The Quantum Structure of Markets: Linking Hamiltonian-Jacobi-Bellman Dynamics to Schrodinger Equation through Feynman Action

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.

Figures (8)

• Figure 1: Forward-looking stochastic trajectories of the market share $X$ originating from the initial state $x_0$ at time $s=0$. The region bounded by the arcs represents the feasible set, within which trajectories may temporarily exit and re-enter over time. The red curve $\{x^o(s)\}$ denotes the optimal trajectory that reaches the terminal state $X(t)$ at time $t$, while the predominance of upward paths motivates the exponential action formulation. The Brownian motion component is treated by time-slicing $[0,t]$, applying Itô's Lemma, and deriving the associated Schrödinger-type representation used to recover the optimal strategy.
• Figure 2: Intuitive flow of the path integral control method for a firm. The nonlinear HJB equation is transformed via exponential change of variables into a linear diffusion representation. Optimal policies are then recovered through importance-weighted Monte Carlo trajectories rather than explicit state-space discretization.
• Figure 3: Simulated market-share trajectory and corresponding Walrasian feedback control under the parameter configuration of Example \ref{['ex2']}. The upper panel reports the evolution of $X(s)$ over standardized time, while the lower panel displays the optimal advertising intensity $u(s)=\phi_w^*(s,X(s))$.
• Figure 4: Monte Carlo simulation for Example \ref{['ex2']}. The left panel shows the empirical distribution of the terminal market share $X(s)$ under the Walrasian feedback strategy, while the right panel reports the distribution of exponential importance weights used in the path integral reweighting scheme.
• Figure 5: Monte Carlo simulation for Example \ref{['ex2']}. The left panel displays a subset of simulated market-share trajectories under the Walrasian feedback control, plotted over standardized time. The right panel reports the empirical distribution of terminal market shares, illustrating the dispersion generated by stochastic diffusion and endogenous advertising dynamics.