Open Access
October 2019 Affine Volterra processes
Eduardo Abi Jaber, Martin Larsson, Sergio Pulido
Ann. Appl. Probab. 29(5): 3155-3200 (October 2019). DOI: 10.1214/19-AAP1477


We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier–Laplace functional in terms of the solution of an associated system of deterministic integral equations of convolution type, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and clarify recent results in the literature on rough volatility models in finance.


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Eduardo Abi Jaber. Martin Larsson. Sergio Pulido. "Affine Volterra processes." Ann. Appl. Probab. 29 (5) 3155 - 3200, October 2019.


Received: 1 December 2017; Revised: 1 March 2019; Published: October 2019
First available in Project Euclid: 18 October 2019

zbMATH: 07155069
MathSciNet: MR4019885
Digital Object Identifier: 10.1214/19-AAP1477

Primary: 60J20
Secondary: 45D05 , 60G22 , 91G20

Keywords: Affine processes , Riccati–Volterra equations , Rough volatility , Stochastic Volterra equations

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.29 • No. 5 • October 2019
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