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Quadratic minimization with portfolio and terminal wealth constraints

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  • Andrew Heunis

Abstract

We address a problem of stochastic optimal control drawn from the area of mathematical finance. The goal is to minimize the expected value of a general quadratic loss function of the wealth at close of trade when there is a specified convex constraint on the portfolio over the trading interval, together with a specified almost-sure lower-bound on the wealth at close of trade. We use a variational approach of Rockafellar which leads naturally to an appropriate vector space of dual variables, a dual functional on the space of dual variables such that the dual problem of maximizing the dual functional is guaranteed to have a solution (i.e. a Lagrange multiplier) when a simple and natural Slater condition holds for the terminal wealth constraint, and obtain necessary and sufficient conditions for optimality of a candidate wealth process. The dual variables are pairs, each comprising an Itô process paired with a member of the adjoint of the space of essentially bounded random variables measurable with respect to the event $$\sigma $$ σ -algebra at close of trade. The necessary and sufficient conditions are used to construct an optimal portfolio in terms of the Lagrange multiplier. The dual problem simplifies to maximization of a concave function over the real line when the portfolio is unconstrained but the terminal wealth constraint is maintained. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Andrew Heunis, 2015. "Quadratic minimization with portfolio and terminal wealth constraints," Annals of Finance, Springer, vol. 11(2), pages 243-282, May.
  • Handle: RePEc:kap:annfin:v:11:y:2015:i:2:p:243-282
    DOI: 10.1007/s10436-014-0254-9
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    References listed on IDEAS

    as
    1. Ralf Korn, 1997. "Some applications of L2-hedging with a non-negative wealth process," Applied Mathematical Finance, Taylor & Francis Journals, vol. 4(1), pages 65-79.
    2. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, October.
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    Cited by:

    1. Tuan Nguyen Dinh, 2023. "Regularity of Multipliers for Multiobjective Optimal Control Problems Governed by Evolution Equations," Journal of Optimization Theory and Applications, Springer, vol. 196(2), pages 762-796, February.
    2. Daniela Neykova & Marcos Escobar & Rudi Zagst, 2015. "Optimal investment in multidimensional Markov-modulated affine models," Annals of Finance, Springer, vol. 11(3), pages 503-530, November.
    3. Dian Zhu & Andrew J. Heunis, 2017. "Quadratic minimization with portfolio and intertemporal wealth constraints," Annals of Finance, Springer, vol. 13(3), pages 299-340, August.

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    More about this item

    Keywords

    Portfolio optimization; Stochastic control; Conjugate duality; Constraints; Lagrange multiplier; Slater condition; C61; C65; G11;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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