First Jump Approximation of a Lévy Driven SDE and an Application to Multivariate ECOGARCH Processes

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(1,1) processes, 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.