We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to −∞ a.s. and the positive tail of the Lévy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Klüppelberg [Stochastic Process. Appl. 64 (1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207-226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Lévy processes. . This reprint differs from the original in pagination and typographic detail. 1 2 C. KLÜPPELBERG, A. E. KYPRIANOU AND R. A. MALLER t > 0, are modelled as a compound Poisson process, yielding the risk processGENERAL LÉVY INSURANCE RISK PROCESSES 3 the ruin probability can be shown to decrease exponentially fast, in fact, proportional to e −νu , as u → ∞. This result has been extended to a Lévy process setting by Bertoin and Doney [6]. If such a "Lundberg coefficient" ν does not exist, as is the case for subexponential and other "convolution equivalent" distributions (see Section 3), estimates of the ruin probability have been derived by Embrechts, Goldie and Veraverbeke [15], Embrechts and Veraverbeke [18] and Veraverbeke [35]; see [16], Section 1.4. It is also of prime interest to understand the way ruin happens. This question has been addressed by Asmussen [1] for the Cramér case, and more recently by Asmussen and Klüppelberg [3] for the subexponential case. They describe the sample path behavior of the process along paths leading to ruin via various kinds of conditional limit theorems. As expected, the Cramér case and the non-Cramér case are qualitatively quite different; see, for example, [2] and [16], Section 8.3.Our aim is to investigate the non-Cramér case in a general Lévy process setting, which clearly reveals the roles of the various assumptions. Our Lévy process X will start at 0 and be assumed to drift to −∞ a.s., but otherwise is quite general. Upward movement of X represents "claim payments," and the drift to −∞ reflects the fact that "premium income" should outweigh claims. "Ruin" will then correspond to passage of X above a specified high level, u, say. In this scenario, heavy-tailedness of the positive side of the distribution of upward jumps models the occurrence of large, possibly ruinous, claims, and has previously been studied in connection with the assumption of a finite mean for the process. But in general we do not want to restrict the process in this way. A higher rate of decrease of the process to −∞ is more desirable from the insurer's point of view, while allowing a heavier tail for the positive part is in keeping with the possibility of even more extreme events, which indeed are observed in recent insurance data.This ...
We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.
We use a discrete-time analysis, giving necessary and sufficient conditions for the almost-sure convergence of ARCH(1) and GARCH(1,1) discrete-time models, to suggest an extension of the ARCH and GARCH concepts to continuous-time processes. Our ‘COGARCH’ (continuous-time GARCH) model, based on a single background driving Lévy process, is different from, though related to, other continuous-time stochastic volatility models that have been proposed. The model generalises the essential features of discrete-time GARCH processes, and is amenable to further analysis, possessing useful Markovian and stationarity properties.
The discrete-time GARCH methodology which hits had such a profound influence on the modelling of heteroscedasticity in time series is intuitively Well motivated in capturing many 'stylized facts' concerning financial series, and is now almost routinely used in a wide range of situations, often including some where the data are not observed at equally spaced intervals of time. However, such data is more appropriately analyzed with a continuous-time model which preserves the essential features of the successful GARCH paradigm. One possible such extension is the diffusion limit of Nelson, but this is problematic in that the discrete-time GARCH model and its continuous-time diffusion limit tire not statistically equivalent. As till alternative, Kluppelberg et al. recently introduced a continuous.-time version of the GARCH (the 'COGARCH' process) which is constructed directly from a background driving Levy process, The present paper how to fit this model to irregularly spaced time series data using discrete-time GARCH methodology, by approximating the COGARCH with an embedded sequence of discrete-time, GARCH series which converges to the continuous-time model in a strong sense (in probability, in the Skorokhod metric), as the discrete/approximating grid grows finer. This property is also especially useful in certain other applications, such as options pricing. The way is then open to using, for the COGARCH, similar statistical techniques to those already worked out for GARCH models and to illustrate this, an empirical investigation using stock index data is carried out
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