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

Abstract

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 \[ dX(t) = ξ+ f(X(t))\, dt + g(X(t))\, dW_t + γ\int_{\mathbb{Z}} z\, N(dt,dz), \] where $W_t$ is a standard Brownian motion, and $N(dt,dz)$ is a Poisson random measure on $({\mathbb{R}}_{+}\times \mathbb{Z}$, $\mathcal{B}({\mathbb{R}}_{+})\otimes \mathcal{Z}$, $\hat{I}»(\hat{I}›\otimes v))$, with $\hat{I}»,{\hat{I}}^3>0$, $\hat{I}›$ 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.

14 pages, 9 figures, 4 tables