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Lévy Score Function and Score-Based Particle Algorithm for Nonlinear Lévy--Fokker--Planck Equations

Yuanfei Huang, Chengyu Liu, Xiang Zhou

TL;DR

This work extends score-based transport methods to nonlinear Lévy–Fokker–Planck equations by introducing a generalized Lévy score that combines diffusion and nonlocal jump contributions. It establishes a probability-flow framework with a fixed-point interpretation, develops sequential Lévy score-based particle algorithms, and proves KL-divergence and discretization-error bounds for the numerical scheme. The approach is validated on diverse applications including biology and finance, demonstrating accurate propagation of probability densities with jump dynamics. Overall, it provides a rigorous, scalable tool for simulating interacting jump-diffusion systems and offers practical guidance for training and error control.

Abstract

The score function for the diffusion process, also known as the gradient of the log-density, is a basic concept to characterize the probability flow with important applications in the score-based diffusion generative modelling and the simulation of Itô stochastic differential equations. However, neither the probability flow nor the corresponding score function for the diffusion-jump process are known. This paper delivers mathematical derivation, numerical algorithm, and error analysis focusing on the corresponding score function in non-Gaussian systems with jumps and discontinuities represented by the nonlinear Lévy--Fokker--Planck equations. We propose the Lévy score function for such stochastic equations, which features a nonlocal double-integral term, and we develop its training algorithm by minimizing the proposed loss function from samples. Based on the equivalence of the probability flow with deterministic dynamics, we develop a self-consistent score-based transport particle algorithm to sample the interactive Lévy stochastic process at discrete time grid points. We provide error bound for the Kullback--Leibler divergence between the numerical and true probability density functions by overcoming the nonlocal challenges in the Lévy score. The full error analysis with the Monte Carlo error and the time discretization error is furthermore established. To show the usefulness and efficiency of our approach, numerical examples from applications in biology and finance are tested.

Lévy Score Function and Score-Based Particle Algorithm for Nonlinear Lévy--Fokker--Planck Equations

TL;DR

This work extends score-based transport methods to nonlinear Lévy–Fokker–Planck equations by introducing a generalized Lévy score that combines diffusion and nonlocal jump contributions. It establishes a probability-flow framework with a fixed-point interpretation, develops sequential Lévy score-based particle algorithms, and proves KL-divergence and discretization-error bounds for the numerical scheme. The approach is validated on diverse applications including biology and finance, demonstrating accurate propagation of probability densities with jump dynamics. Overall, it provides a rigorous, scalable tool for simulating interacting jump-diffusion systems and offers practical guidance for training and error control.

Abstract

The score function for the diffusion process, also known as the gradient of the log-density, is a basic concept to characterize the probability flow with important applications in the score-based diffusion generative modelling and the simulation of Itô stochastic differential equations. However, neither the probability flow nor the corresponding score function for the diffusion-jump process are known. This paper delivers mathematical derivation, numerical algorithm, and error analysis focusing on the corresponding score function in non-Gaussian systems with jumps and discontinuities represented by the nonlinear Lévy--Fokker--Planck equations. We propose the Lévy score function for such stochastic equations, which features a nonlocal double-integral term, and we develop its training algorithm by minimizing the proposed loss function from samples. Based on the equivalence of the probability flow with deterministic dynamics, we develop a self-consistent score-based transport particle algorithm to sample the interactive Lévy stochastic process at discrete time grid points. We provide error bound for the Kullback--Leibler divergence between the numerical and true probability density functions by overcoming the nonlocal challenges in the Lévy score. The full error analysis with the Monte Carlo error and the time discretization error is furthermore established. To show the usefulness and efficiency of our approach, numerical examples from applications in biology and finance are tested.
Paper Structure (14 sections, 3 theorems, 65 equations, 9 figures, 2 algorithms)

This paper contains 14 sections, 3 theorems, 65 equations, 9 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Probability Flow of Jump-Diffusion and Lévy Score Function
  3. Mckean--Vlasov SDE driven by Lévy noise
  4. Lévy--Fokker-Planck equation and Lévy score function
  5. Isotropic Lg-stable Lévy process
  6. Probability flow for probability density function
  7. Score-based Particle Approach: Theoretical and Numerical Analysis
  8. The Kullback–Leibler divergence error
  9. Sequential Lévy score-based particle method
  10. Error analysis of Algorithm \ref{['algorithm']}
  11. Proof of Theorem \ref{['theo:KL']}
  12. Proof of Theorem \ref{['theo:error']}
  13. Numerical Experiments
  14. Conclusions

Key Result

Theorem 3.2

\newlabeltheo:KL0 For interaction kernels subass:bounded (bounded) and subass:BS (Biot-Savart), let $p^{\theta}_{t}(x)$ denote the solution to the transport equation eqn:transport, and let $\widehat{p}_{t}(x)$ denote the solution to the Lévy--Fokker--Planck equation eqn:FPE. Suppose Assumption ass where $\bar{C}$ is a constant given in eqn:barC.

Figures (9)

  • Figure 1: [Example \ref{['example1']}] Probability flows of \ref{['exa:1dfinite']}. The left panel illustrates the temporal evolution of the probability distribution as a heat map, overlaid with two selective stochastic trajectories based on the Monte Carlo simulation (red) and the two deterministic trajectories based on the transport map (white). The middle and right panels compare the probability distributions $\hat{p}_t(x)$ from the Monte Carlo simulation and the proposed method in the time-state space.
  • Figure 2: [Example \ref{['example1']}] The TV distance between $P^{\mathrm{MC}}$ and $P^{\mathrm{S}}$ for \ref{['exa:1dfinite']}.
  • Figure 3: [Example \ref{['example2']}] Probability flows of \ref{['eqn:exampe2']}.
  • Figure 4: [Example \ref{['example2']}] The TV distance between $P^{\mathrm{MC}}$ and $P^{\mathrm{S}}$ for \ref{['eqn:exampe2']}.
  • Figure 5: [Example \ref{['example3']}] The Probability flows of \ref{['exa:doublewell']} along the axes: $x$ (top), and $y$ (bottom).
  • ...and 4 more figures

Theorems & Definitions (13)

  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Lemma 3.6
  • Proof 1
  • Proof 2: Proof of Theorem \ref{['theo:KL']}
  • Proof 3: Proof of Theorem \ref{['theo:error']}
  • Example 1: 1D Jump-diffusion with finite jump activity
  • ...and 3 more