# Jump Processes with Deterministic and Stochastic Controls

## TL;DR

This paper develops a comprehensive theory for 1D jump processes influenced by deterministic and stochastic controls, providing analytical solutions and applications to environmental risk assessment.

## Contribution

It introduces a novel formulation of the master equation for jump processes with control, including analytical steady-state solutions.

## Key findings

- Analytical steady-state solutions for controlled jump processes.
- Application to crop-failure risk assessment in environmental geophysics.
- A new theoretical framework for systems with combined deterministic and stochastic jumps.

## Abstract

We consider the dynamics of a 1D system evolving according to a deterministic drift and randomly forced by two types of jumps processes, one representing an external, uncontrolled forcing and the other one a control that instantaneously resets the system according to specified protocols (either deterministic or stochastic). We develop a general theory, which includes a different formulation of the master equation using antecedent and posterior jump states, and obtain an analytical solution for steady state. The relevance of the theory is illustrated with reference to stochastic irrigation to assess probabilistically crop-failure risk, a problem of interest for environmental geophysics.