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February 2002 Minimizing Shortfall Risk and Applications to Finance and Insurance Problems
Huyên Pham
Ann. Appl. Probab. 12(1): 143-172 (February 2002). DOI: 10.1214/aoap/1015961159


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.


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Huyên Pham. "Minimizing Shortfall Risk and Applications to Finance and Insurance Problems." Ann. Appl. Probab. 12 (1) 143 - 172, February 2002.


Published: February 2002
First available in Project Euclid: 12 March 2002

zbMATH: 1015.93071
MathSciNet: MR1890060
Digital Object Identifier: 10.1214/aoap/1015961159

Primary: 60G44 , 60H05 , 60H30 , 90A46 , 93E20

Keywords: duality theory , finance and insurance , optional decomposition under constraints , Semimartingales , Shortfall risk minimization

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.12 • No. 1 • February 2002
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