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2013 | OriginalPaper | Chapter

Path Properties and Regularity of Affine Processes on General State Spaces

Authors : Christa Cuchiero, Josef Teichmann

Published in: Séminaire de Probabilités XLV

Publisher: Springer International Publishing

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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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previous chapter Approximation and Stability of Solutions of SDEs Driven by a Symmetric α Stable Process with Non-Lipschitz Coefficients

next chapter Langevin Process Reflected on a Partially Elastic Boundary II

Here, K ^∗ denotes the dual cone.

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 \({\mathbb{R}}^{n}\).

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

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.

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

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

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.

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

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Title: Path Properties and Regularity of Affine Processes on General State Spaces
Authors: Christa Cuchiero
Josef Teichmann
Publisher: Springer International Publishing
Book: Séminaire de Probabilités XLV
Print ISBN: 978-3-319-00320-7

Electronic ISBN: 978-3-319-00321-4

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-3-319-00321-4_8