## Journal of Applied Probability

- J. Appl. Probab.
- Volume 42, Number 2 (2005), 379-392.

### 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.

#### Article information

**Source**

J. Appl. Probab. Volume 42, Number 2 (2005), 379-392.

**Dates**

First available in Project Euclid: 14 June 2005

**Permanent link to this document**

http://projecteuclid.org/euclid.jap/1118777177

**Digital Object Identifier**

doi:10.1239/jap/1118777177

**Mathematical Reviews number (MathSciNet)**

MR2145483

**Zentralblatt MATH identifier**

1079.60038

**Subjects**

Primary: 60G17: Sample path properties

Secondary: 60G55: Point processes 91B30: Risk theory, insurance 91B32: Resource and cost allocation 62P05: Applications to actuarial sciences and financial mathematics

**Keywords**

Ruin theory sample path property optimal reserve allocation multidimensional risk process risk measure

#### Citation

Loisel, Stéphane. Differentiation of some functionals of risk processes, and optimal reserve allocation. J. Appl. Probab. 42 (2005), no. 2, 379--392. doi:10.1239/jap/1118777177. http://projecteuclid.org/euclid.jap/1118777177.