# Approximation of the first passage time distribution for the birth-death processes

## TL;DR

This paper introduces a new approximation method for the first passage time distribution in birth-death processes, utilizing properties of these processes, Keilson's theorem, and Riemann sums, with applications to complex models including order-book dynamics.

## Contribution

It presents a novel approximation technique for first passage times in birth-death processes using closed-form expressions derived from theoretical properties.

## Key findings

- Method effectively approximates first passage time distributions.
- Applied to three birth-death processes and a long-range memory order-book model.
- Facilitates comparison between models with spurious and true long-range memory.

## Abstract

We propose a general method to obtain approximation of the first passage time distribution for the birth-death processes. We rely on the general properties of birth-death processes, Keilson's theorem and the concept of Riemann sum to obtain closed-form expressions. We apply the method to the three selected birth-death processes and the sophisticated order-book model exhibiting long-range memory. We discuss how our approach contributes to the competition between spurious and true long-range memory models.