4.3 Compound Poisson processes

Definition of compound Poisson processes

A compound Poisson process tracks the cumulative effect of random events that arrive according to a Poisson process, where each event carries a random "size" or "impact." Think of an insurance company: claims arrive randomly over time, and each claim has a different dollar amount. The compound Poisson process gives you the running total.

Formally, let $N(t)$ be a Poisson process with rate $\lambda$, and let $X_1, X_2, \ldots$ be independent and identically distributed (i.i.d.) random variables, also independent of $N(t)$. The compound Poisson process is:

$S(t) = \sum_{i=1}^{N(t)} X_i$

Here $S(t)$ represents the total accumulated value of all events up to time $t$. When $N(t) = 0$, the sum is zero by convention.

Poisson process for event arrivals

The event count $N(t)$ is an ordinary (homogeneous) Poisson process:

• Events occur independently of one another.
• The rate $\lambda$ (average number of events per unit time) is constant.
• The number of events in any interval of length $t$ follows a Poisson distribution with parameter $\lambda t$.
• Inter-arrival times are exponentially distributed with parameter $\lambda$, so the expected time between events is $\frac{1}{\lambda}$.

Independent and identically distributed jump sizes

Each event $i$ carries a random variable $X_i$ representing its magnitude. These jump sizes must satisfy two conditions:

• Identically distributed: every $X_i$ is drawn from the same distribution $F_X$.
• Independent: the $X_i$ are mutually independent and independent of the arrival process $N(t)$.

The distribution $F_X$ can be anything appropriate for the application: exponential for claim sizes, gamma for service times, lognormal for financial losses, etc. The independence between $N(t)$ and the $X_i$ is what makes the compound Poisson process analytically tractable.

Properties of compound Poisson processes

Moment generating function and PGF

The key analytical tool is the moment generating function (MGF). If $M_X(\theta) = E[e^{\theta X}]$ is the MGF of each jump $X_i$, then the MGF of $S(t)$ has a clean closed form:

$M_{S(t)}(\theta) = E[e^{\theta S(t)}] = \exp\!\bigl(\lambda t \bigl(M_X(\theta) - 1\bigr)\bigr)$

This follows from conditioning on $N(t)$ and using the Poisson PGF. For discrete-valued jumps, you can equivalently work with the probability generating function (PGF):

$G_{S(t)}(z) = G_{N(t)}\!\bigl(G_X(z)\bigr) = \exp\!\bigl(\lambda t \bigl(G_X(z) - 1\bigr)\bigr)$

Both forms let you extract moments by differentiation and identify the distribution of $S(t)$ in many cases.

Moments of compound Poisson processes

Let $\mu_X = E[X_i]$ and $\sigma_X^2 = \text{Var}(X_i)$. Using the law of total expectation and total variance (conditioning on $N(t)$):

• Mean: $E[S(t)] = \lambda t \, \mu_X$
• Variance: $\text{Var}(S(t)) = \lambda t \, E[X_i^2] = \lambda t \,(\sigma_X^2 + \mu_X^2)$

The variance formula deserves a closer look. It comes from the Eve's law (law of total variance) decomposition:

$\text{Var}(S(t)) = E[\text{Var}(S \mid N)] + \text{Var}(E[S \mid N]) = \lambda t \, \sigma_X^2 + \lambda t \, \mu_X^2$

The first term captures randomness in jump sizes; the second captures randomness in the number of jumps.

Stationary and independent increments

The compound Poisson process inherits the stationary and independent increments property from the underlying Poisson process. Concretely:

• For any $s > 0$, the increment $S(t+s) - S(t)$ has the same distribution as $S(s)$.
• Increments over non-overlapping time intervals are independent.
• Given the current value $S(t)$, future increments don't depend on the path before time $t$.

This is sometimes loosely called the "memoryless property," though that term more precisely refers to the exponential distribution. What matters here is that the process "resets" statistically at every point in time, which greatly simplifies calculations.

Examples of compound Poisson processes

Aggregate claims in insurance

An insurance company receives claims at rate $\lambda = 10$ per month. Each claim amount $X_i$ follows a lognormal distribution with mean $\mu_X = \$5{,}000$ and variance $\sigma_X^2$. The total claims paid out by month $t$ is $S(t) = \sum_{i=1}^{N(t)} X_i$. The expected total payout over one month is $E[S(1)] = 10 \times 5{,}000 = \$50{,}000$. This model is foundational for premium setting and reserve calculations.

Cumulative damage models

In reliability engineering, a machine experiences random shocks at rate $\lambda$, and each shock causes damage $X_i$ (perhaps Weibull- or exponentially distributed). The system fails when $S(t)$ exceeds a threshold $D$. The compound Poisson framework lets you compute the distribution of time to failure and optimize maintenance schedules.

Inventory demand modeling

Customer orders arrive at rate $\lambda$, and each order requests a random quantity $X_i$. The compound Poisson process $S(t)$ gives total demand over $[0, t]$, which is useful for setting reorder points and estimating stock-out probabilities. This model is especially appropriate for low-frequency, high-volume order patterns, like spare parts or wholesale goods.

Generalizations of compound Poisson processes

Marked Poisson processes

A marked Poisson process attaches a random "mark" (label or attribute) to each event. The mark could encode event type, severity, location, or any other characteristic. A compound Poisson process is actually a special case: the mark is the jump size $X_i$, and you sum the marks. In the general marked process, you might analyze the marks without summing them.

Compound Cox processes

A Cox process (or doubly stochastic Poisson process) replaces the constant rate $\lambda$ with a random intensity process $\Lambda(t)$. A compound Cox process then sums i.i.d. jumps over this random-rate arrival process. This is useful when the event rate itself fluctuates unpredictably, as in financial markets (where trading intensity varies) or epidemiology (where infection rates change over time).

Compound renewal processes

A compound renewal process replaces the exponential inter-arrival times of the Poisson process with a general distribution (gamma, Weibull, lognormal, etc.). You lose the convenient Poisson structure and independent increments, but you gain flexibility to model arrivals where the exponential assumption is unrealistic. Analysis typically relies on renewal theory rather than the Poisson MGF formulas.

Simulation of compound Poisson processes

Simulating a compound Poisson process on $[0, T]$ is straightforward. Here's the procedure:

1. Generate the number of events. Draw $N \sim \text{Poisson}(\lambda T)$. This gives the total event count on $[0, T]$.
2. Generate event times. Draw $N$ uniform random variables $U_1, \ldots, U_N$ on $[0, T]$ and sort them. These are the arrival times. (Alternatively, generate exponential inter-arrival times and take cumulative sums, stopping when you exceed $T$.)
3. Generate jump sizes. For each event $i = 1, \ldots, N$, sample $X_i$ from the jump size distribution $F_X$ using inverse transform sampling, acceptance-rejection, or a built-in generator.
4. Compute the process. The value of $S(t)$ at any time $t$ is the cumulative sum of all $X_i$ whose arrival times fall in $[0, t]$.

Repeating this procedure many times gives you Monte Carlo samples of $S(T)$, from which you can estimate means, variances, tail probabilities, and other quantities of interest.

Parameter estimation for compound Poisson processes

Given observed data (event times and jump sizes), you need to estimate the rate $\lambda$ and the parameters of the jump size distribution $F_X$.

Method of moments estimation

Match theoretical moments to sample moments:

1. Estimate $\lambda$ from the observed number of events per unit time: $\hat{\lambda} = \frac{\text{total events}}{\text{observation period}}$.
2. Estimate $\mu_X$ and $\sigma_X^2$ from the sample mean and variance of the observed jump sizes.
3. If jump sizes aren't directly observed (only $S(t)$ increments are), use $E[S(t)] = \lambda t \mu_X$ and $\text{Var}(S(t)) = \lambda t (\sigma_X^2 + \mu_X^2)$ to solve for the unknowns.

This approach is simple and fast but can be statistically inefficient, especially with small samples.

Maximum likelihood estimation

MLE constructs the likelihood from the observed data and maximizes it:

• If both arrival times and jump sizes are observed, the likelihood factors cleanly: a Poisson process likelihood for the arrivals times a product of $f_X(x_i)$ for the jumps. You can estimate $\lambda$ and the jump distribution parameters separately.
• If only aggregate increments $S(t_{k+1}) - S(t_k)$ are observed, the likelihood involves the distribution of compound Poisson increments, which often lacks a closed form. Numerical optimization or the FFT-based approach to computing the compound distribution is then needed.

MLE estimators are asymptotically efficient and consistent, but may require iterative numerical methods.

Bayesian inference approaches

Bayesian estimation places prior distributions on $\lambda$ and the jump size parameters, then updates them with observed data via Bayes' theorem:

• A common choice is a Gamma prior for $\lambda$ (conjugate to the Poisson likelihood).
• Priors for jump size parameters depend on the assumed family $F_X$.
• The posterior is computed analytically (in conjugate cases) or via MCMC sampling.

Bayesian methods naturally quantify parameter uncertainty through the posterior distribution and allow you to incorporate domain expertise through informative priors.

Applications of compound Poisson processes

Risk theory and ruin probabilities

The classical Cramér-Lundberg model describes an insurer's surplus as:

$U(t) = u + ct - S(t)$

where $u$ is initial capital, $c$ is the premium income rate, and $S(t)$ is the compound Poisson claim process. Ruin occurs if $U(t) < 0$ for some $t > 0$. The ruin probability depends on the relationship between premium income and expected claims, and the tail behavior of the claim size distribution. For exponentially distributed claims with mean $\mu_X$, the ruin probability has the explicit form $\psi(u) = \frac{\lambda \mu_X}{c} \exp\!\left(-\frac{c - \lambda \mu_X}{c \mu_X} u\right)$, provided $c > \lambda \mu_X$ (the net profit condition).

Reliability analysis and shock models

Systems subject to random shocks degrade according to $S(t)$. The system fails at the first time $S(t)$ exceeds a damage threshold $D$. Compound Poisson models let you compute the distribution of time-to-failure, optimize inspection intervals, and compare maintenance policies (e.g., age-based vs. condition-based replacement). These models appear in aerospace, power systems, and manufacturing.

Inventory management and demand modeling

With compound Poisson demand, you can derive the distribution of total demand over a lead time, which directly feeds into reorder point and safety stock calculations. For example, if orders arrive at rate $\lambda = 3$ per week with mean order size $\mu_X = 50$ units, expected weekly demand is $150$ units, and the variance of weekly demand is $3(\sigma_X^2 + 2500)$. This variance drives safety stock decisions.