Stochastic Control Problems in Mathematical Finance

9319816 Karatzas这个项目将重点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的新技术。这些问题包括(1)不完全市场和/或有约束条件下或有债权的最优化、对冲和估值问题；(2)不完全或“有效完全”市场中的经济均衡问题。关于(1)，预计具有随机资金流的投资组合优化问题将需要泛函分析和凸分析的非标准技术。将定价问题转化为一类新的随机控制问题，其中终端报酬函数具有随机分量。与(2)有关的问题包括建立某些指数局部鞅的鞅性质，以及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”，即有可能准确计算此类工具的现值，从而计算其价格。事实上，情况并非如此；除其他因素外，由于投资限制、交易成本、不同的利率、不同的信息模式等原因，市场是“不完美的”或“不完整的”。因此，开发在不完全市场中为金融工具定价的新方法是非常重要的；本项目将研究这种方法。与此类似，标准经济均衡理论提供了确定金融资产价格的方法，从而使个体代理人的效用最大化和“市场畅通”(即，供求相等)--但同样，只有在完美市场的背景下才能如此。这个项目将发展一个更普遍的均衡理论，同时也通过能够处理不完全市场而更适用。

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.