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Path Properties and Regularity of Affine Processes on General State Spaces

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Book cover Séminaire de Probabilités XLV

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2078))

Abstract

We provide a new proof for regularity of affine processes on general state spaces by methods from the theory of Markovian semimartingales. On the way to this result we also show that the definition of an affine process, namely as stochastically continuous time-homogeneous Markov process with exponential affine Fourier–Laplace transform, already implies the existence of a càdlàg version. This was one of the last open issues in the fundaments of affine processes.

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Notes

  1. 1.

    Here, K  ∗  denotes the dual cone.

  2. 2.

    In Definition 2.1 affine processes on the canonical state space \(D = \mathbb{R}_{+}^{m} \times {\mathbb{R}}^{n-m}\) are considered, whereas in Definition 12.1 the state space D can be an arbitrary subset of \({\mathbb{R}}^{n}\).

  3. 3.

    Due to relation (9) and the continuity of (t, u)↦h(x; t, u), this is also implied by the weaker condition (7).

  4. 4.

    As mentioned at the beginning of Sect.  2 , ∞ corresponds to a “point at infinity” and D Δ ∪{∞} is the one-point compactification of D Δ . If the state space D is compact, we do not adjoin {∞} and only consider a sequence with values in D Δ.

  5. 5.

    Note that these subsequences depend on u. For notational convenience we however suppress the dependence on u.

  6. 6.

    Note that due to the measurable projection theorem, \(\widetilde{\Omega } \in {\mathcal{F}}^{x}\).

  7. 7.

    In the case of affine processes, this would be implied by the differentiability of Φ and ψ with respect to t, which we only prove in Sect. 6 using the results of this paragraph.

  8. 8.

    Here, \(\mathcal{O}(\mathcal{F}_{t})\) denotes the \((\mathcal{F}_{t})\)-optional σ-algebra.

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Acknowledgements

We thank Georg Grafendorfer and Enno Veerman for discussions and helpful comments. Both authors gratefully acknowledge the financial support by the ETH Foundation.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090, Wien, Austria

    Christa Cuchiero

  2. Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

    Josef Teichmann

Authors
  1. Christa Cuchiero
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  2. Josef Teichmann
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Correspondence to Josef Teichmann .

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Editors and Affiliations

  1. , Laboratoire de Mathématiques, Université de Versailles-St Quentin, avenue des États-Unis 45, Versailles, 78035, France

    Catherine Donati-Martin

  2. IECN, Campus scientifique, BP 239, Nancy-Université, INRIA, Vandoeuvre-lès-Nancy, 54506, France

    Antoine Lejay

  3. Laboratoire de Mathématiques, Université de Versailles-St-Quentin, avenue des Etats-Unis 45, Versailles, 78035, France

    Alain Rouault

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Cuchiero, C., Teichmann, J. (2013). Path Properties and Regularity of Affine Processes on General State Spaces. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_8

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