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 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: mathbbZ → ℝ, the diffusion coefficient g: mathbbZ → ℝ. The model of interest is given by [ dX(t) = xi + f(X(t)), dt + g(X(t)), dWt + γ int_mathbbZ z, N(dt,dz), ] where Wt is a standard Brownian motion, and N(dt,dz) is a Poisson random measure on (ℝ+ × mathbbZ, B (ℝ+) otimes Z, λ( Lambda otimes v)), with λ, γ > 0, Lambda being the Lebesgue measure on ℝ+, and v a finite measure on the measurable space (mathbbZ, 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.