Neural MJD: Neural Non-Stationary Merton Jump Diffusion for Time Series Prediction

TL;DR

Neural MJD tackles non-stationary time-series with abrupt jumps by proposing a neural-parameterized non-stationary Merton Jump Diffusion (MJD) model. It fuses a time-inhomogeneous Itô diffusion with a time-inhomogeneous compound Poisson jump process, whose parameters are learned from past data and context via a neural network, and introduces likelihood truncation with a provable error bound plus an Euler-Maruyama with restart inference scheme. The authors provide theoretical results on truncation error decay and weak-convergence improvements, and demonstrate superior predictive performance and uncertainty quantification on synthetic data and real-world business and financial datasets relative to state-of-the-art baselines. This approach offers a scalable, interpretable framework for forecasting under non-stationarity and jumps, with practical impact in finance, commerce, and beyond.

Abstract

While deep learning methods have achieved strong performance in time series prediction, their black-box nature and inability to explicitly model underlying stochastic processes often limit their generalization to non-stationary data, especially in the presence of abrupt changes. In this work, we introduce Neural MJD, a neural network based non-stationary Merton jump diffusion (MJD) model. Our model explicitly formulates forecasting as a stochastic differential equation (SDE) simulation problem, combining a time-inhomogeneous Itô diffusion to capture non-stationary stochastic dynamics with a time-inhomogeneous compound Poisson process to model abrupt jumps. To enable tractable learning, we introduce a likelihood truncation mechanism that caps the number of jumps within small time intervals and provide a theoretical error bound for this approximation. Additionally, we propose an Euler-Maruyama with restart solver, which achieves a provably lower error bound in estimating expected states and reduced variance compared to the standard solver. Experiments on both synthetic and real-world datasets demonstrate that Neural MJD consistently outperforms state-of-the-art deep learning and statistical learning methods.

Figures (7)

• Figure 1: The overview of Neural MJD. Our model captures discontinuous jumps in time-series data and uncovers the underlying non-stationary SDEs from historical sequences and context information. Our method enables numerical simulations for future forecasting along time.
• Figure 1: Quantitative results on the synthetic dataset.
• Figure 2: Comparison of numerical simulations with and without restart strategy during inference.
• Figure 3: Qualitative result on the synthetic dataset.
• Figure 4: Neural MJD training pipeline. The symbol $\rho$ represents the MJD parameters $\{\mu_\tau, \sigma_\tau, \lambda_\tau, \nu_\tau, \gamma_\tau\}$ in our model.

Theorems & Definitions (11)

• Theorem 4.1
• Proposition 4.1
• Definition A.1
• Definition B.1
• Theorem C.1
• Lemma C.1: Theorem 2 in short2013improved
• Lemma C.2: Bounds on the Standard Normal CDF
• proof
• proof
• Proposition C.2