arXiv subjects

Aaron Zeff Palmer

Publications and source records attributed to Aaron Zeff Palmer.

11 recordsLinked to original sources

arXiv2023

The lifted functional approach to mean field games with common noise

We introduce a new path-by-path approach to mean field games with common noise that recovers duality at the pathwise level. We verify this perspective by explicitly solving some difficult examples with linear-quadratic data, including control in the volatility coefficient of the common noise as well as the constraint of partial information. As an application, we establish the celebrated separation principle in the latter context. In pursuing this program, we believe we have made a crucial contribution to clarifying the notion of regular solution in the path dependent PDE literature.

math.OC↗

arXiv2023

The role of information in high dimensional stochastic optimal control

The stochastic optimal control of many agents is an important problem in various fields. We investigate the problem of partial observations, where the state of each agent is not fully observed and the control must be decided based on noisy observations. This results in a high-dimensional Markov decision process that is impractical to handle directly. However, in the limit as the number of agents approaches infinity, a finite-dimensional mean-field optimal control problem emerges, which coincides with the problem of full information. Our main contribution is to investigate a central limit theorem for the Gaussian fluctuations of the mean-field optimal control. Our findings show that partial observations play an essential role in the fluctuations, in contrast to the mean-field limit. We establish a method that uses an approximate Kalman filter, which is straightforward to compute even when the number of states is large. This provides some theoretical evidence of the efficacy of Kalman filter methods that are commonly used across a range of practical applications. We demonstrate our results with two examples: an epidemic model with observations of positive tests and a simple two-state model that exhibits a phase transition at which point the fluctuations diverge.

math.OC↗

arXiv2023

The sharp interface limit of an Ising game

The Ising model of statistical physics has served as a keystone example of phase transitions, thermodynamic limits, scaling laws, and many other phenomena and mathematical methods. We introduce and explore an Ising game, a variant of the Ising model that features competing agents influencing the behavior of the spins. With long-range interactions, we consider a mean-field limit resulting in a nonlocal potential game at the mesoscopic scale. This game exhibits a phase transition and multiple constant Nash-equilibria in the supercritical regime. Our analysis focuses on a sharp interface limit for which potential minimizing solutions to the Ising game concentrate on two of the constant Nash-equilibria. We show that the mesoscopic problem can be recast as a mixed local/nonlocal space-time Allen-Cahn type minimization problem. We prove, using a $Γ$-convergence argument, that the limiting interface minimizes a space-time anisotropic perimeter type energy functional. This macroscopic scale problem could also be viewed as a problem of optimal control of interface motion. Sharp interface limits of Allen-Cahn type functionals have been well studied. We build on that literature with new techniques to handle a mixture of local derivative terms and nonlocal interactions. The boundary conditions imposed by the game theoretic considerations also appear as novel terms and require special treatment.

math.AP↗

arXiv2020

Optimal Stopping of Stochastic Transport Minimizing Submartingale Costs

Given a stochastic state process $(X_t)_t$ and a real-valued submartingale cost process $(S_t)_t$, we characterize optimal stopping times $τ$ that minimize the expectation of $S_τ$ while realizing given initial and target distributions $μ$ and $ν$, i.e., $X_0\sim μ$ and $X_τ\sim ν$. A dual optimization problem is considered and shown to be attained under suitable conditions. The optimal solution of the dual problem then provides a contact set, which characterizes the location where optimal stopping can occur. The optimal stopping time is uniquely determined as the first hitting time of this contact set provided we assume a natural structural assumption on the pair $(X_t, S_t)_t$, which generalizes the twist condition on the cost in optimal transport theory. This paper extends the Brownian motion settings studied in [15, 16] and deals with more general costs.

math.PR↗

arXiv2020

Hidden convexity in a problem of nonlinear elasticity

We study compressible and incompressible nonlinear elasticity variational problems in a general context. Our main result gives a sufficient condition for an equilibrium to be a global energy minimizer, in terms of convexity properties of the pressure in the deformed configuration. We also provide a convex relaxation of the problem together with its dual formulation, based on measure-valued mappings, which coincides with the original problem under our condition.

math.AP↗

arXiv2020

A solution to the Monge transport problem for Brownian martingales

We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable, as well as a stochastic version of the standard "twist condition" frequently used in deterministic Monge transport theory. This setting includes in particular the case of the distance cost $c(x,y)=|x-y|$. We prove existence and uniqueness of the solution and characterize it as the first time Brownian motion hits a barrier that is determined by solutions to a corresponding dual problem.

math.AP↗

arXiv2019

The Dyson and Coulomb games

We introduce and investigate certain $N$ player dynamic games on the line and in the plane that admit Coulomb gas dynamics as a Nash equilibrium. Most significantly, we find that the universal local limit of the equilibrium is sensitive to the chosen model of player information in one dimension but not in two dimensions. We also find that players can achieve game theoretic symmetry through selfish behavior despite non-exchangeability of states, which allows us to establish strong localized convergence of the N-Nash systems to the expected mean field equations against locally optimal player ensembles, i.e., those exhibiting the same local limit as the Nash-optimal ensemble. In one dimension, this convergence notably features a nonlocal-to-local transition in the population dependence of the $N$-Nash system.

math.AP↗

arXiv2019

Stochastic optimal transport with free end time

We consider a stochastic transportation problem between two prescribed probability distributions (a source and a target) over processes with general drift dependence and with free end times. First, and in order to establish a dual principle, we associate two equivalent formulations of the primal problem in order to guarantee its convexity and lower semi-continuity with respect to the source and target distributions. We exhibit an equivalent Eulerian formulation, whose dual variational principle is given by Hamilton-Jacobi-Bellman type variational inequalities. In the case where the dependence on the drift is bounded, regularity results on the minimizers of the Eulerian problem then enable us to prove attainment in the corresponding dual problem. We also address attainment when the drift component of the cost defining Lagrangian $L$ is superlinear $L \approx |u|^p$ with $1<p<2$, in which case the setting is reminiscent of our approach -- in a previous work -- on deterministic controlled transport problems with free end time. We finally address criteria under which the optimal drift and stopping time are unique, namely strict convexity in the drift component and monotonicity in time of the Lagrangian.

math.OC↗

arXiv2019

PDE Methods For Optimal Skorokhod Embeddings

We consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. PDEs and a free boundary problem approach are used to address the problem in general dimensions with space-time inhomogeneous costs given by Lagrangian integrals along the paths. We introduce an Eulerian---mass flow---formulation of the problem, whose dual is given by Hamilton-Jacobi-Bellman type variational inequalities. Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space-time. This complements and provides a constructive PDE alternative to recent results of Beiglböck, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.

math.AP↗

arXiv2019

Optimal Controlled Transports with Free End Times Subject to Import/Export Tariffs

We analyze controlled mass transportation plans with free end-time that minimize the transport cost induced by the generating function of a Lagrangian within a bounded domain, in addition to costs incurred as export and import tariffs at entry and exit points on the boundary. We exhibit a dual variational principle à la Kantorovich, that takes into consideration the additional tariffs. We then show that the primal optimal transport problem has an equivalent Eulerian formulation whose dual involves the resolution of a Hamilton-Jacobi-Bellman quasi-variational inequality with non-homogeneous boundary conditions. This allows us to prove existence and to describe the solutions for both the primal optimization problem and its Eulerian counterpart.

math.AP↗

arXiv2018

Optimal Transport with Controlled Dynamics and Free End Times

We consider optimal transport problems where the cost is optimized over controlled dynamics and the end time is free. Unlike the classical setting, the search for optimal transport plans also requires the identification of optimal "stopping plans," and the corresponding Monge-Kantorovich duality involves the resolution of a Hamilton-Jacobi-Bellman quasi-variational inequality. We discuss both Lagrangian and Eulerian formulations of the problem, and its natural connection to Pontryagin's maximum principle. We also exhibit a purely dynamic situation, where the optimal stopping plan is a hitting time of a barrier given by the free boundary problem associated to the dual variational inequality. This problem was motivated by its stochastic counterpart, which will be studied in a companion paper.

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