Home Statistical Dictionary Compound Poisson Process

Compound Poisson Process

A process that sums up random-sized jumps arriving according to a Poisson process, used whenever both how often events happen and how large each event is matter, such as total insurance claims over time.

Stochastic ProcessesIntermediateFree

In Plain English

A plain Poisson process counts events, one, two, three, treating every arrival as identical. A compound Poisson process adds a second layer of randomness: each arriving event also brings its own random size, and what accumulates over time is the running total of those sizes, not just the count. This is exactly the structure of an insurance company's total claims, claims arrive at random times (a Poisson process) and each claim has a random dollar amount (the compounding), and the total payout is the compound Poisson process's value at time $t$ .

Definition

A compound Poisson process is defined as $X (t) = \sum_{i = 1}^{N (t)} Y_{i}$ , where ${N (t)}$ is a Poisson process with rate $λ$ , and ${Y_{i}}$ are i.i.d. random variables (the jump sizes), independent of $N (t)$ , drawn from some distribution $F_{Y}$ . The process $X (t)$ has mean $E [X (t)] = λ t E [Y]$ and variance $Var (X (t)) = λ t E [Y^{2}]$ (both following directly from conditioning on $N (t)$ and applying the law of total expectation/variance). A compound Poisson process is a Levy process (it has stationary, independent increments) and is the canonical example of a pure-jump process, its paths are piecewise constant, jumping by $Y_{i}$ at each of the $N (t)$ arrival times. In actuarial science, the compound Poisson process is the standard model for aggregate claims in ruin theory, where the classical Cramer-Lundberg model studies an insurer's surplus as premium income minus this compound Poisson claims process.

Formula

X (t) = i = 1 \sum N (t) Y_{i}, E [X (t)] = λ t E [Y]

Notation

N (t)

—The Poisson process counting the number of arrivals by time $t$

Y_{i}

—The $i$-th jump size, i.i.d. from a distribution $F_Y$, independent of $N(t)$

Properties

The compound Poisson process's mean and variance formulas, $λ t E [Y]$ and $λ t E [Y^{2}]$ , both follow from conditioning on the random number of jumps $N (t)$ and applying the law of total expectation and variance, a standard, widely reused technique whenever a total is built from a random number of random-sized terms.
The Cramer-Lundberg ruin model in actuarial science represents an insurer's surplus over time as $U (t) = u + c t - X (t)$ , initial capital plus steady premium income minus the compound Poisson claims process, and the probability of ruin (surplus ever going negative) is a central quantity actuaries compute from this exact construction.
The special case where jump sizes $Y_{i}$ are always exactly 1 reduces the compound Poisson process back to an ordinary Poisson process, making the plain Poisson process a degenerate special case of the compound version, useful for seeing how the two models relate.

At a Glance

Formula

X (t) = \sum_{i = 1}^{N (t)} Y_{i}

: random sum of random-sized jumps

Pathspiecewise constant, jumping by

Y_{i}

at each of the Poisson arrival times

Process familya Levy process, and the canonical pure-jump example

Classic applicationaggregate insurance claims in actuarial ruin theory

Last updated September 8, 2026← Back to the Dictionary