Mathematical Sciences: Stochastic Analysis in Nonlinear Financial Markets

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9503582 Cvitanic This project, which is funded through the Applied Mathematics Program, the Statistics and Probability Program, and the Economics Program, will investigate various mathematical problems of stochastic analysis and control that arise in the context of modern theory and practice of nonlinear financial markets. These include (i) questions of optimization, hedging and pricing of contingent claims in markets with transaction costs, as well as in markets in which the prices can depend in a nonlinear fashion on the investment strategy and wealth of the agent; (ii) nonstandard models for asset prices. Mathematical questions related to (i) include existence and uniqueness of certain forward-backward stochastic differential equations, viscosity solutions to certain partial differential equations and variational inequalities, as well as a non-standard stochastic control problem in which the terminal loss function has a random component. With regard to (ii), it is expected that a new kind of stochastic calculus will have to be developed, one which goes beyond the semimartingales framework. One of the reasons for the present gap between the theory of finance and the reality of the stock market is the prevailing assumption in theoretical work that the market is perfect in the sense that every financial contract can be priced by calculating exactly its present value. This is, however, not the case in reality due to different types of market friction" such as transaction costs, different interest rates, presence of large investors who can influence the asset prices, and so forth Thus, it is very important to develop new methods for pricing financial instruments in imperfect markets, some of which will be the focus of this research. In a similar vein, the risk one undertakes when buying or selling financial instruments is typically much larger than predicted by the perfect-market theory (as recent scandals involving the trading of options confirm). This project will develop a theory of risk-hedging in imperfect markets that is more general and at the same time more applicable to realistic financial markets than current theory. ***