Mathematical Sciences: Singular Control in Mathematical Finance and Related Problems

当交易成本的存在使得不断调整投资组合的成本过高时，金融领域就会出现一个奇异的随机控制问题。建议用相关的Hamilton-Jacobi- Bellman方程的粘度解分析来检验这类问题。将特别关注价值函数和交易策略对小额交易成本的敏感性。它也被提议发展非马尔可夫方法来处理这些问题，沿着对偶和鞅方法用于处理其他类型的市场不完全性。著名的布莱克-斯科尔斯期权定价公式假设交易可以在没有交易成本的情况下完成，并且由该公式构建的对冲投资组合不断调整所持有的股票数量。当Black-Scholes模型中引入非零交易成本时，期权的对冲组合实际上变得毫无用处。本文提出构建包含交易成本存在的交替连续时间模型，并确定交易成本的大小如何影响这些模型中的期权价格和对冲组合。

DMS-9500626 Shreve A singular stochastic control problem arises in finance when the presence of transaction costs makes constant portfolio adjustment prohibitively expensive. It is proposed to examine such problems using viscosity solution analysis of the relevant Hamilton-Jacobi- Bellman equation. Particular interest will be paid to sensitivity of value functions and trading strategies to small transaction costs. It is also proposed to develop non-Markov methods for such problems, along the lines of duality and martingale methods used to handle other types of market incompleteness. The celebrated Black-Scholes formula for option pricing assumes that trading can be done without transactions costs, and the hedging portfolio constructed by this formula constantly adjusts the amount of stock held. When a non-zero transaction cost is introduced to the Black-Scholes model, the hedging portfolio for an option become practically useless. It is proposed to construct alternate continuous- time models which incorporate the presence of transaction costs, and to determine how the size of transaction costs influence option prices and hedging portfolios in these models.