Duality for optimal consumption under no unbounded profit with bounded risk
Duality for optimal consumption under no unbounded profit with bounded risk
复制标题
无无限利润和有限风险下最优消费的二元性
DOI：
10.1214/21-aap1767
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
M. Monoyios
中科院分区：
文献类型：
--
作者：
M. Monoyios
在不满足无界利润和有界风险的半鞅不完全市场（NUPBR）中，我们给出了无限水平上最优消费的对偶性的确定性处理。而不是基地的对偶域（本地）鞅平减指数，我们使用一类上鞅平减指数，使通货紧缩的财富加上累积通货紧缩的消费是一个上鞅的所有容许的消费计划。这就产生了一个强的对偶性，因为由平减指数支配的过程的扩大的对偶域是自然封闭的，而不需要调用它的封闭性。这样，我们就自动达到了平减指数集的两极。我们通过证明局部鞅平减指数支配的过程集在我们的对偶域中是稠密的来完成这幅图，确认我们已经确定了自然的对偶空间。除了最优消费和平减指数，我们还讨论了最优财富过程。在最优情形下，收缩财富是一个上鞅和一个势，而收缩财富加上累积收缩消费是一个一致可积鞅。这是终端财富问题中相应特征的自然概括，其中最优的紧缩财富是一致可积鞅。我们没有使用涉及等价局部鞅测度的构造。这是很自然的，因为这样的措施通常不存在于无限的地平线上，我们正在NUPBR下工作，这并不需要它们的存在。与终端财富问题相比，对偶证明的结构揭示了一个有趣的特征。在那里，对偶域是$L^{1}$-有界的，但在这里，原始域有这个性质，因此在对偶证明中的许多步骤显示了原始域和对偶域的角色明显颠倒，与Kramkov和Schachermayer的证明相比。
We give a definitive treatment of duality for optimal consumption over the infinite horizon, in a semimartingale incomplete market satisfying no unbounded profit with bounded risk (NUPBR). Rather than base the dual domain on (local) martingale deflators, we use a class of supermartingale deflators such that deflated wealth plus cumulative deflated consumption is a supermartingale for all admissible consumption plans. This yields a strong duality, because the enlarged dual domain of processes dominated by deflators is naturally closed, without invoking its closure. In this way we automatically reach the bipolar of the set of deflators. We complete this picture by proving that the set of processes dominated by local martingale deflators is dense in our dual domain, confirming that we have identified the natural dual space. In addition to the optimal consumption and deflator, we characterise the optimal wealth process. At the optimum, deflated wealth is a supermartingale and a potential, while deflated wealth plus cumulative deflated consumption is a uniformly integrable martingale. This is the natural generalisation of the corresponding feature in the terminal wealth problem, where deflated wealth at the optimum is a uniformly integrable martingale. We use no constructions involving equivalent local martingale measures. This is natural, given that such measures typically do not exist over the infinite horizon and that we are working under NUPBR, which does not require their existence. The structure of the duality proof reveals an interesting feature compared with the terminal wealth problem. There, the dual domain is $L^{1}$-bounded, but here the primal domain has this property, and hence many steps in the duality proof show a marked reversal of roles for the primal and dual domains, compared with the proofs of Kramkov and Schachermayer.
无无界利润、有界风险下的中间消耗最优投资
影响因子：
1
作者：
Chau, Huy N.;Cosso, Andrea;Fontana, Claudio;Mostovyi, Oleksii
通讯作者：
Mostovyi, Oleksii