Mathematical Sciences: Stochastic Analysis & Modeling in Financial Mathematics

DMS-9732810 金融数学中的随机分析与建模 主要研究者：IOANNIS KARATZAS和JAKSA CVITANIC 研究提出了各种开放的随机分析和控制的数学问题，这些问题出现在现代金融市场理论的背景下。这些问题中的大多数是在一个连续时间模型的背景下制定的市场与一个无风险和几个风险证券，并涉及 (i)单代理优化、多代理均衡、套期保值和市场中未定权益的定价问题，市场的价格可能取决于“大投资者”的投资战略;（二）有交易成本的市场中的优化、最便宜的套期保值和定价;（三）任意不完全市场中的公平定价问题，包括有投资组合约束的市场;（iv）在各种模型中最大化对冲的可能性;（v）研究新的博弈型期权;（vi）风险的动态度量;（vii）战略市场博弈中的静态均衡;（viii）随机分析的相关问题，如某些正倒向随机微分方程的存在性和唯一性，某些偏微分方程和变分不等式的粘性解的存在性和唯一性，解决损失函数具有随机分量的非标准随机控制问题。预计已知的工具，从随机分析和鞅理论，凸对偶理论，泛函分析，随机控制和粘性解偏微分方程和变分不等式，将证明有价值的解决这些问题，而且，新的工具将不得不开发解决非标准问题出现。因此，该项目应导致对这些领域的理论和应用方面的理解的进步。 拟议的研究的重要性与以下事实有关，即金融市场理论与实践之间的主要差距之一是大多数模型中普遍假设市场是完美的，这意味着每一个金融合同都可以以独特的方式定价，其风险原则上可以完全控制（对冲）。然而，事实并非如此;市场是“不完美的”或“不完整的”，除其他外，这是由于投资限制、交易成本、借贷利率不同、信息模式不同、存在能够影响价格的“大型”或“知情”投资者等。发展新的方法来研究不完全市场中的风险和金融工具的定价方式是非常重要的，本建议中提出了其中一些方法。特别是，这些方法应证明对新兴市场有用，如再保险、灾难保险、能源衍生品、信用风险衍生品和其他流动性不足的市场。

DMS-9732810 STOCHASTIC ANALYSIS AND MODELING IN FINANCIAL MATHEMATICS Principal Investigators: IOANNIS KARATZAS and JAKSA CVITANIC Research is proposed on various open mathematical problems of stochastic analysis and control, which arise in the context of the modern theory for financial markets. Most of these problems are formulated in the context of a continuous-time model for a market with one riskless and several risky securities, and involve (i) questions of single-agent optimization, multi-agent equilibrium, hedging and pricing of contingent claims in markets with prices that can depend on the investment strategy of a "large investor"; (ii) optimization, least expensive hedging, and pricing, in markets with transaction costs; (iii) questions of fair pricing in arbitrary incomplete markets, including markets with portfolio constraints; (iv) maximization of the probability of hedging, in various models; (v) study of new, game-type options; (vi) dynamic measures of risk; (vii) stationary equilibrium in strategic market games; (viii) related questions of stochastic analysis such as existence and uniqueness of certain Forward-Backward Stochastic Differential Equations, existence and uniqueness of viscosity solutions to certain PDE's and variational inequalities, solving non-standard stochastic control problems where the loss function has a random component. It is expected that known tools from stochastic analysis and martingale theory, convex duality theory, functional analysis, stochastic control and viscosity solutions to partial differential equations and variational inequalities, will prove valuable in the resolution of these questions, but also that new tools will have to be developed to solve nonstandard problems that arise. Thus, the project should result in the advancement of the understanding of both the theoretical and applied aspects of these fields. The significance of the proposed research is related to the fact that one of the main gaps between theory and practice of financial markets is the prevailing assumption in most of the models that the markets are perfect, implying that every financial contract can be priced in a unique way, and its risk can, in principle be fully controlled (hedged). In reality, however, this is not the case; markets are "imperfect" or "incomplete" due, among other things, to investment constraints, transaction costs, different interest rates for borrowing and lending, different patterns of information, presence of "large" or "informed" investors who can influence prices, etc. Thus, it is very important to develop new methods for studying the risks and the ways of pricing financial instruments in imperfect markets, some of which are suggested in this proposal. In particular, these methods should prove useful in newly developing markets such as re-insurance, catastrophic insurance, energy derivatives, credit-risk derivatives and other insufficiently liquid markets.