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 Research是在现代金融市场理论的背景下出现的各种公开的随机分析和控制的数学问题上提出的。这些问题大多是在一个具有一个无风险和多个风险证券的市场的连续时间模型的背景下提出的，涉及(I)单主体优化、多主体均衡、套期保值和价格取决于“大投资者”投资策略的市场中未定权益的定价问题；(Ii)在有交易成本的市场中的最优化、最低成本套期保值和定价问题；(Iii)任意不完全市场(包括具有投资组合约束的市场)中的公平定价问题；(Iv)在各种模型中套期保值概率的最大化；(V)研究新的博弈型期权；(Vi)动态风险度量；(Vii)战略市场博弈中的平稳均衡；(Viii)随机分析的相关问题，例如某些正倒向随机微分方程解的存在唯一性，某些偏微分方程粘性解的存在唯一性和变分不等式，解决损失函数具有随机分量的非标准随机控制问题。人们期望从随机分析和鞅理论、凸对偶理论、泛函分析、随机控制以及偏微分方程和变分不等式的粘性解中得到的已知工具，将被证明在解决这些问题中是有价值的，但也将不得不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。拟议研究的意义与这样一个事实有关，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的普遍假设，即市场是完美的，这意味着每一份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，实际情况并非如此；市场是“不完美的”或“不完整的”，原因包括投资限制、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“大的”或“知情的”投资者等等。因此，开发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，本提案中提出了其中一些建议。特别是，在再保险、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等新兴市场，这些方法应该被证明是有用的。

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.