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Title: Simulation of a general class of $\alpha$-stable processes Authors:  Malcolm Egan - Universite Blaise Pascal (France) [presenting]
Nourddine Azzaoui - Universite Clermont Auvergne (France)
Gareth Peters - University of New South Wales (Australia)
Abstract: The heavy-tail and extremal dependence properties of $\alpha$-stable processes have lead to their extensive use in fields ranging from finance to engineering. In these fields, the stochastic integral representation plays an important role both in characterizing $\alpha$-stable processes as well as for the purposes of simulation and parameter estimation. In order use the stochastic integral representation, constraints on the random measure must be imposed. A key constraint is the independently scattered condition, where orthogonal increments of the random measure are independent. A key feature of the independently scattered condition is that the covariation is both left and right additive, which allows for simulation and estimation of this class of processes. Recently, a new generalization of the independently scattered condition has been introduced, which also preserves the left and right additivity of the covariation. This new generalization allows the characteristic function a wide class of $\alpha$-stable processes to be determined by a bimeasure. We deal with the problem of simulating from the bimeasure characterization of $\alpha$-stable processes. In particular, we prove conditions under which the bimeasure leads to a positive definite characteristic function for the case of a two-dimensional skeleton. Based on this result, we then propose a method to construct and simulate $n-$dimensional skeletons, for arbitrary $n > 2$.