An efficient Wasserstein-distance approach for reconstructing jump-diffusion processes using parameterized neural networks

TL;DR

This paper introduces a neural network-based method to efficiently reconstruct multidimensional jump-diffusion processes by analyzing Wasserstein distances, with improved performance when prior drift information is available.

Contribution

It proposes a novel temporally decoupled Wasserstein-distance approach for jump-diffusion process reconstruction using parameterized neural networks, incorporating prior drift information.

Findings

Effective reconstruction demonstrated across multiple examples.

Performance improves with prior knowledge of the drift function.

Provides bounds on process discrepancies based on Wasserstein distance.

Abstract

We analyze the Wasserstein distance ($W$-distance) between two probability distributions associated with two multidimensional jump-diffusion processes. Specifically, we analyze a temporally decoupled squared $W_2$-distance, which provides both upper and lower bounds associated with the discrepancies in the drift, diffusion, and jump amplitude functions between the two jump-diffusion processes. Then, we propose a temporally decoupled squared $W_2$-distance method for efficiently reconstructing unknown jump-diffusion processes from data using parameterized neural networks. We further show its performance can be enhanced by utilizing prior information on the drift function of the jump-diffusion process. The effectiveness of our proposed reconstruction method is demonstrated across several examples and applications.