Optimization of a dynamic profit function using Euclidean path integral

We introduce a novel Euclidean path integral control approach to derive optimal strategies for firms operating within a Walrasian economic framework, aiming at achieving Pareto optimality, while also considering non-cooperative feedback Nash Equilibrium. Our methodology is rooted in the formulation of a stochastic control problem that incorporates forward-looking stochastic dynamics. One distinctive feature of our approach is its independence from the requirement of a value function for determining optimal strategies. Instead, we employ a computation method based on a continuously differentiable Itô process. An additional advantage of our method lies in its effectiveness in tackling generalized non-linear market dynamics, where the construction of a Hamiltonian–Jacobi–Bellman equation can be very challenging. Given our focus on a significant number of firms, our approach can be readily compared to mean-field games based on forward Fokker–Plank and backward Hamiltonian–Jacobi–Bellman equations. The central contribution of our research is the establishment of a non-cooperative feedback Nash equilibrium, which can be systematically compared with solutions resulting from mean-field interactions. We provide numerous illustrative examples to demonstrate the efficacy of our approach relative to the Pontryagin maximum principle.