Neural Networks for Tamed Milstein Approximation of SDEs with Additive Symmetric Jump Noise Driven by a Poisson Random Measure

This work aims to estimate the drift and diffusion functions in stochastic differential equations (SDEs) driven by a particular 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 non-parametric function approximators. Estimation is carried out in a non-parametric fashion for the drift function $f:\mathbb{Z}\to \mathbb{R}$, the diffusion coefficient $g:\mathbb{Z}\to \mathbb{R}$. The model of interest is given by $$dX(t)=\xi +f(X(t))dt+g(X(t))d{W}_t+\gamma {\int }_{\mathbb{Z}}zN(dt,dz),$$ where $W_t$ is a standard Brownian motion, and $N(dt,dz)$ is a Poisson random measure on $({\mathbb{R}}_{+}\times \mathbb{Z}$, $\mathcal{B}({\mathbb{R}}_{+})\otimes \mathcal{Z}$, $\lambda (\mathrm{\Lambda }\otimes v))$, with $\lambda ,\gamma >0$, $\mathrm{\Lambda }$ being the Lebesgue measure on ${\mathbb{R}}_{+}$, and $v$ a finite measure on the measurable space $(\mathbb{Z},\mathcal{Z})$. Neural networks are used as non-parametric 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.