# Affine Volterra processes

@article{Jaber2019AffineVP, title={Affine Volterra processes}, author={Eduardo Abi Jaber and Martin Larsson and Sergio Pulido}, journal={The Annals of Applied Probability}, year={2019} }

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…

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## References

SHOWING 1-10 OF 31 REFERENCES

### Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations

- Mathematics
- 2008

### The affine transform formula for affine jump-diffusions with a general closed convex state space

- Mathematics
- 2010

We establish existence of exponential moments and the validity of the affine transform formula for affine jump-diffusions with a general closed convex state space. This extends known results for…

### Corrections to: Extending the martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E. 's

- Mathematics
- 1999

We extend the definition of Walsh's martingale measure stochastic integral so as to be able to solve stochastic partial differential equations whose Green's function is not a function but a Schwartz…

### Affine processes with compact state space

- Mathematics
- 2017

The behavior of affine processes, which are ubiquitous in a wide range of applications, depends crucially on the choice of state space. We study the case where the state space is compact, and prove…

### Sample Path Properties of Volterra Processes

- Mathematics
- 2011

We consider the regularity of sample paths of Volterra processes. These processes are defined as stochastic integrals $$ M(t)=\int_{0}^{t}F(t,r)dX(r), \ \ t \in \mathds{R}_{+}, $$ where $X$ is a…

### The characteristic function of rough Heston models

- Mathematics
- 2016

It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to…

### Regularity Properties of Some Stochastic Volterra Integrals with Singular Kernel

- Mathematics
- 2002

We prove the Hölder continuity of some stochastic Volterra integrals, with singular kernels, under integrability assumptions on the integrand. Some applications to processes arising in the analysis…

### Affine Diffusions with Non-Canonical State Space

- Mathematics
- 2010

Multidimensional affine diffusions have been studied in detail for the case of a canonical state space. We present results for general state spaces and provide a complete characterization of all…

### Uniqueness for Volterra-type stochastic integral equations

- Mathematics
- 2015

We study uniqueness for a class of Volterra-type stochastic integral equations. We focus on the case of non-Lipschitz noise coefficients. The connection of these equations to certain degenerate…

### Continuous martingales and Brownian motion

- Mathematics
- 1990

0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.-…