Fractional Skellam processes with applications to finance

Alexander Kerss 1 , Nikolai Leonenko 1 , and Alla Sikorskii 2
  • 1 Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4 YH, UK
  • 2 Department of Statistics and Probability, Michigan State University, 619 Red Cedar Road, East Lansing, MI, 48824, USA

Abstract

The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.

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