This project addresses topics related to stochastic controls, games and portfolios, including the study of relative arbitrage in stochastic portfolio theory, where one seeks simple, descriptive conditions that allow for arbitrage relative to a large equity market, and then tries to describe the nature of the most efficient such arbitrage; problems of stochastic control with discretionary stopping, and stochastic games with features of both stopping and control; problems of stochastic control of bounded variation or "singular" type; and problems of stochastic control and/or stopping under partial observations. In recent years, the investigator and his collaborators have made considerable progress in identifying simple, descriptive conditions on observable characteristics, such as diversity or sufficient intrinsic volatility, which allow the construction of simple (based solely on observable quantities) portfolios that can outperform a large equity market. This project seeks to build on these results in order to understand the nature of optimal, that is, "least-expensive", arbitrages of this type, and their connections to stochastic analysis (exit measures for supermartingales, degenerate diffusions), parabolic partial differential equations (propagation of qualitative properties such as convexity), and the fine structure of financial markets (the onset of arbitrage opportunities, or of "bubbles", and its implications for pricing and for hedging). Additionally, the investigator has gained considerable understanding of stochastic control problems with discretionary stopping, when a "controller" (a player who affects the dynamics of the game) and a "stopper" (a player who decides the duration of the game) cooperate to minimize an expected cost. This project embarks on an effort to understand non-cooperative versions of such games, both in zero-sum and in non-zero-sum contexts. A rich theory seems to emerge, and we intend fully to pursue its development as described in the proposal. Furthermore, research will focus on such stochastic optimization problems in the presence of unobservable parameters, modeled in a Bayesian framework by means of random variables with known prior distributions and continuous updating. Such problems are notoriously hard to solve explicitly, but we have ambitious plans in this direction and some preliminary results.
Stochastic portfolio theory is a relatively novel mathematical framework for analyzing portfolio behavior and equity market structure; it is descriptive as opposed to normative, is consistent with observable characteristics of actual portfolios and real markets, and provides a theoretical tool (with insights into questions of arbitrage, construction of portfolios with controlled behavior, etc.) which is also useful for practical applications. Optimization problems that involve features of both stochastic control and optimal stopping arise, for instance, in the study of target-tracking models, where one has to stay as close as possible to a certain target by spending fuel, to declare when one has arrived "sufficiently close", and then to decide whether to engage the target or not. Problems of combined optimal stochastic control/stopping also arise in Mathematical Finance: in the context of computing the upper- and lower-hedging prices of American contingent claims under portfolio constraints; in portfolio/consumption problems with an embedded "retirement" option; in the study of dynamic measures for managing risk; in the context of dynamically consistent utilities; and in stochastic games of the principal/agent type. Stochastic control with partial observations has implications for the adaptive sequential detection of change-points, for signal processing, for finance, and for other fields of application where learning about unknown parameters and dynamic system optimization have to take place simultaneously, and in real time.
