Mathematical Sciences: Singular Control in Mathematical Finance and Related Problems

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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.