First jump approximation of a Lévy-driven SDE and an application to multivariate ECOGARCH processes

Abstract

The first jump approximation of a pure jump Lévy process, which converges to the Lévy process in the Skorokhod topology in probability, is generalised to a multivariate setting and an infinite time horizon. It is shown that it can generally be used to obtain “first jump approximations” of Lévy-driven stochastic differential equations, by establishing that it has uniformly controlled variations. Applying this general result to multivariate exponential continuous time GARCH processes of order (1, 1), it is shown that there exists a sequence of piecewise constant processes determined by multivariate exponential GARCH(1, 1) processes in discrete time which converge in probability in the Skorokhod topology to the continuous time process.