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TL;DR

A counting process \( N(t) \) where events occur independently at a constant rate \( \lambda \). The number of events in an interval of length t follows \( \text{Poisson}(\lambda t) \), and inter-arrival times are exponentially distributed.

By Valenke Exam Prep Team·Last updated 2026-06-03

Poisson Process

A counting process \( N(t) \) where events occur independently at a constant rate \( \lambda \). The number of events in an interval of length t follows \( \text{Poisson}(\lambda t) \), and inter-arrival times are exponentially distributed.

Why it matters for interviews

Models trade arrivals, jump events in asset prices (jump-diffusion models), and order flow in market microstructure. Understanding its properties is essential for modeling discrete events in continuous time.

Definition and Mathematical Foundation

A counting process \( N(t) \) where events occur independently at a constant rate \( \lambda \). The number of events in an interval of length t follows \( \text{Poisson}(\lambda t) \), and inter-arrival times are exponentially distributed.

Application in Quantitative Finance

Models trade arrivals, jump events in asset prices (jump-diffusion models), and order flow in market microstructure. Understanding its properties is essential for modeling discrete events in continuous time.

Related Concepts

Poisson Process

Related Terms

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Frequently Asked Questions

What is the memoryless property?▼
The exponential inter-arrival times have no memory: \( P(T > s+t | T > s) = P(T > t) \). The time until the next event does not depend on how long you have already waited.
What is a compound Poisson process?▼
A Poisson process where each event carries a random size (jump). Used in Merton's jump-diffusion model where asset prices experience random jumps at Poisson-distributed times.
How does the Poisson process relate to the binomial distribution?▼
The Poisson distribution is the limit of Binomial(n, p) as \( n \to \infty \), \( p \to 0 \), with \( np = \lambda \). This connects discrete counting to continuous-time modeling.