Abstract
We address a problem of stochastic optimal control in mathematical finance, namely quadratic hedging with constraints on both the portfolio invested and the wealth process. Quadratic hedging involves the minimization of a quadratic loss criterion. Constraints on the portfolio are essentially control constraints while constraints on the wealth process are state constraints, so the problem amounts to stochastic optimal control with the combination of control and state constraints. Few results are available on general problems of this kind. However, our particular problem has the nice properties of being convex, with simple linear dynamics and the state constraint in the form of a one-sided almost-sure inequality. These are key to the application of a powerful variational method of Rockafellar for abstract problems of convex programming. We construct an optimal portfolio by means of this approach.
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