9319816 Karatzas This project will focus on various mathematical problems arising in continuous-time finance that require the development of new techniques in stochastic control theory. These include (1) questions of optimization, hedging, and valuation of contingent claims in incomplete markets and/or under constraints; and (2) questions of economic equilibrium in incomplete or "effectively complete" markets. With regard to (1), it is expected that the problem of portfolio optimization with random endowment streams will require non-standard techniques from functional and convex analysis. Questions of pricing are formulated in terms of a new kind of stochastic control problem where the terminal reward function has a random component. Issues related to (2) include the establishment of the martingale property for certain exponential local martingales, and the representation property of a given set of martingales as stochastic integrals with respect to a second, given set. Existing methods for pricing financial instruments (for example, contracts whose value will be revealed at some future date) assume that financial markets are "complete" of "perfect" in the sense that it is possible to exactly calculate the present value, and hence the price, of such instruments. In reality, this is not the case; markets are "imperfect" or "incomplete" due, among other things, to investment constraints, transaction costs, different interest rates, different patterns of information, etc. Thus, it is very important to develop new methods for pricing financial instruments in imperfect markets; such methodologies will be investigated in this project. In a similar vein, standard economic equilibrium theory provides ways to determine prices for financial assets so that individual agents' utilities are maximized and "markets clear" (that is, supply equals demand) - but again, only in the context of perfect markets. This project will develop an equilibrium theory which is more g eneral and at the same time also more applicable by being able to deal with imperfect markets.
