Boundary behaviour of the Volterra square-root process

2026-06-05

AI Summary

This study investigates the boundary behavior of Volterra square-root processes on the non-negative real line, with a focus on whether the process hits zero and the associated probabilistic properties. By establishing a time-dependent Feller condition and leveraging comparison principles for Volterra integral equations, generalized Riemann–Liouville fractional differential equations, and asymptotic analysis via Laplace transforms, the work systematically characterizes boundary behavior and moment properties under different kernel functions. The main contributions are as follows: for regular kernels, the process is shown to stay strictly positive over finite time horizons and to possess finite negative-order moments; for rough kernels, the process hits zero with positive probability, inducing an atom at the boundary, yet its limiting distribution retains finite negative exponential moments. These results provide rigorous criteria for the existence of equivalent martingale measures in Volterra Heston models.

Abstract

In this work, we study the boundary behaviour of the Volterra square- root process on $\mathbb{R}_+$. For regular Volterra kernels, we establish a time-dependent Feller condition that guarantees that the process does not hit zero on $[0, T]$, and prove finiteness of negative $p$-moments. For rough kernels that are regularly varying at zero, we show that the process necessarily hits zero with positive probability, and that its law has an atom at the boundary. Finally, for the limit distribution, we show that finiteness of negative moments is determined by the long-time asymptotics of the associated resolvent. In particular, while in the rough case the process has an atom at zero, its limit distribution has finite negative exponential moments. Our proofs are based on comparison principles for Volterra integral equations and generalized Riemann-Liouville fractional equations. The latter provide us with upper and lower bounds for the solution of the associated Volterra Riccati equation, and hence also on the asymptotics of the Laplace transform. As an application, we study the structure of equivalent martingale measures in the Volterra Heston model. For the rough case, we show that equivalent martingale measures exist only under very restrictive assumptions on the drift under the real-world measure.

Problem

• Volterra square-root process
• boundary behaviour
• Feller condition
• negative moments
• rough kernels

Innovation

• Volterra square-root process
• rough kernels
• Feller condition
• fractional equations
• equivalent martingale measures