Conjugate duality in problems of constrained utility maximization
Conjugate duality in problems of constrained utility maximization
复制标题
约束效用最大化问题中的共轭对偶性
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
A. Heunis
中科院分区：
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作者：
C. Labbé;A. Heunis
我们证明了铋的一种简单而优雅的方法[J.数学。分析应用，44(1973)，pp.384-404]将共轭对偶应用于Bolza的凸性问题，直接适用于数学金融中具有投资组合约束的效用最大化问题。这给出了一个相关的对偶问题的直接构造，以及欧拉-拉格朗日关系和横截性关系，然后利用这些关系建立了关于对偶问题的解的最优投资组合的存在性。该方法是完全综合的，并且不需要关于无约束优化的虚拟完全市场的相当困难的先验假设，后者一直是在有交易约束的效用最大化问题中综合最优投资组合的标准方法。它还补充了罗杰斯的二元性合成[数学讲稿，编号。LNM-1814，Springer-Verlag，纽约，2003，第95-131页]和Klein和Rogers[数学。金融学，17(2007)，第225-247页]关于市场不完善情况下效用最大化的一般问题。
We show that a simple and elegant method of Bismut [J. Math. Analysis Appl., 44 (1973), pp. 384–404] for applying conjugate duality to convex problems of Bolza adapts directly to problems of utility maximization with portfolio constraints in mathematical finance. This gives a straightforward construction of an associated dual problem together with Euler–Lagrange and transversality relations, which are then used to establish existence of optimal portfolios in terms of solutions of the dual problem. The approach is completely synthetic, and does not require the rather difficult a priori hypothesis of a fictitious complete market for unconstrained optimization, which has been the standard approach for synthesizing optimal portfolios in problems of utility maximization with trading constraints. It also complements a duality synthesis of Rogers [Lecture Notes in Mathematics, No. LNM-1814, Springer-Verlag, New York, 2003, pp. 95–131] and Klein and Rogers [Math. Finance, 17 (2007), pp. 225–247] for general problems of utility maximization with market imperfections.