Stochastic Controls, Portfolios, and Competing Particle Systems

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In recent years good progress has been made in identifying instances of simple, descriptive conditions on observable characteristics of large equity markets, under which it is possible to outperform the market using simple portfolios based solely on observables. Nevertheless, a satisfactory general theory has not emerged yet: Is there a "canonical" way to understand the existing examples? Are they special cases of a much more general construction? Under what conditions is such arbitrage possible over arbitrary time horizons, and what is the "best possible" such arbitrage under specific market structures? The investigator and his collaborators study these issues and their connections to stochastic analysis and partial differential equations. They also study stable competing particle systems, that is, multidimensional diffusions interacting through their ranks in a manner giving rise to invariant measures that are in broad agreement with stability properties observed in large equity markets over decades. The stochastic analysis of such systems presents interesting challenges, such as the solvability of the stochastic equations that implement such diffusions; the study of multiple collisions, of their associated collision local times, of their invariant distributions, of time reversal; and the estimation of parameters in the resulting models for practical implementation. Furthermore, they work on stochastic optimization problems of the control and stopping type in the presence of unobservable parameters, modeled in a Bayesian framework by means of random variables with known prior distributions and continuous updating. This work 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. The investigator studies problems in stochastic controls, portfolios, and competing particle systems. These include: + 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; + The related study of the distribution of the time-to-explosion for diffusion processes; + The study of stochastic differential equations with "singular" (generalized) drift, determined by local time; + Problems of stochastic control or stopping under partial observations ("adaptive control"); and + The study of (rank-based) systems of competing Brownian particles, whose dynamics at any given time depend on the empirical measure of their configuration.