Roland K. Price & Zoran Vojinovic Urban Hydroinformatics: Data, Models and Decision Support for Integrated Urban Water Management by January 2011. IWA Publishing, London. 552 pages. Paperback ISBN: 9781843392743, £105.00/US$189.00/€ 141.75

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A central limit theorem for realised power and bipower variations of continuous semimartingales

Central limit theorem, quadratic variation, bipower variation --Central limit theorem,quadratic variation,bipower variation

A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales

Consider a semimartingale of the form Y_{t}=Y_0+\int _0^{t}a_{s}ds+\int _0^{t}_{s-} dW_{s}, where a is a locally bounded predictable process and (the "volatility") is an adapted right--continuous process with left limits and W is a Brownian motion. We define the realised bipower variation process V(Y;r,s)_{t}^n=n^{((r+s)/2)-1} \sum_{i=1}^{[nt]}|Y_{(i/n)}-Y_{((i-1)/n)}|^{r}|Y_{((i+1)/n)}-Y_{(i/n)}|^{s}, where r and s are nonnegative reals with r+s>0. We prove that V(Y;r,s)_{t}n converges locally uniformly in time, in probability, to a limiting process V(Y;r,s)_{t} (the "bipower variation process"). If further is a possibly discontinuous semimartingale driven by a Brownian motion which may be correlated with W and by a Poisson random measure, we prove a central limit theorem, in the sense that \sqrt(n) (V(Y;r,s)^n-V(Y;r,s)) converges in law to a process which is the stochastic integral with respect to some other Brownian motion W', which is independent of the driving terms of Y and \sigma. We also provide a multivariate version of these results.

A methodology for probabilistic real-time forecasting – an urban case study

Copyright © IWA Publishing 2013. The definitive peer-reviewed and edited version of this article is published in Journal of Hydroinformatics, Volume 15 (3), pp. 751-762 (2013), DOI:10.2166/hydro.2012.031 and is available at www.iwapublishing.comThe phenomenon of urban flooding due to rainfall exceeding the design capacity of drainage systems is a global problem and can have significant economic and social consequences. The complex nature of quantitative precipitation forecasts (QPFs) from numerical weather prediction (NWP) models has facilitated a need to model and manage uncertainty. This paper presents a probabilistic approach for modelling uncertainty from single-valued QPFs at different forecast lead times. The uncertainty models in the form of probability distributions of rainfall forecasts combined with a sewer model is an important advancement in real-time forecasting at the urban scale. The methodological approach utilized in this paper involves a retrospective comparison between historical forecasted rainfall from a NWP model and observed rainfall from rain gauges from which conditional probability distributions of rainfall forecasts are derived. Two different sampling methods, respectively, a direct rainfall quantile approach and the Latin hypercube sampling based method were used to determine the uncertainty in forecasted variables (water level, volume) for a test urban area, the city of Aarhus. The results show the potential for applying probabilistic rainfall forecasts and their subsequent use in urban drainage forecasting for estimation of prediction uncertainty