This project explores information-theoretic ideas for solving model selection problems in Mathematical Finance. Typically, these problems consist in the specification of a diffusion measure, or more generally, a measure on path-space, which describes the future states of the market. The data used in the inversion consists of expected values of functionals, which correspond to observed prices of "benchmark" securities. This problem is studied from the point of view of partial differential equations and Monte-Carlo simulation. We focus on model selection criteria based on minimizing the Kullback-Leibler entropy distance between the unknown probability and a Bayesian prior. In the special case of diffusion processes, this leads to a constrained stochastic control problem that can be solved via Lagrange multipliers. In the case of Monte Carlo simulation, one must construct appropriate weighted measures, in a technique which is reminiscent of "importance sampling." These problems and their generalizations to non-linear constraints will be studied in a unified way using methods of Applied Mathematics and Numerical Analysis. The goal is to achieve a better understanding of the question of model selection in Financial Economics, which is crucial for the management of financial risk by quantitative methods.
Non-Technical Description
This research proposal deals with the pricing and hedging of complex financial instruments called derivative securities. The technology derived here, which is based on the mathematical fields of probability, statistics and numerical analysis, is used to develop accurate tools for pricing and managing complex financial instruments. The motivation for this research comes from the fact that quantitative finance offers many challenging mathematical and computer-related problems. This is a consequence of the so-called "globalization'' phenomenon that links different financial markets and economies throughout the world. The current proposal deals with new mathematical methods for fine-tuning these models. Our aim is to better understand how they work and how they represent financial risk. By bringing to bear robust statistical methods and powerful mathematical techniques, we expect to shed light on pricing and risk-management systems and to develop better models that can be shared with the financial industry. This is an important application of Mathematics to a new area of research: quantitative finance. The proposal is part of a greater effort at New York University's Courant Institute in the field of finance and markets. So far, we have been successful in training young scientists that enter the business arena with a unique set of professional skills. This suggests that such research is both directly and indirectly suitable in terms of the larger picture of the national economy.
