Strategic Dynamics of Firms via Path Integral Control

In this paper, we introduce a Euclidean path integral control approach to determine optimal strategies for firms operating under a Walrasian system, Pareto optimality, and a non-cooperative feedback Nash equilibrium. Our framework formulates a Lagrangian control problem with forward-looking stochastic dynamics, eliminating the need for a value function to derive optimal strategies. The method relies on a continuously differentiable Itô process generated by integrating factors, providing a computationally feasible alternative to solving complex market dynamics. Our approach facilitates the analysis of generalized nonlinear market dynamics, where constructing a Hamilton–Jacobi–Bellman (HJB) equation is particularly challenging. Similar to the Feynman–Kac approach, our solutions are not unique. Given the large number of firms considered, our method draws comparisons with mean-field game approach. The primary contribution of this work is the derivation of a non-cooperative feedback Nash equilibrium, offering a comparative perspective against solutions generated by mean-field interactions. We illustrate the effectiveness of our approach through various examples, contrasting it with the Pontryagin maximum principle.