The Annals of Applied Probability

Minimizing Shortfall Risk and Applications to Finance and Insurance Problems

Huyên Pham

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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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1015961159

Digital Object Identifier
doi:10.1214/aoap/1015961159

Mathematical Reviews number (MathSciNet)
MR1890060

Zentralblatt MATH identifier
1015.93071

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

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

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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