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