带有交易成本和递归效用的投资组合优化
Portfolio Optimization under Transaction Costs with Recursive Preferences
AI总结：
本文研究了带比例交易成本和Epstein-Zin随机微分效用的Merton投资-消费问题，通过仅依赖原始方法且无需额外假设的完整验证论证，并引入用财富影子份额参数化的新思想，简化并深化了包括幂效用特例在内的分析。
摘要：
Merton投资-消费问题在金融学和随机控制领域都是基础性的。该问题的一个重要扩展引入了交易成本，这在金融视角上高度相关，但在控制视角上具有挑战性，因为解现在涉及奇异控制。另一个重要扩展是从可加效用转向随机微分效用（SDU），它允许时间偏好与风险偏好分离。在本文中，我们研究了带有比例交易成本和Epstein-Zin SDU的Merton问题的扩展版本。我们完整刻画了使问题适定的所有参数组合（可能依赖于交易成本水平），并提供了完整的验证论证，该论证不依赖任何额外技术假设且仅使用原始方法。由于对偶方法失效，SDU的情形需要新的数学技术。即使在（可加）幂效用的特殊情况下，我们的论证也比现有文献中的论证显著更简单、更优雅且更深远。这意味着我们能轻松分析该问题中以往极具挑战性的方面，包括比较静态分析、迄今需单独处理的边界情况，以及超越小交易成本机制的情形。一个关键且新颖的想法是用财富的影子份额对消费和价值函数进行参数化，这可能具有更广泛的适用性。
英文摘要：
The Merton investment-consumption problem is fundamental, both in the field of finance, and in stochastic control. An important extension of the problem adds transaction costs, which is highly relevant from a financial perspective but also challenging from a control perspective because the solution now involves singular control. A further significant extension takes us from additive utility to stochastic differential utility (SDU), which allows time preferences and risk preferences to be disentangled. In this paper, we study this extended version of the Merton problem with proportional transaction costs and Epstein-Zin SDU. We fully characterise all parameter combinations for which the problem is well posed (which may depend on the level of transaction costs) and provide a full verification argument that relies on no additional technical assumptions and uses primal methods only. The case with SDU requires new mathematical techniques as duality methods break down. Even in the special case of (additive) power utility, our arguments are significantly simpler, more elegant and more far-reaching than the ones in the extant literature. This means that we can easily analyse aspects of the problem which previously have been very challenging, including comparative statics, boundary cases which heretofore have required separate treatment and the situation beyond the small transaction cost regime. A key and novel idea is to parametrise consumption and the value function in terms of the shadow fraction of wealth, which may be of much wider applicability.
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