Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.04417 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912472742494208 |
|---|---|
| author | Ramirez-Gonzalez, Jose-Hermenegildo Sun, Ying |
| author_facet | Ramirez-Gonzalez, Jose-Hermenegildo Sun, Ying |
| contents | 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 $λ, γ> 0$, $Λ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Neural Networks for Tamed Milstein Approximation of SDEs with Additive Symmetric Jump Noise Driven by a Poisson Random Measure Ramirez-Gonzalez, Jose-Hermenegildo Sun, Ying Machine Learning 60H10, 68T07 I.2.6; G.3 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 $λ, γ> 0$, $Λ$ 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. |
| title | Neural Networks for Tamed Milstein Approximation of SDEs with Additive Symmetric Jump Noise Driven by a Poisson Random Measure |
| topic | Machine Learning 60H10, 68T07 I.2.6; G.3 |
| url | https://arxiv.org/abs/2507.04417 |