论文标题
二元性可用于随机终止收入的最佳消费
Duality for optimal consumption with randomly terminating income
论文作者
论文摘要
我们建立了一个严格的二元理论，在没有有限的风险的无限利润下，在有收入流的存在下，无限的地平线问题是可以在成倍分布的时间内随机终止的最佳消费问题，而与资产价格无关。因此，我们缩小了Vellekoop和Davis遇到的双重性差距，在黑色choles市场中的一个版本中。二元理论的许多经典原则都存在，显着的例外是，零初始财富的边际效用是有限的。我们用作双重变量，一类超级马丁的缩水器，例如缩水财富以及超过收入超过收入的累积缩水消费是一款超级智能。我们表明，在我们的双重域中，折扣本地的Martingale缩水器的空间是密集的，因此双重问题也可以表示为与折扣的本地Martingale Feflators相比。我们表征了最佳财富过程，表明最佳的缩水财富是一种潜在的腐烂到零，而缩水财富加上收入的累积缩减消费是最佳的统一整合的martingale。我们将分析应用于Vellekoop和Davis示例，并提供数值解决方案。
We establish a rigorous duality theory, under No Unbounded Profit with Bounded Risk, for an infinite horizon problem of optimal consumption in the presence of an income stream that can terminate randomly at an exponentially distributed time, independent of the asset prices. We thus close a duality gap encountered by Vellekoop and Davis in a version of this problem in a Black-Scholes market. Many of the classical tenets of duality theory hold, with the notable exception that marginal utility at zero initial wealth is finite. We use as dual variables a class of supermartingale deflators such that deflated wealth plus cumulative deflated consumption in excess of income is a supermartingale. We show that the space of discounted local martingale deflators is dense in our dual domain, so that the dual problem can also be expressed as an infimum over the discounted local martingale deflators. We characterise the optimal wealth process, showing that optimal deflated wealth is a potential decaying to zero, while deflated wealth plus cumulative deflated consumption over income is a uniformly integrable martingale at the optimum. We apply the analysis to the Vellekoop and Davis example and give a numerical solution.
