# Analysis and data-driven reconstruction of bivariate jump-diffusion processes

## TL;DR

This paper introduces a data-driven, non-parametric method to reconstruct and analyze bivariate jump-diffusion processes, enabling parameter estimation and process understanding from synthetic data.

## Contribution

It presents a novel approach for reconstructing bivariate jump-diffusion processes using higher-order Kramers-Moyal coefficients, validated with synthetic data.

## Key findings

- Successful reconstruction of jump-diffusion processes from synthetic data
- Effective parameter estimation using higher-order Kramers-Moyal coefficients
- Identification of limitations due to data scarcity and parameter imbalance

## Abstract

We introduce the bivariate jump-diffusion process, comprising two-dimensional diffusion and two-dimensional jumps, that can be coupled to one another. We present a data-driven, non-parametric estimation procedure of higher-order (up to 8) Kramers-Moyal coefficients that allows one to reconstruct relevant aspects of the underlying jump-diffusion processes and to recover the underlying parameters. The procedure is validated with numerically integrated data using synthetic bivariate time series from continuous and discontinuous processes. We further evaluate the possibility of estimating the parameters of the jump-diffusion model via data-driven analyses of the higher-order Kramers-Moyal coefficients, and the limitations arising from the scarcity of points in the data or disproportionate parameters in the system.