Mathematical Sciences: Stochastic Analysis & Modeling in Financial Mathematics

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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.