Feller's test for explosions of stochastic Volterra equations

TL;DR

This paper develops a Feller's test to determine explosion behavior in one-dimensional stochastic Volterra processes with convolution kernels, extending classical results to path-dependent, memory-including models.

Contribution

It introduces a new sufficient condition for explosion in stochastic Volterra processes, especially for models with memory, and discusses approximation methods for singular kernels.

Findings

Derived a Feller's test for stochastic Volterra processes.

Illustrated results with Volterra square-root, Jacobi, and power-type diffusions.

Discussed approximation of fractional kernels in finance applications.

Abstract

This paper provides a Feller's test for explosions of one-dimensional continuous stochastic Volterra processes of convolution type. The study focuses on dynamics governed by nonsingular kernels, which preserve the semimartingale property of the processes and introduce memory features through a path-dependent drift. In contrast to the classical path-independent case, the sufficient condition derived in this study for a Volterra process to remain in the interior of an interval is generally more restrictive than the necessary condition. The results are illustrated with three specifications of the dynamics: the Volterra square-root diffusion, the Volterra Jacobi process and the Volterra power-type diffusion. For the Volterra square-root diffusion, also known as the Volterra CIR process, the paper presents a detailed discussion on the approximation of the singular fractional kernel with a sum of exponentials, a method commonly employed in the mathematical finance literature.