arXiv subjects

Martin Bergerhausen

Publications and source records attributed to Martin Bergerhausen.

3 recordsLinked to original sources

arXiv2026

Stability results for distribution-dependent stochastic Volterra equations

We investigate stability properties of distribution-dependent stochastic Volterra equations with respect to changes in the coefficients, the Volterra kernels, and the initial condition. Under Lipschitz continuity assumptions on the coefficients, we first derive quantitative stability estimates for strong solutions with explicit error bounds. We then prove a general convergence theorem for strong solutions under substantially weaker assumptions, replacing Lipschitz continuity by a continuity assumption together with uniform linear growth of the approximating sequence. Finally, we study the associated distribution-dependent Volterra local martingale problem and prove the stability of its solutions under convergence of the coefficients, kernels, and initial distributions.

math.PR↗

arXiv2026

Weak solutions to distribution-dependent stochastic Volterra equations

We prove the existence of weak solutions for distribution-dependent stochastic Volterra equations under linear growth and continuity conditions on the coefficients and mild regularity assumptions on the kernels, including singular kernels. To this end, we formulate an associated local martingale problem and establish its connection with weak solutions. Moreover, we derive continuity and integrability properties of the solutions.

math.PR↗

arXiv2024

Neural stochastic Volterra equations: learning path-dependent dynamics

Stochastic Volterra equations (SVEs) serve as mathematical models for the time evolutions of random systems with memory effects and irregular behaviour. We introduce neural stochastic Volterra equations as a physics-inspired architecture, generalizing the class of neural stochastic differential equations, and provide some theoretical foundation. Numerical experiments on various SVEs, like the disturbed pendulum equation, the generalized Ornstein--Uhlenbeck process, the rough Heston model and a monetary reserve dynamics, are presented, comparing the performance of neural SVEs, neural SDEs and Deep Operator Networks (DeepONets).

cs.LG↗