  This paper proposes a stochastic approach to model temperature dynamic and
study related risk measures. The dynamic of temperatures can be modelled by a
mean-reverting process such as an Ornstein-Uhlenbeck one. In this study, we
estimate the parameters of this process thanks to daily observed suprema of
temperatures, which are the only data gathered by some weather stations. The
expression of the cumulative distribution function of the supremum is obtained
thanks to the law of the hitting time. The parameters are estimated by a least
square method quantiles based on this function. Theoretical results, including
mixing property and consistency of model parameters estimation, are provided.
The parameters estimation is assessed on simulated data and performed on real
ones. Numerical illustrations are given for both data. This estimation will
allow us to estimate risk measures, such as the probability of heat wave and
the mean duration of an heat wave.
