Inverse Problems in Mathematical Finance: Exploring Bayesian Model Selection Algorithms

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Technical DescriptionThis project explores information-theoretic ideas for solving modelselection problems in Mathematical Finance. Typically, these problemsconsist in the specification of a diffusion measure, or more generally, ameasure 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. Thisproblem is studied from the point of view of partial differentialequations and Monte-Carlo simulation. We focus on model selectioncriteria based on minimizing the Kullback-Leibler entropy distance betweenthe unknown probability and a Bayesian prior. In the special case ofdiffusion processes, this leads to a constrained stochastic controlproblem that can be solved via Lagrange multipliers. In the case of MonteCarlo simulation, one must construct appropriate weighted measures, in atechnique which is reminiscent of "importance sampling." These problemsand their generalizations to non-linear constraints will be studied in aunified way using methods of Applied Mathematics and Numerical Analysis.The goal is to achieve a better understanding of the question of modelselection in Financial Economics, which is crucial for the management offinancial risk by quantitative methods.Non-Technical DescriptionThis research proposal deals with the pricing and hedging of complexfinancial instruments called derivative securities. The technology derivedhere, which is based on the mathematical fields of probability, statisticsand numerical analysis, is used to develop accurate tools for pricing andmanaging complex financial instruments. The motivation for this researchcomes from the fact that quantitative finance offers many challengingmathematical and computer-related problems. This is a consequence of theso-called "globalization'' phenomenon that links different financialmarkets and economies throughout the world. The current proposal dealswith new mathematical methods for fine-tuning these models. Our aim is tobetter understand how they work and how they represent financial risk. Bybringing to bear robust statistical methods and powerful mathematicaltechniques, we expect to shed light on pricing and risk-management systemsand to develop better models that can be shared with the financialindustry. This is an important application of Mathematics to a new areaof research: quantitative finance. The proposal is part of a greatereffort at New York University's Courant Institute in the field of financeand markets. So far, we have been successful in training young scientiststhat enter the business arena with a unique set of professional skills.This suggests that such research is both directly and indirectly suitablein terms of the larger picture of the national economy.