xt and a model for {Xt} which incorporates long-range dependence in squared returns. Our approach mirrors that for the (symmetric) Student process model of Heyde and Leonenko (2005), to which the present work is intended as a complement and a sequel. One intention is to compare, partly on the basis of fitting to data, versions of the general model wherein the returns have either (symmetric) t-distributions or variance-gamma distributions.

#### Article information

Source
J. Appl. Probab. Volume 43, Number 2 (2006), 441-453.

Dates
First available in Project Euclid: 8 July 2006

http://projecteuclid.org/euclid.jap/1152413733

Digital Object Identifier
doi:10.1239/jap/1152413733

Mathematical Reviews number (MathSciNet)
MR2248575

Zentralblatt MATH identifier
1103.62103

#### Citation

Finlay, Richard; Seneta, Eugene. Stationary-increment student and variance-gamma processes. J. Appl. Probab. 43 (2006), no. 2, 441--453. doi:10.1239/jap/1152413733. http://projecteuclid.org/euclid.jap/1152413733.

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