WIAS Preprint No. 1297, (2008)

Holomorphic transforms with application to affine processes


  • Belomestny, Denis
  • Kampen, Jörg
  • Schoenmakers, John G. M.
    ORCID: 0000-0002-4389-8266

2010 Mathematics Subject Classification

  • 60J25 91B28


  • Itô-Lévy processes, holomorphic transforms, affine processes




In a rather general setting of Itô-Lévy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multidimensional affine Itô-Lévy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.

Appeared in

  • Journal of Functional Analysis, Vol. 257, 4, (2009) pp. 1222-1250

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