Stochastic differential equations driven by additive Volterra-Lévy and Volterra-Gaussian noises (Q6193451)

The authors explored the existence and uniqueness of solutions to stochastic differential equations (SDEs) with Volterra processes driven by Lévy noise. They began by examining the smoothness properties of these processes, particularly focusing on Volterra-Gaussian processes which extended the compact interval representation of fractional Brownian motion to stochastic equations involving such processes. Key findings and contributions of the article included: 1. The authors investigate the continuity and Hölder properties of Volterra-Lévy processes, establishing moment upper bounds for increments to apply the Kolmogorov-Chentsov theorem. 2. They conduct a detailed study of Volterra-Gaussian processes arising when the Lévy process was a Brownian motion, introducing two types of kernels extending the Molchan-Golosov kernel of fractional Brownian motion. 3. The authors derive inverse operators for both types of Volterra-Gaussian processes in terms of generalized fractional integrals and derivatives for Sonine pairs. 4. They apply these results to study stochastic differential equations with Volterra-Lévy processes, including both deterministic analogs and stochastic equations, investigating solvability under various conditions on the coefficients and underlying processes. 5. Furthermore, they extend these results to stochastic differential equations with Volterra-Gaussian processes, proving the existence and uniqueness of weak and strong solutions under certain assumptions on the coefficients' growth and Hölder continuity. The structure of the paper is logically organized, starting with the theoretical framework and smoothness properties of Volterra processes, then proceeding to analyze the existence and uniqueness of solutions to SDEs with both Volterra-Lévy and Volterra-Gaussian processes. In conclusion, the article significantly contributes to the understanding of stochastic processes involving Volterra integrals and Lévy noise. It provides valuable insights into the smoothness properties of these processes and establishes conditions for the existence and uniqueness of solutions to associated stochastic differential equations. Moreover, the study of Volterra-Gaussian processes enriches the understanding of fractional Brownian motion and extendes it to broader classes of stochastic equations. Overall, the findings presented in the article are essential for researchers and practitioners working in the field of stochastic analysis and mathematical modeling of complex systems. For the entire collection see [Zbl 1515.60023].