The existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether it is drifting to $+\infty$, $-\infty$, or oscillating. Whenever the Lévy process drifts to $+\infty$, we prove that the $\kappa$th moment of the first passage time (conditioned to be finite) exists if and only if the $(\kappa+1)$th moment of the Lévy jump measure exists. This generalizes a result shown earlier by Delbaen for Cramér–Lundberg risk processes. Whenever the Lévy process drifts to $-\infty$, we prove that all moments of the first passage time exist, while for an oscillating Lévy process we derive conditions for non-existence of the moments, and in particular we show that no integer moments exist.

We obtain series expansions of the q-scale functions of arbitrary spectrally negative Lévy processes, including processes with infinite jump activity, and use these to derive various new examples of explicit q-scale functions. Moreover, we study smoothness properties of the q-scale functions of spectrally negative Lévy processes with infinite jump activity. This complements previous results of Chan et al. (Prob. Theory Relat. Fields 150, 2011) for spectrally negative Lévy processes with Gaussian component or bounded variation.

Financial data are as a rule asymmetric, although most econometric models are symmetric. This applies also to continuous-time models for high-frequency and irregularly spaced data. We discuss some asymmetric versions of the continuous-time GARCH model, concentrating then on the GJR-COGARCH model. We calculate higher-order moments and extend the first-jump approximation. These results are prerequisites for moment estimation and pseudo maximum likelihood estimation of the GJR-COGARCH model parameters, respectively, which we derive in detail.

For a given bivariate Lévy process (U t , L t )t≥0, distributional properties of the stationary solutions of the stochastic differential equation dV t = V t-dU t + dL t are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.