Stochastic Control Problems in Mathematical Finance

这个项目将集中研究连续时间金融中出现的各种数学问题，这些问题需要发展随机控制理论中的新技术。这些问题包括：(1)不完全市场和/或约束条件下或有债权的优化、套期保值和估值问题；(2)不完全市场或“有效完全”市场的经济均衡问题。关于(1)，预计随机禀赋流的投资组合优化问题将需要来自函数和凸分析的非标准技术。定价问题是根据一种新的随机控制问题来表述的，其中终端奖励函数具有随机成分。与(2)相关的问题包括建立某些指数局部鞅的鞅性质，以及将给定鞅集合表示为关于第二个给定集合的随机积分的表示性质。现有的金融工具定价方法（例如，其价值将在未来某个日期披露的合同）假设金融市场是“完整的”或“完美的”，因为有可能精确计算出这些工具的现值，从而计算出价格。在现实中，情况并非如此；由于投资限制、交易成本、不同的利率、不同的信息模式等因素，市场是“不完美的”或“不完全的”。因此，在不完善的市场中开发新的金融工具定价方法是非常重要的；本项目将对这些方法进行研究。同样，标准的经济均衡理论也提供了确定金融资产价格的方法，以使个体代理人的效用最大化和“市场出清”（即供给等于需求）——但同样，这仅在完美市场的背景下。这个项目将发展一个均衡理论，这是更普遍的，同时也更适用于能够处理不完美的市场。

9319816 Karatzas This project will focus on various mathematical problems arising in continuous-time finance that require the development of new techniques in stochastic control theory. These include (1) questions of optimization, hedging, and valuation of contingent claims in incomplete markets and/or under constraints; and (2) questions of economic equilibrium in incomplete or "effectively complete" markets. With regard to (1), it is expected that the problem of portfolio optimization with random endowment streams will require non-standard techniques from functional and convex analysis. Questions of pricing are formulated in terms of a new kind of stochastic control problem where the terminal reward function has a random component. Issues related to (2) include the establishment of the martingale property for certain exponential local martingales, and the representation property of a given set of martingales as stochastic integrals with respect to a second, given set. Existing methods for pricing financial instruments (for example, contracts whose value will be revealed at some future date) assume that financial markets are "complete" of "perfect" in the sense that it is possible to exactly calculate the present value, and hence the price, of such instruments. In reality, this is not the case; markets are "imperfect" or "incomplete" due, among other things, to investment constraints, transaction costs, different interest rates, different patterns of information, etc. Thus, it is very important to develop new methods for pricing financial instruments in imperfect markets; such methodologies will be investigated in this project. In a similar vein, standard economic equilibrium theory provides ways to determine prices for financial assets so that individual agents' utilities are maximized and "markets clear" (that is, supply equals demand) - but again, only in the context of perfect markets. This project will develop an equilibrium theory which is more g eneral and at the same time also more applicable by being able to deal with imperfect markets.