Stochastic Portfolios, Controls, and Interacting Particles

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This award will focus on three areas of importance in financial mathematics: (1) stochastic portfolio theory, (2) stochastic control, and (3) entropic gradient flows. Stochastic portfolio theory is a mathematical framework for analyzing portfolio behavior and equity market structure. It is descriptive as opposed to normative, consistent with observable characteristics of actual portfolios equity markets, and a theoretical tool useful for practical applications. In addition, it provides insights into questions of outperformance and arbitrage of market-structure stability, and of portfolio construction that results in controlled behavior over long-time horizons for purposes of institutional (for example, foundation, endowment, pension) investing. Stochastic control with partial observations arises in the sequential detection of change-points, in signal processing, in finance, in other contexts where learning about unknown parameters, and where dynamic optimization must take place simultaneously and in real time. Optimization problems that involve features of both control and stopping arise, for instance, in 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. It also arises in situations where one has to resolve the tension between exploration (learning about unobservable quantities) and exploitation (taking an actions that costs now, but yields benefits later). The third focus of this project, the study of entropic gradient flows, complements the second law of thermodynamics and pertains to phenomena where the entropy of the current configuration, relative to the steady-state, not only decreases as the system approaches equilibrium, but does so in the most efficient manner possible – the state follows a path of steepest possible descent for the entropy, in terms of an appropriate distance on phase space. This has important implications for the training of neural networks, for gradient descent algorithms in stochastic optimization, and for optimal portfolio liquidation. The award will provide graduate students with the opportunity for research experiences. The PI will develop a theory of portfolios that does not rely on the existence of equivalent martingale measures and allows the outperformance of one portfolio by another. This theory will be based on local martingale deflators, the optional decomposition theorem, and appropriate functional-analytic tools. It will exclude only very egregious forms of what is commonly called arbitrage, and help develop a simple, streamlined mathematical framework for finance with complete answers to the basic questions of hedging and portfolio optimization. This theory will cover markets with an arbitrary number of assets (such as bond markets, or those with splits and mergers), open markets (with a fixed, finite number of companies but of a composition varying with capitalization, such as S&P 500), portfolio constraints, model uncertainty, and market stability via models based on systems of diffusions interacting through their ranks. The PI will work on stochastic optimization problems that combine features of optimal control, stopping, and filtering of unobservable parameters or signals. Such problems are notoriously hard to solve explicitly and little theory exists for them. Finally, the project will explore the full range of applicability for a tool the PI developed recently, using time-reversal and optimal transport techniques: the trajectorial approach to the Otto calculus. The PI will study its relevance for diffusions of McKean-Vlasov type, the training of deep neural networks, and the steepest descent of the relative entropy along flows of Markov processes, including on discrete structures with the help of appropriately weighted Sobolev norms.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.