## The Annals of Applied Probability

### Minimizing Shortfall Risk and Applications to Finance and Insurance Problems

Huyên Pham

#### Abstract

We consider a controlled process governed by $X^{x, \theta} = x + \int \theta dS + H^{\theta}$, where $S$ is a semimartingale, $\Theta$ the set of control processes . is a convex subset of $L(S)$ and ${H^{\theta} :\theta \in \Theta}$ is a concave family of adapted processes with finite variation. We study the problem of minimizing the shortfall risk defined as the expectation of the shortfall $(B - X_T^{x, \theta})_+$ weighted by some loss function, where $B$ is a given nonnegative measurable random variable. Such a criterion has been introduced by Föllmer and Leukert [Finance Stoch. 4 (1999) 117–146] motivated by a hedging problem in an incomplete financial market context:$\Theta = L(S)$ and $H^{\theta} \equiv 0$. Using change of measures and optional decomposition under constraints, we state an existence result to this optimization problem and show some qualitative properties of the associated value function. A verification theorem in terms of a dual control problem is established which is used to obtain a quantitative description of the solution. Finally, we give some applications to hedging problems in constrained portfolios, large investor and reinsurance models.

#### Article information

Source
Ann. Appl. Probab., Volume 12, Number 1 (2002), 143-172.

Dates
First available in Project Euclid: 12 March 2002

https://projecteuclid.org/euclid.aoap/1015961159

Digital Object Identifier
doi:10.1214/aoap/1015961159

Mathematical Reviews number (MathSciNet)
MR1890060

Zentralblatt MATH identifier
1015.93071

#### Citation

Pham, Huyên. Minimizing Shortfall Risk and Applications to Finance and Insurance Problems. Ann. Appl. Probab. 12 (2002), no. 1, 143--172. doi:10.1214/aoap/1015961159. https://projecteuclid.org/euclid.aoap/1015961159

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• CNRS, UMR 7599 UFR MATHÉMATIQUES, CASE 7012 UNIVERSITÉ PARIS 7 2 PLACE JUSSIEU 75251 PARIS CEDEX 05 FRANCE E-MAIL: pham@gauss.math.jussieu.fr