{"ID":2897918,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.04417","arxiv_id":"2507.04417","title":"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 $(\\mathbb{R}_{+} \\times \\mathbb{Z}$, $\\mathcal{B} (\\mathbb{R}_{+}) \\otimes \\mathcal{Z}$, $λ( Λ\\otimes v))$, with $λ, γ\u003e 0$, $Λ$ 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.","short_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 func...","url_abs":"https://arxiv.org/abs/2507.04417","url_pdf":"https://arxiv.org/pdf/2507.04417v2","authors":"[\"Jose-Hermenegildo Ramirez-Gonzalez\",\"Ying Sun\"]","published":"2025-07-06T15:13:31Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\"]","methods":"[\"Diffusion Model\"]","has_code":false}