Ornstein-Uhlenbeck process and its supremum : theorical results and application to the climatic risk

Abstract

Forecasting and assessing the risk of heat waves is a crucial public policy stake. Evaluate the probability of heat waves and their severity can be possible by knowing the temperature in continuous time. However, daily extremes (maxima and minima) might be the only available data. The Ornstein-Uhlenbeck process is commonly used to model temperature dynamic. An estimation of the process parameters using only daily observed suprema of temperatures is proposed here. This new approach is based on a least square minimization using the cumulative distribution function of the supremum. Risk measures related to heat waves are then obtained numerically. In order to calculate explicitly those risk measures, it can be useful to have the joint law of the Ornstein-Uhlenbeck process and its supremum. The study is _rst limited to the joint density / distribution of the endpoint and supremum of the Ornstein-Uhlenbeck process. This probability admits a density, solution of the Fokker-Planck equation and explicitly obtained as an expansion involving parabolic cylinder functions. The proof of the density expression relies on a decomposition on a Hilbert basis of the space via a spectral method. We also study the oscillating Ornstein-Uhlenbeck process, which drift parameter is piecewise constant depending on the sign of the process. The Laplace transform of this process hitting time is determined and we also calculate the probability for the process to be positive on a fixed time.