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

TL;DR

This paper introduces a neural network-based framework for estimating drift and diffusion functions in SDEs driven by Lévy processes with jumps, using a Tamed-Milstein scheme for non-parametric modeling of complex stochastic dynamics.

Contribution

It combines neural networks with the Tamed-Milstein scheme to non-parametrically estimate functions in SDEs with jump noise, addressing complex nonlinear dynamics without restrictive assumptions.

Findings

Effective non-parametric estimation of drift and diffusion functions.

Flexible modeling of systems with jumps and discontinuities.

Potential for improved inference in Lévy-driven SDEs.

Abstract

This work aims to estimate the drift and diffusion functions in stochastic differential equations (SDEs) driven by a particular class of L\'evy 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 \[ dX(t) = \xi + f(X(t))\, dt + g(X(t))\, dW_t + \gamma \int_{\mathbb{Z}} z\, N(dt,dz), \] where $W_t$ is a standard Brownian motion, and $N(dt,dz)$ is a Poisson random measure on $(\mathbb{R}_{+} \times \mathbb{Z}$, $\mathcal{B} (\mathbb{R}_{+}) \otimes \mathcal{Z}$, $\lambda( \Lambda \otimes v))$, with $\lambda, \gamma > 0$, $\Lambda$ 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 L\'evy processes.