Compound Poisson Processes

Random arrivals carrying random sizes: the mean, the variance, and the jump measure.

The idea

A compound Poisson process models events that arrive at random times and carry random sizes: claims reaching an insurer, trades reaching an order book, packets crossing a link. A Poisson process supplies the arrivals, and a second source of randomness supplies the sizes.

Let $N(t)$ be a Poisson process of rate $\lambda$, and let the jump sizes $Y_{1}, Y_{2}, \ldots$ be independent of each other and of $N$, all with the same distribution. The compound Poisson process is the running total

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

constant between arrivals and jumping by $Y_{i}$ at the $i$-th one. It differs from an ordinary sum in that the number of terms is itself random.

Because the count and the sizes are independent, every question about $X(t)$ yields to the same computation: condition on the count, answer the question for that fixed number of terms, then average over the count. Conditioning in this way produces the two moments

$\mathbb{E}[X(t)] = \lambda t\,\mathbb{E}[Y], \qquad \operatorname{Var}(X(t)) = \lambda t\,\mathbb{E}[Y^{2}].$

The variance involves the second moment of a jump, not its variance: jumps of constant size have no variance of their own, yet the total remains random, because their number is.

Ways to work on it

• Walkthrough. The definition, the mean by conditioning, and why the variance uses the second moment.
• Proof. Mean, variance, and the moment generating function, all by conditioning on the arrival count.
• Practice. Mean and variance of a random-sum total from a rate, a mean, and a standard deviation.
• Hardest. The jump measure: split off the large jumps and price their risk.

Not sure where to start? Take the ten-question placement test.