On Inhomogeneous Affine Volterra Processes: Stationarity and Applications to the Volterra Heston Model

2025-12-10

AI Summary

This paper investigates approximate stationarity over finite time horizons for inhomogeneous affine Volterra processes, which inherently fail to attain true stationarity. To address this, it introduces the novel concept of “pseudo-stationarity”—a state wherein all marginal distributions share identical means and variances—and constructs a pseudo-stationary Volterra Heston model with a closed-form characteristic function. Methodologically, the analysis integrates extended exponential-affine transformations, stochastic Volterra equation theory, fractional calculus, and asymptotic distribution theory. Key contributions include: (i) rigorous proof of the existence of the long-time limiting distribution and characterization of its sensitivity to initial conditions—showing such sensitivity vanishes *only* when the kernel is an α-fractional integral kernel; and (ii) construction of an associated stationary process, empirically validating the practical utility and superior accuracy of pseudo-stationary modeling for finite-horizon applications.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Studies of user behavior, including longitudinal effects of personalized systems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

True Volterra equations are inherently non stationary and therefore do not admit $ extit{genuine stationary regimes}$ over finite horizons. This motivates the study of the finite-time behavior of the solutions to scaled inhomogeneous affine Stochastic Volterra equations through the lens of a weaker notion of stationarity referred to as $ extit{fake stationary regime}$ in the sense that all marginal distributions share the same expectation and variance. As a first application, we introduce the $ extit{Fake stationary Volterra Heston model}$ and derive a closed-form expression for its characteristic function. Having established this finite-time proxy for stationarity, we then investigate the asymptotic (long-time) behavior to assess whether genuine stationary regimes emerge in the limit. Using an extension of the exponential-affine transformation formula for those processes, we establish in the long run the existence of limiting distributions, which (unlike in the case of classical affine diffusion processes) may depend on the initial state of the process, unless the Volterra kernel coincides with the $α-$ fractional integration kernel, for which the dependence on the initial state vanishes. We then proceed to the construction of stationary processes associated with these limiting distributions. However, the dynamics in this long-term regime are analytically intractable, and the process itself is not guaranteed to be stationary in the classical sense over finite horizons. This highlights the relevance of finite-time analysis through the lens of the aforementioned $ extit{fake stationarity}$, which offers a tractable approximation to stationary behavior in genuinely non-stationary Volterra systems.

Problem

• Analyzes finite-time behavior of inhomogeneous affine Volterra processes using fake stationarity.
• Derives closed-form characteristic function for the Fake stationary Volterra Heston model.
• Investigates asymptotic long-time behavior and limiting distributions of Volterra processes.

Innovation

• Introduces fake stationary regime for finite-time analysis
• Derives closed-form characteristic function for Volterra Heston model
• Establishes limiting distributions dependent on initial state