The Annals of Applied Probability

Minimizing Shortfall Risk and Applications to Finance and Insurance Problems

Huyên Pham

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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

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

First available in Project Euclid: 12 March 2002

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Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control 60G44: Martingales with continuous parameter 60H05: Stochastic integrals 60H30: Applications of stochastic analysis (to PDE, etc.) 90A46

Shortfall risk minimization semimartingales optional decomposition under constraints duality theory finance and insurance


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.

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