Inhomogeneous and Compound Poisson Processes

Generalising the Poisson process, an inhomogeneous version lets the event rate vary over time or space, while a compound version attaches independent random sizes to each event.

Definition

An inhomogeneous Poisson process is a counting process with independent increments whose count in a region is Poisson distributed with mean given by the integral of a non-constant intensity, and a compound Poisson process is the sum of independent identically distributed random jumps occurring at the events of a Poisson process.

Scope

This topic covers the inhomogeneous Poisson process defined by a varying intensity function and cumulative mean measure, the time-change that maps it to a standard Poisson process, the compound Poisson process formed by summing independent marks at Poisson event times, its mean, variance, and characteristic function, and applications to insurance risk and shot noise.

Core questions

• How does a varying intensity function generalise the constant-rate process?
• How can an inhomogeneous process be transformed into a homogeneous one by a time change?
• How are the mean and variance of a compound Poisson sum computed?
• How do these processes model insurance claims and shot noise?

Key theories

Time-change to standard Poisson

Rescaling time by the cumulative intensity function turns an inhomogeneous Poisson process into a standard rate-one Poisson process, which both characterises the inhomogeneous process and provides a simulation method by inversion or thinning.

Compound Poisson distribution

The sum of a Poisson-distributed number of independent jumps has mean and variance expressible through the jump distribution, and its characteristic function is the exponential of the rate times the jump characteristic function minus one, linking it to infinitely divisible laws.

Clinical relevance

Inhomogeneous Poisson processes model time-varying arrival rates such as daily traffic or seasonal disease incidence, while compound Poisson processes are the classical model of aggregate insurance claims in the Cramer-Lundberg risk theory and of shot noise in physics and signal processing.

History

Lundberg introduced the compound Poisson risk model in 1903 and Cramer developed its ruin theory in the 1930s, while inhomogeneous Poisson processes and their thinning-based simulation, formalised by Lewis and Shedler in 1979, became standard tools for modelling time-varying event rates.

Key figures

• Filip Lundberg
• Harald Cramer
• John Kingman

Seminal works

• kingman1993

Frequently asked questions

What is the difference between inhomogeneous and compound Poisson processes?

An inhomogeneous process keeps unit jumps but lets the event rate vary in time or space, whereas a compound process keeps a Poisson number of events but gives each a random size.

How is a compound Poisson process used in insurance?

It models total claims as a Poisson number of independent claim amounts; the resulting aggregate is the basis of classical ruin theory, which studies the probability that accumulated claims exceed reserves.