Optimization of a dynamic profit function using Euclidean path integral

Paramahansa Pramanik () and Alan M. Polansky ()
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Paramahansa Pramanik: University of South Alabama
Alan M. Polansky: Northern Illinois University

SN Business & Economics, 2024, vol. 4, issue 1, 1-20

Abstract: Abstract 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.

Keywords: Dynamic profit; Euclidean path integral; Walrasian system; Pareto optimality; Non-cooperative feedback Nash equilibrium; Stochastic differential games; Primary 93E20; Secondary 49N90 (search for similar items in EconPapers)
Date: 2024
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s43546-023-00602-5

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