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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References
Dubovitskii, A.Y., Mil’yutin, A.A.: Necessary conditions for a weak extremum in problems of optimal control with mixed inequality constraints. Zhur. Vychislitel. Mat. Mat-Fys. 8, 725–779 (1968). (transl: USSR Comp. Math. Math. Phys. 8, 24–98 (1968))
Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
Heunis, A.J.: Quadratic minimization with portfolio and terminal wealth constraints. Ann. Financ. 11, 243–282 (2015)
Makowski, K., Neustadt, L.W.: Optimal control with mixed control-phase variable equality and inequality constraints. SIAM J. Control Optim. 12, 184–228 (1974)
Labbé, C., Heunis, A.J.: Convex duality in constrained mean-variance portfolio optimization. Adv. Appl. Probab. 39, 77–104 (2007)
Lim, A.E.B., Zhou, X.Y.: Mean-variance portfolio selection with random parameters in a complete market. Math. Oper. Res. 27, 101–120 (2002)
Markowitz, H.: Portfolio selection. J. Financ. 7, 77–91 (1952)
Rockafellar, R.T.: Conjugate Duality and Optimization, (CBMS-NSF Ser. No. 16). SIAM, Philadelphia (1974)
Rockafellar, R.T., Wets, J.B.: Stochastic convex programming: singular multipliers and extended duality. Pac. J. Math. 62, 507–522 (1976)
Zhu, D., Heunis, A.J.: Quadratic minimization with portfolio and intertemporal wealth constraints. Ann. Financ. (to appear)
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Heunis, A. (2018). Quadratic Hedging with Mixed State and Control Constraints. In: Tempo, R., Yurkovich, S., Misra, P. (eds) Emerging Applications of Control and Systems Theory. Lecture Notes in Control and Information Sciences - Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-319-67068-3_28
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DOI: https://doi.org/10.1007/978-3-319-67068-3_28
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