论文标题
在没有无限利润的情况下，具有限制风险的二元性可用于最佳消费
Duality for optimal consumption under no unbounded profit with bounded risk
论文作者
论文摘要
在半明星不完整的市场中，我们对二元性的最佳消费提供了确切的二元性处理，从而满足了有界风险（NUPBR）的无限利润。我们没有将双重域基于（本地）Martingale缩进器以基于双重的超级马丁代理缩水器，例如，缩水财富加上累积的放气消费量是所有可接受的消费计划的超级智能消费。这会产生强大的二元性，因为由缩水器主导的流程的扩大双重结构域自然而然地关闭，而无需调用其闭合。通过这种方式，我们会自动到达缩水器集的双极性。我们通过证明由当地的Martingale缩进器主导的一组过程在我们的双重域中茂密，从而确认我们已经确定了自然的双重空间，从而完成了这张照片。除了最佳消费和缩进器外，我们还表征了最佳财富过程。最佳的是，放气财富是一种超级马丁的和潜力，而缩气的财富加上累积的放气消费量是统一的整合群落。这是终端财富问题中相应特征的自然概括，其中最佳的缩水财富是一个统一的整合马丁纳尔。我们不使用涉及等效的当地赛车措施的结构。鉴于这种措施通常不存在于无限的地平线，并且我们在Nupbr下工作，这是很自然的，这不需要它们的存在。与终端财富问题相比，二元性证明的结构揭示了一个有趣的特征。在那里，双域是$ l^{1} $ - 有限的，但是在这里，原始域具有此属性，因此，双重性证明中的许多步骤显示了与Kramkov和Schachermayer的证明相比，对原始和双重域的角色显着逆转。
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.
