Neural networks for the Tamed–Milstein approximation of SDEs with additive symmetric jumps

This work aims to estimate the drift and diffusion functions in stochastic differential equations (SDEs) driven by a special class of Lévy processes with finite jump intensity, using neural networks. We propose a framework that integrates the Tamed-Milstein scheme with neural networks employed as nonparametric function approximators. Estimation is carried out in a nonparametric fashion for the drift function f:R→R and the diffusion coefficient g:R→R. The model of interest is given by dX(t)=f(X(t))dt+g(X(t))dWt+γ∫ZzN(dt,dz), where Wt is a standard Brownian motion and N(dt,dz) is a Poisson random measure on (R+×Z,B(R+)⊗Z,λ(Λ⊗v)), with λ,γ>0, Λ denoting the Lebesgue measure on R+, and v a finite symmetric measure on the measurable space (Z,Z). Neural networks are used as nonparametric function approximators, enabling the modeling of complex nonlinear dynamics without assuming restrictive functional forms. The proposed methodology constitutes a flexible alternative for inference in systems with state-dependent noise and discontinuities driven by Lévy processes.