Differentiation of some functionals of risk processes, and optimal reserve allocation

Abstract

For general risk processes, we introduce and study the expected time-integrated negative part of the process on a fixed time interval. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below 0, studied by Dos Reis, and the probability of ruin. We carry out differentiation of other functionals of one-dimensional and multidimensional risk processes with respect to the initial reserve level. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve that minimizes one of these risk measures, illustrate the variety of fields of application and the benefits deriving from an efficient and effective use of such tools.

Properties of a risk measure derived from the expected area in red

“…, we now show that the optimal risk limit allocation leads to solutions that are similar to the ones of the optimal initial reserve allocation problem considered in Loisel (2005).…”

“…This paper studies a new risk measure derived from the expected area in red introduced in Loisel (2005). Specifically, we derive various properties of a risk measure defined as the smallest initial capital needed to ensure that the expected time-integrated negative part of the risk process on a fixed time interval [0, T ] (T can be infinite) is less than a given predetermined risk limit.…”

“…However, all those risk measures are either drawn from the infinite-time ruin theory or involve the behavior of the risk process between ruin times and recovery times. This is why Loisel (2005) introduces a risk measure based on a fixed time interval, finite or infinite, i.e. the expected time-integrated negative part of general risk processes on a fixed time interval [0, T ], also called the expected area in red.…”

Impact of Climate Change on Heat Wave Risk

“…Our starting point for the construction of our risk measure is a quantity introduced in Loisel [3] related to a risk process in insurance. Let T > 0, we thus consider first, which we call "the expected area in red" for (X t ), be defined as:…”

“…Following the work of [3], it is possible to differentiate functions A X and B X,Y with respect to x and y in order to see how climate change may impact those risk measures. We begin with a sensitivity analysis of A with respect to threshold x. Theorem 7.…”