Splitting methods for stochastic Hodgkin-Huxley type systems and a localized fundamental mean-square convergence theorem

par Anna Melnykova

mardi 28 avr. 2026, 14:00 → 15:00 Europe/Paris

Existing fundamental theorems for mean-square convergence of numerical methods for stochastic differential equations (SDEs) require globally or one-sided Lipschitz continuous coefficients, while strong convergence results under merely local Lipschitz conditions are largely restricted to Euler-Maruyama type methods. To address these limitations, we introduce a novel localized version of the fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients, which naturally arise in a wide range of applications.

The proposed convergence framework is then applied to the splitting schemes approximating the solution of conditionally linear SDEs (a key example of one being stochastic Hodgkin-Huxley model). In addition, we establish key structure-preserving properties of the splitting methods, in particular state-space preservation and geometric ergodicity. Numerical experiments support the theoretical results and demonstrate that the proposed splitting schemes significantly outperform Euler-Maruyama type methods in preserving the qualitative features of the model.

 

Joint work with Irene Tubikanec (JKU Linz) and Pierre Étoré (Université Grenoble Alpes)