Abstract
We consider stochastic (partial) differential equations appearing as Markovian lifts of affine Volterra processes with jumps from the point of view of the generalized Feller property which was introduced in, e.g., Dörsek and Teichmann (A semigroup point of view on splitting schemes for stochastic (partial) differential equations, 2010. arXiv:1011.2651). In particular, we provide new existence, uniqueness and approximation results for Markovian lifts of affine rough volatility models of general jump diffusion type. We demonstrate that in this Markovian light the theory of stochastic Volterra processes becomes almost classical.
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The authors are grateful for the support of the ETH Foundation. Christa Cuchiero thanks the Forschungsinstitut für Mathematik for its support of a research stay at ETH Zürich in fall 2016. She additionally gratefully acknowledges financial support by the Vienna Science and Technology Fund (WWTF) under Grant MA16-021. Both authors are very grateful to Sergio Pulido for helping us to improve Sect. 5.2 and to an anonymous referee whose comments helped to improve the article along several lines.
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Cuchiero, C., Teichmann, J. Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case. J. Evol. Equ. 20, 1301–1348 (2020). https://doi.org/10.1007/s00028-020-00557-2
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DOI: https://doi.org/10.1007/s00028-020-00557-2
Keywords
- Variation of constant formula
- Stochastic partial differential equations
- Affine processes
- Stochastic Volterra processes
- Rough volatility models