The Numeraire in Stochastic Finance Theory

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The optimal wealth process corresponding to the expected logarithmic utility maximization problem is probably the most popular and important investment strategy, due to the abundance of its optimality properties. One of the most prominent characteristics of the log-optimal strategy is its remarkable numeraire property: every nonnegative wealth process, when discounted by the log-optimal one, becomes a supermartingale. For this last reason, the log-optimal wealth process is commonly referred to as "the numeraire". The present project investigates issues related to the numeraire in financial market modeling. Focus is given in the following areas: (1) deep connections between the existence of the numeraire and market viability, which will result in strengthening and simplifying the theoretical foundations of Stochastic Finance; (2) several ramifications of the numeraire property in non-traditional models; (3) enlargement of the list of optimality properties of the numeraire; (4) closed-form expressions of general equilibria in incomplete financial markets with logarithmic investors possessing heterogeneous beliefs; (5) continuity and differentiability of the numeraire with respect to market inputs, such as statistical probabilities and information.This project provides a new approach in analyzing the appropriateness of theoretical market models to successfully reflect real-world phenomena, as well as to investigate deeper properties of financial markets. The power of this approach is its all-encompassing nature and its simplicity, since it solely depends on the existence and structure of a singe remarkable investment opportunity in the market, which is the numeraire. The broader impact that the project will have is twofold: (1) it will facilitate cross-fertilization between the scientific areas of Probability and Finance, with rich interplay of ideas flowing both ways from one field to the other; (2) it will cast the pedagogical methodology of teaching Financial Mathematics under a new perspective. In fact, a monograph incorporating this new approach in Stochastic Finance is expected to emerge after the project has been carried out.