Poisson random measures: an application to simulations of Lévy and max-stable processes

This thesis is about simulations of two classes of stochastic processes: Lévy processes and max-stable processes. The aim of a simulation is to find an algorithm that computes the possible values of a realization of a stochastic process X onto a finite number n of points. This means that the outcome of a sampling algorithm is a random vector that we would like to be representative of the whole process. In particular. we would like the random vector our simulation)to inherit the statistical properties of X. Then, the most ambitious goal that the simulation has the same distribution of the restriction of the stochastic process X onto the evaluation points. We call this an exact simulation. Finding such algorithm is, in the vast majority of cases, not possible. In the case of the Lévy processes, the independence of increments let us simplify the problem from the multivariate case (computing sample-paths) to the univariate case (computing the independent increments). Problems arise from the fact that we will assume to only know the characteristic function of increments, not their distribution. Fortunately the distribution of the increments of a Lévy process has a semi-parametric form, depending only on a triplet (c,σ,μ) where c,σ are numbers and μ is a potentially infinite measure, called the Lévy measure, which is a sort of mean measure of a Poisson process. If there are no jumps the problem is reduced to sampling a normal random variable, which is a problem solved decades ago from Box and Muller. The only problem left is to simulate the jumps from μ. A partial solution to this problem is described in this thesis. In fact, we can find some representations of the jumps in the form of a Poisson process with, in general, infinite number of points. A truncation is needed for the feasibility of this method on a computer and that will always generate some error. Our strategy will be to find a way of letting the biggest jumps appear first in such a way that the error is minimized. In this case there is no chance of getting an exact simulation, however it is possible to estimate the order of convergence of the error. In the case of max-stable processes, the problem is different. In fact, simulating the marginal distributions is in general easy, since they belong to three well known distributions (Gumbel, Weibull, Fréchet) with known cumulative distribution functions. The problem here is dealing with the mutual dependence between points of the simulation. Even in this situation, our strategy involves finding a representation of the process as a Poisson random measure with infinite points. Differently form the Lévy case, since here we are dealing with component-wise maxima instead of sums, there is hope that truncation will not generate errors. The problem is that in general it is not possible to know for sure the number of iterations that guarantees our simulation to be exact. In the thesis we show two approaches: the first one is changing measure in such a way that it is possible during the algorithm to know exactly whether a truncation leads to an exact-simulation. A second approach is computing at each iteration the probability that a truncation leads to an error and stopping the algorithm when this probability is considered small enough.