## Minimizing shortfall risk and applications to finance and insurance problems.(English)Zbl 1015.93071

The author studies the problem of minimizing the shortfall risk defined as the expectation of the shortfall $$(B-X_{T}^{x,\theta})_+$$ (weighted by some loss function) with the following controlled process: $X_t^{x,\theta}=x+\int_0^t\theta_u dS_u+H_t^{\theta},$ where $$B$$ is a given nonnegative measurable random variable, $$S$$ is a semimartingale, $$\Theta$$ is the set of the control process, $$\theta$$ is a convex subset of $$L(S)$$ and $$(H^{\theta}:\theta\in \Theta)$$ is a concave family of adapted processes with finite variation, $$t\in [0,T].$$
An existence result to this optimization problem is stated, and some qualitative properties of the associated value function are shown. A verification theorem in terms of a dual control problem is established. Some applications to hedging problems in constrained portfolios, large investor and reinsurance models are given.
The approach for solving these control problems uses probabilistic methods rather than PDE methods via the Bellman equation. This allows relaxing the assumption of a Markov state process required in the PDE approach.

### MSC:

 93E20 Optimal stochastic control 91B30 Risk theory, insurance (MSC2010) 60G44 Martingales with continuous parameter 60H05 Stochastic integrals 49K45 Optimality conditions for problems involving randomness 91G80 Financial applications of other theories
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### References:

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