# Driving an Ornstein--Uhlenbeck Process to Desired First-Passage Time Statistics

## TL;DR

This paper investigates how to control the first-passage time distribution of an Ornstein-Uhlenbeck process with two boundaries, enabling independent adjustment of mean and variability through specific control strategies.

## Contribution

It reveals the scale invariance of the FPT distribution with respect to the drift and proposes methods to tune control parameters for desired FPT moments and distribution shape.

## Key findings

- FPT distribution is scale invariant with respect to drift
- Mean and CV of FPT can be controlled independently
- Optimal control parameters can minimize distribution distance

## Abstract

First-passage time (FPT) of an Ornstein-Uhlenbeck (OU) process is of immense interest in a variety of contexts. This paper considers an OU process with two boundaries, one of which is absorbing while the other one could be either reflecting or absorbing, and studies the control strategies that can lead to desired FPT moments. Our analysis shows that the FPT distribution of an OU process is scale invariant with respect to the drift parameter, i.e., the drift parameter just controls the mean FPT and doesn't affect the shape of the distribution. This allows to independently control the mean and coefficient of variation (CV) of the FPT. We show that that increasing the threshold may increase or decrease CV of the FPT, depending upon whether or not one of the threshold is reflecting. We also explore the effect of control parameters on the FPT distribution, and find parameters that minimize the distance between the FPT distribution and a desired distribution.