a Poisson random measure
/ PWA-sohn /
A Poisson random measure is the high-dimensional generalization of the Poisson process: instead of counting points on a time line, it counts points scattered in any measured space, with the number of points in disjoint regions independent and Poisson distributed. It is the precise object that records all the jumps of a Lévy process at once — each jump is a point in (time, jump-size) space — and it is the integrator in the Lévy-Itô decomposition.
Given a sigma-finite measure mu (the intensity) on a space E, a Poisson random measure N is a random counting measure such that (i) for any measurable A, N(A) is Poisson distributed with mean mu(A) (and N(A) = infinity a.s. if mu(A) = infinity), and (ii) for disjoint A_1, ..., A_n the counts N(A_1), ..., N(A_n) are independent. For a Lévy process the jumps form a Poisson random measure on (0, infinity) x (R\{0}) with intensity dt nu(dx): the number of jumps of size in B occurring in time interval (s, t] is Poisson with mean (t-s) nu(B). One integrates test functions against it: integral of f(s, x) N(ds, dx) literally sums f over the jump points, and the master formula E[exp(integral f dN)] = exp(integral (e^f - 1) d mu) (the exponential / Campbell formula) generates all the Laplace functionals.
The Poisson random measure is the rigorous foundation of the entire jump theory: the compensated measure N - dt nu gives the small-jump martingale, the Lévy-Itô decomposition is an integral against N, and stochastic calculus for jumps (the Itô formula with jumps) is written in terms of integrals against N and its compensator. The subtlety to keep straight: when mu is infinite (as near zero for an infinite Lévy measure) the total number of points is infinite, so raw integrals integral f dN need not converge — one must use the compensated measure for unbounded-near-zero integrands, which is precisely why small jumps require compensation.
Plot every jump of a Lévy process as a dot at coordinates (time of jump, size of jump). The resulting scatter of dots is a realization of a Poisson random measure on (0, infinity) x (R\{0}) with intensity dt nu(dx): dots are dense near size 0 if nu is infinite there, and sparse for large sizes. Counting dots in a box (s,t] x B gives a Poisson((t-s)nu(B)) variable.
Every Lévy jump is a point; the cloud is a Poisson random measure.
When the intensity is infinite (infinite Lévy measure near zero) there are infinitely many points, so integral f dN against an integrand not vanishing fast enough at 0 diverges — use the compensated measure N - dt nu. The Campbell formula E[e^(integral f dN)] = e^(integral (e^f - 1) d mu) is the workhorse.