Davis和Norman问题的渐近性和对偶性
Asymptotics and Duality for the Davis and Norman Problem
摘要：
我们重新讨论了在具有成比例交易成本的布莱克-斯科尔斯模型中，在无限时间范围内消费的期望对数效用最大化的问题，正如戴维斯和诺曼的开创性论文[Math. OperationResearch，15，1990]中所研究的那样。类似于Kallsen和Muhle-Karbe [Ann. Appl. Probab.，20，2010]，我们通过确定影子价格来解决这个问题，影子价格是一个无摩擦的价格过程，其值在买卖价差中，这导致了相同的优化问题。然而，我们使用不同的参数化，这有利于计算和验证。此外，对于小的交易成本，我们确定分数泰勒展开的任意阶的边界的非贸易区域和价值函数。这扩展了Janecek和Shreve [Finance Stoch.，8，2004]，谁确定了这些幂级数的主导项。
英文摘要：
We revisit the problem of maximizing expected logarithmic utility from consumption over an infinite horizon in the Black-Scholes model with proportional transaction costs, as studied in the seminal paper of Davis and Norman [Math. Operation Research, 15, 1990]. Similarly to Kallsen and Muhle-Karbe [Ann. Appl. Probab., 20, 2010], we tackle this problem by determining a shadow price, that is, a frictionless price process with values in the bid-ask spread which leads to the same optimization problem. However, we use a different parametrization, which facilitates computation and verification. Moreover, for small transaction costs, we determine fractional Taylor expansions of arbitrary order for the boundaries of the no-trade region and the value function. This extends work of Janecek and Shreve [Finance Stoch., 8, 2004], who determined the leading terms of these power series.
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