This paper presents a new model for characterising temporal dependence in exceedances above a threshold. The model is based on the class of trawl processes, which are stationary, infinitely divisible stochastic processes. The model for extreme values is constructed by embedding a trawl process in a hierarchical framework, which ensures that the marginal distribution is generalised Pareto, as expected from classical extreme value theory. We also consider a modified version of this model that works with a wider class of generalised Pareto distributions, and has the advantage of separating marginal and temporal dependence properties. The model is illustrated by applications to environmental time series, and it is shown that the model offers considerable flexibility in capturing the dependence structure of extreme value data.

We propose a parsimonious stochastic model for characterising the distributional and temporal properties of rainfall. The model is based on an integrated OrnsteinUhlenbeck process driven by the Hougaard Lévy process. We derive properties of this process and propose an extended model which generalises the Ornstein-Uhlenbeck process to the class of continuous-time ARMA (CARMA) processes. The model is illustrated by fitting it to empirical rainfall data on both daily and hourly time scales. It is shown that the model is sufficiently flexible to capture important features of the rainfall process across locations and time scales. Finally we study an application to the pricing of rainfall derivatives which introduces the market price of risk via the Esscher transform. We first give a result specifying the risk-neutral expectation of a general moving average process. Then we illustrate the pricing method by calculating futures prices based on empirical daily rainfall data, where the rainfall process is specified by our model.

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