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Markovian lifts of positive semidefinite affine Volterra type processes

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  • Christa Cuchiero
  • Josef Teichmann

Abstract

We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein Uhlenbeck processes whose state space are matrix valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multivariate Hawkes type processes. We apply these affine covariance processes for multivariate (rough) volatility modeling and introduce a (rough) multivariate Volterra Heston type model.

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  • Christa Cuchiero & Josef Teichmann, 2019. "Markovian lifts of positive semidefinite affine Volterra type processes," Papers 1907.01917, arXiv.org, revised Sep 2019.
    • Handle: RePEc:arx:papers:1907.01917
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    References listed on IDEAS

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    1. Eduardo Abi Jaber & Omar El Euch, 2018. "Markovian structure of the Volterra Heston model," Working Papers hal-01716696, HAL.
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    4. Philipp Doersek & Josef Teichmann, 2010. "A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations," Papers 1011.2651, arXiv.org.
    5. Christa Cuchiero & Damir Filipovi'c & Eberhard Mayerhofer & Josef Teichmann, 2009. "Affine processes on positive semidefinite matrices," Papers 0910.0137, arXiv.org, revised Apr 2011.
    6. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    7. Elisa Alòs & Yan Yang, 2014. "A closed-form option pricing approximation formula for a fractional Heston model," Economics Working Papers 1446, Department of Economics and Business, Universitat Pompeu Fabra.
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