56 research outputs found
On the Depletion Problem for an Insurance Risk Process: New Non-ruin Quantities in Collective Risk Theory
The field of risk theory has traditionally focused on ruin-related
quantities. In particular, the socalled Expected Discounted Penalty Function
has been the object of a thorough study over the years. Although interesting in
their own right, ruin related quantities do not seem to capture path-dependent
properties of the reserve. In this article we aim at presenting the
probabilistic properties of drawdowns and the speed at which an insurance
reserve depletes as a consequence of the risk exposure of the company. These
new quantities are not ruin related yet they capture important features of an
insurance position and we believe it can lead to the design of a meaningful
risk measures. Studying drawdowns and speed of depletion for L\'evy insurance
risk processes represent a novel and challenging concept in insurance
mathematics. In this paper, all these concepts are formally introduced in an
insurance setting. Moreover, using recent results in fluctuation theory for
L\'evy processes, we derive expressions for the distribution of several
quantities related to the depletion problem. Of particular interest are the
distribution of drawdowns and the Laplace transform for the speed of depletion.
These expressions are given for some examples of L\'evy insurance risk
processes for which they can be calculated, in particular for the classical
Cramer-Lundberg model.Comment: 23 pages, 4 figure
Triangulating stable laminations
We study the asymptotic behavior of random simply generated noncrossing
planar trees in the space of compact subsets of the unit disk, equipped with
the Hausdorff distance. Their distributional limits are obtained by
triangulating at random the faces of stable laminations, which are random
compact subsets of the unit disk made of non-intersecting chords coded by
stable L\'evy processes. We also study other ways to "fill-in" the faces of
stable laminations, which leads us to introduce the iteration of laminations
and of trees.Comment: 34 pages, 5 figure
Optimality of doubly reflected Lévy processes in singular control
We consider a class of two-sided singular control problems. A controller either increases or decreases a given spectrally negative Lévy process so as to minimize the total costs comprising of the running and controlling costs where the latter is proportional to the size of control. We provide a sufficient condition for the optimality of a double barrier strategy, and in particular show that it holds when the running cost function is convex. Using the fluctuation theory of doubly reflected Lévy processes, we express concisely the optimal strategy as well as the value function using the scale function. Numerical examples are provided to confirm the analytical results
On maximum increase and decrease of Brownian motion
Article accepté aux Annales de l'Institut henri Poincaré - série B - 2007International audienceThe joint distribution of maximum increase and decrease for Brownian motion up to an independent exponential time is computed. This is achieved by decomposing the Brownian path at the hitting times of the infimum and the supremum before the exponential time. It is seen that an important element in our formula is the distribution of the maximum decrease for the three dimensional Bessel process with drift started from 0 and stopped at the first hitting of a given level. From the joint distribution of the maximum increase and decrease it is possible to calculate the correlation coefficient between these at a fixed time and this is seen to be -0.47936...
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