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.

We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations, yielding theoretical estimates of learning rates which, unlike previous works, become increasingly tighter when the regularity of the unknown drift and diffusion coefficients becomes higher. Our method being kernel-based, offline pre-processing may in principle be profitably leveraged to enable efficient numerical implementation.

Stochastic differential equations (SDEs) are used to describe a wide variety of complex stochastic dynamical systems. Learning the hidden physics within SDEs is crucial for unraveling fundamental understanding of the stochastic and nonlinear behavior of these systems. We propose a flexible and scalable framework for training deep neural networks to learn constitutive equations that represent hidden physics within SDEs. The proposed stochastic physics-informed neural network framework (SPINN) relies on uncertainty propagation and moment-matching techniques along with state-of-the-art deep learning strategies. SPINN first propagates stochasticity through the known structure of the SDE (i.e., the known physics) to predict the time evolution of statistical moments of the stochastic states. SPINN learns (deep) neural network representations of the hidden physics by matching the predicted moments to those estimated from data. Recent advances in automatic differentiation and mini-batch gradient descent are leveraged to establish the unknown parameters of the neural networks. We demonstrate SPINN on three benchmark in-silico case studies and analyze the framework's robustness and numerical stability. SPINN provides a promising new direction for systematically unraveling the hidden physics of multivariate stochastic dynamical systems with multiplicative noise.

In this paper a new dissimilarity measure to identify groups of assets dynamics is proposed. The underlying generating process is assumed to be a diffusion process solution of stochastic differential equations and observed at discrete time. The mesh of observations is not required to shrink to zero. As distance between two observed paths, the quadratic distance of the corresponding estimated Markov operators is considered. Analysis of both synthetic data and real financial data from NYSE/NASDAQ stocks, give evidence that this distance seems capable to catch differences in both the drift and diffusion coefficients contrary to other commonly used metrics.