# Jump Locations of Jump-Diffusion Processes with State-Dependent Rates

## TL;DR

This paper introduces a framework for analyzing jump locations in jump-diffusion processes with state-dependent jump rates, revealing their relationship with the overall process distribution and demonstrating applications with non-monotonic behaviors.

## Contribution

It develops an iterative map approach to compute jump location distributions in state-dependent jump-diffusions, linking jump locations to the stationary density of the process.

## Key findings

- Derived an iterative method for jump location distributions.
- Linked long-term jump location distribution to stationary density.
- Showed non-monotonic dependence of jump behavior on diffusion strength.

## Abstract

We propose a general framework for studying jump-diffusion systems driven by both Gaussian noise and a jump process with state-dependent intensity. Of particular natural interest are the jump locations: the system evaluated at the jump times. However, the state-dependence of the jump rate provides direct coupling between the diffusion and jump components, making disentangling the two to study individually difficult. We provide an iterative map formulation of the sequence of distributions of jump locations. Computation of these distributions allows for the extraction of the interjump time statistics. These quantities reveal a relationship between the long-time distribution of jump location and the stationary density of the full process. We provide a few examples to demonstrate the analytical and numerical tools stemming from the results proposed in the paper, including an application that shows a non-monotonic dependence on the strength of diffusion.