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We study a new risk measure inspired from risk theory with a heat wave risk analysis motivation. We show that this risk measure and its sensitivities can be computed in practice for relevant temperature stochastic processes. This is in particular useful for measuring the potential impact of climate change on heat wave risk. Numerical illustrations are given.
Climate change has now been accepted as a proven fact. The recent 5th Intergovernmental Panel on Climate Change (IPCC) Assessment Report [
Our main goal here is to propose a generic risk measure in the context of climate change. Given a stochastic process that describes the temporal evolution of a climate indicator, we aim at quantifying an extreme and risky behavior of this indicator, without keeping a binary view (hitting some threshold or not, for example). The indicator may be any climate process that induces a risk that may be impacted by climate change: temperature, rainfall, sea level, snow level,
Let
A heat wave corresponds to a period during which the maximum temperature is very high.
Our starting point for the construction of our risk measure is a quantity introduced in Loisel [
Temperature has an important seasonal component. However, the importance of the seasonality of temperature processes depends on the point of view. The literature on the modeling of temperature processes distinguishes two types of models: models for large periods of time and models for smaller period of time. In the first case, one needs to introduce some seasonality component. In the second case, the seasonality is not the major part. When studying heat wave risk, we focus on a very small period of time, from several weeks to a couple of months. In this context, the seasonality can reasonably be neglected.
The other important observed property of temperature processes is the meanreverting property. A natural process to model seasonally adjusted temperature processes appears to be an Ornstein–Uhlenbeck process, which has been extensively used in the literature.
The authors of [
We have chosen the following model: let
In this section, we will study the properties of risk measures
The proofs are rather direct and omitted here.
These results confirm the intuition that, in various ways, if the probability to get high values of the maximum and minimum temperature process is increased due to climate change, then the risk measures,
As pointed out by one referee, one of the (claimed) consequences of climate change is not only an increase of temperature levels, but also, a tendency for more extreme weather. This can be interpreted as an increase of both the average level and variance of the temperature processes. To see the impact of a change in both levels and variability, we use the concept of increasing convex ordering.
Note that, in particular, if
The following results precisely show how this translates in terms of the parameters of the Gaussian models considered in this paper.
Following the work of [
☐
For the new risk measure,
We now carry out sensitivity analysis of
Then, we obtain that:
With Fubini and using the independence between
Then, applying Lebesgue’s theorem, we obtain the result. ☐
A similar analysis may be conducted with nonconstant coefficients, leading to more sophisticated temperature processes.
In this section, we aim at computing Equations (
First of all, let us remark that:
Now, since
The first one (Model 1) is the classical OUprocess Equation (
The second one (Model 2) is the OUprocess,
In their paper, Benth and Šaltytė Benth [
Although the process Equation (
Fitted parameters in the model of Benth and Šaltytė Benth [



0  1 



4 













4  2 



In order to compare the two models and since we are mostly interested in the summer period, we choose
Ameasure for both Model 1 Equation (
Ameasure for both Model 1 Equation (
For the
Ameasure for both Model 1 Equation (
Ameasure for Model 2 Equation (
Let us briefly recall the model we used here described in
For the process,
Bmeasure for the Model described in
Bmeasure for the model described in
Bmeasure for the model described in
Bmeasure for the model described in
This paper is the first step to tackling the problem of measuring climate risk. We have shown that risk measures
A heat wave is defined by “Météo France” as a sequence of at least three consecutive days for which the highest temperature is larger than a highlevel temperature and the lowest temperature is greater than a lowlevel temperature (both high and the lowlevel temperatures depend on the geographical zone). For
For some events, the extreme behavior of several quantities at the same time can breed a major risk. For example, both high wind and high temperature have to be taken into account in wildfire risk. Therefore, we could imagine quantities, like, for two processes,
This work has been supported by the Chair “Actuariat Responsable: gestion des risques naturels et changements climatiques”, funded by Generali since 2010.
The authors declare no conflict of interest.