[Paper Review] Optimization of a Dynamic Profit Function using Euclidean Path Integral

Paramahansa Pramanik, Alan M. Polansky|arXiv (Cornell University)|Feb 21, 2020

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper introduces a novel application of Euclidean path integral methods to optimize dynamic profit functions under Walrasian, Pareto, and non-cooperative Nash equilibrium conditions. By leveraging Feynman-type path integration and Wick rotation, the approach solves nonlinear stochastic differential games—such as the Merton-Garman-Hamiltonian system—where traditional Pontryagin methods fail, enabling closed-form solutions for non-additive convex strategies under linear constraints and stochastic dynamics.

ABSTRACT

A Euclidean path integral is used to find an optimal strategy for a firm under a Walrasian system, Pareto optimality and a non-cooperative feedback Nash Equilibrium. We define dynamic optimal strategies and develop a Feynman type path integration method to capture all non-additive convex strategies. We also show that the method can solve the non-linear case, for example Merton-Garman-Hamiltonian system, which the traditional Pontryagin maximum principle cannot solve in closed form. Furthermore, under Walrasian system we are able to solve for the optimal strategy under a linear constraint with a linear objective function with respect to strategy.

Motivation & Objective

• To develop a path integral-based method for optimizing dynamic profit functions in stochastic environments where traditional optimal control techniques fail.
• To extend the applicability of optimal control to nonlinear systems, such as the Merton-Garman-Hamiltonian system, by using Euclidean path integrals.
• To derive optimal strategies under three economic equilibrium frameworks: Walrasian, Pareto, and non-cooperative feedback Nash equilibrium.
• To establish convergence and existence conditions for the path integral formulation in stochastic differential game settings with bounded, complete functional spaces.
• To solve profit maximization problems under linear constraints and linear objectives using a non-perturbative path integral approach.

Proposed method

• Utilizes a Feynman-type path integral formulation to represent the expectation of the cumulative profit integral over time, integrating over all possible market share and strategy trajectories.
• Applies Wick rotation to transform the stochastic process into a Euclidean action functional, enabling the use of quantum field theory-inspired path integral techniques.
• Defines the Euclidean action functional $\mathcal{A}_{0,t}(x)$ as the core of the path integral, derived from the profit and dynamics functions, ensuring convergence under smoothness and bounded derivative conditions.
• Employs a regularization scheme using $\omega_\varepsilon(x)$ functions to ensure absolute convergence of the path integral, with the limit taken as $\varepsilon \to 0$.
• Introduces local transition functions $\Psi_{s,s+\Delta s}(x)$ over small time intervals to recursively build the global path integral solution, ensuring consistency with stochastic dynamics.
• Imposes smoothness and bounded derivative conditions (Assumptions 1 and 2) on the profit and action functionals to guarantee existence and convergence of the path integral representation.

Experimental results

Research questions

• RQ1 Can a path integral formulation based on Euclidean action functionals solve nonlinear stochastic differential games in dynamic profit maximization where the Pontryagin maximum principle fails?
• RQ2 How can the Feynman path integral method be adapted to capture non-additive convex strategies in a stochastic dynamic optimization setting?
• RQ3 Under what conditions does the Euclidean path integral converge for a dynamic profit function under Walrasian, Pareto, and Nash equilibrium constraints?
• RQ4 Can the method derive closed-form optimal strategies for the Merton-Garman-Hamiltonian system, a nonlinear stochastic control problem?
• RQ5 How do linear constraints on the strategy and linear objective functions affect the solvability and structure of the path integral formulation?

Key findings

• The path integral formulation successfully solves the Merton-Garman-Hamiltonian system in closed form, a problem that cannot be solved using the classical Pontryagin maximum principle.
• The method provides a consistent framework for deriving optimal strategies under Walrasian equilibrium, where each firm faces identical market dynamics and aims to maximize expected profit over time.
• For Pareto optimality, the path integral approach maximizes a weighted sum of profits across firms, with the optimal strategy derived from a single global action functional.
• In the non-cooperative feedback Nash equilibrium, the method yields a consistent solution where each firm's optimal strategy is derived under the assumption that others' strategies are fixed.
• The convergence of the path integral is rigorously established under Assumptions 1 and 2, which require smoothness and bounded derivatives of the profit and action functionals.
• The Euclidean action functional $\mathcal{A}_{0,t}(x)$, after Wick rotation, ensures that the conditional expectation term remains non-negative when $G[s,x(s),u(s)] \geq 0$, preserving physical and economic interpretability.