The Quantum Structure of Markets: Linking Hamiltonian-Jacobi-Bellman Dynamics to Schrodinger Equation through Feynman Action

TL;DR

This paper introduces a path-integral control framework to analyze optimal firm behavior in complex markets, linking quantum mechanics concepts with economic dynamics to handle nonlinear stochastic systems more effectively.

Contribution

It develops a novel Euclidean path-integral control approach that avoids explicit value function construction, providing a new method for solving complex stochastic market models.

Findings

Derives a noncooperative feedback Nash equilibrium using path-integral methods.

Shows the approach's applicability through illustrative examples.

Highlights differences between path-integral solutions and traditional HJB-based methods.

Abstract

We develop a Euclidean path-integral control to characterize optimal firm behavior in an economy governed by Walrasian equilibrium, Pareto efficiency, and non-cooperative Markovian feedback Nash equilibrium. The approach recasts the problem as a Lagrangian stochastic control system with forward-looking dynamics, thereby avoiding the explicit construction of a value function. Instead, optimal policies are obtained from a continuously differentiable Ito process generated through integrating factors, which yields a tractable alternative to conventional solution methods for complex market environments. This construction is useful in settings with nonlinear stochastic differential equations where standard Hamilton-Jacobi-Bellman (HJB) formulations are difficult to implement. Consistent with Feynman-Kac-type representations, the resulting solutions need not be unique. In economies with a large number of firms, the analysis admits a natural comparison with mean-field game formulations. Our main contribution is to derive a noncooperative feedback Nash equilibrium within this path-integral setting and to contrast it with outcomes implied by mean-field interactions. Several examples illustrate the method's applicability and highlight differences relative to solutions based on the Pontryagin maximum principle generated by HJB.