🤖 AI Summary
This work addresses the challenge of characterizing optimal firm behavior in nonlinear stochastic market environments, where traditional Hamilton–Jacobi–Bellman (HJB) methods are often intractable. The authors introduce a Euclidean path integral control framework that reformulates the equilibrium problem as a forward-looking Lagrangian stochastic control system. By leveraging Itô processes and integrating factors, the approach directly generates optimal strategies without explicitly constructing a value function. For the first time within this framework, a non-cooperative feedback Nash equilibrium is derived and contrasted with mean-field game solutions, revealing fundamental differences from the Pontryagin maximum principle. Combining the Feynman–Kac representation with mean-field approximations, the method yields computationally tractable equilibria for large-scale stochastic markets, with numerical examples demonstrating both its efficacy and its marked divergence from classical HJB solutions.
📝 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.