Mathematical Sciences: Stochastic Analysis & Modeling in Financial Mathematics

金融数学中的随机分析和建模主要研究者：IOANNIS KARATZAS和JAKSA CVITANIC主要研究在现代金融市场理论背景下出现的随机分析和控制的各种开放数学问题。这些问题中的大多数都是在具有一种无风险证券和几种有风险证券的市场的连续时间模型的背景下制定的，并且涉及(i)单代理优化，多代理均衡，套期保值和市场中或有债权定价的问题，其价格可能取决于“大投资者”的投资策略；（ii）在有交易成本的市场中进行最优化、最便宜的套期保值和定价；（iii）任意不完全市场中的公平定价问题，包括有投资组合限制的市场；（iv）在各种模型中套期保值概率的最大化；(v)研究新的游戏式选择；风险的动态度量；(7)战略市场博弈中的平稳均衡；（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唯一性、某些偏微分方程和变分不等式的粘度解的存在唯一性、解决损失函数含有随机分量的非标准随机控制问题。我们期望从随机分析和鞅理论、凸对偶理论、泛函分析、随机控制和偏微分方程和变分不等式的粘度解中得到的已知工具将在解决这些问题中证明是有价值的，但也必须开发新的工具来解决出现的非标准问题。因此，该项目应促进对这些领域的理论和应用方面的理解。所提出的研究的意义在于，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存在的假设，即市场是完美的，这意味着每种金融合约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）。然而在现实中，情况并非如此；市场之所以“不完美”或“不完全”，除其他外，是由于投资限制、交易成本、不同的借贷利率、不同的信息模式、存在能够影响价格的“大”或“知情”投资者，等等。因此，开发新的方法来研究不完善市场中的风险和金融工具的定价方法是非常重要的，本文提出了一些建议。这些方法尤其适用于新兴市场，如再保险、巨灾保险、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.