On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients

Paper: arXiv 2402.19203

Authors: Aurélien Alfonsi, Guillaume Szulda

Abstract

We consider one-dimensional stochastic Volterra equations with jumps for which we establish conditions upon the convolution kernel and coefficients for the strong existence and pathwise uniqueness of a non-negative càdlàg solution. By using the approach recently developed in arXiv:2302.07758, we show the strong existence by using a nonnegative approximation of the equation whose convergence is proved via a variant of the Yamada–Watanabe approximation technique. We apply our results to Lévy-driven stochastic Volterra equations. In particular, we are able to define a Volterra extension of the so-called alpha-stable Cox–Ingersoll–Ross process, which is especially used for applications in Mathematical Finance.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents high-level measure-theoretic probability and stochastic analysis, establishing theoretical existence and uniqueness for a non-negative Volterra extension of the CIR process. However, as a purely theoretical probability paper, it lacks empirical backtesting, real market data, or implementation code.

Research Flowchart

  flowchart TD
    A[Research Goal: Prove existence & uniqueness of non-negative solutions for stochastic Volterra equations with jumps] --> B[Inputs: Convolution kernel, non-Lipschitz coefficients & Lévy jump noise]
    B --> C[Methodology: Non-negative approximation of the Volterra equation]
    C --> D[Analysis: Prove convergence via Yamada-Watanabe approximation technique]
    D --> E[Key Outcome 1: Established strong existence & pathwise uniqueness of càdlàg solutions]
    D --> F[Key Outcome 2: Defined Volterra extension of alpha-stable Cox-Ingersoll-Ross process for Math Finance]