Poisson process

For a Poisson process with a constant intensity $\lambda$, the probability that the event occurs exactly $n$ times in time $t$ is given by the next formula. The mean number of occurrences is $\lambda t$¹.

$$P(N_t = n) = \frac{e^{-\lambda t}(\lambda t)^n}{n!}, \qquad n = 0, 1, 2, \dots$$

Consider a hypothetical Poisson process with intensity $\lambda = 1$ (one event per year on average) and count the events over one year. The probability of 0 events is $e^{-1} = 0.3679$, that of 1 event is also 0.3679, that of 2 is 0.1839, and that of 3 is 0.0613. Even with a mean of 1, there is a probability of about 37% that nothing happens in a year, and about 26% that it happens two or more times.

A Poisson process has no size for each event; it only counts. To also represent the size of a jump, a value of another random variable is added or multiplied at each counted event. This form is called a compound Poisson process¹.

Notes

1. 久田祥史 (Yoshifumi Hisata), "ジャンプ拡散過程を用いたオプション価格付けモデルについて" (On option pricing models using jump-diffusion processes), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (June 2003), Appendix 1 (pages 70–71) (in Japanese), equation (A-3) and the definition of a compound Poisson process. The mean and the example probabilities are calculated from the formula. ↩ ↩²