Abstract: We investigate an inverse problem arising in a stochastic production–consumption market with interacting heterogeneous agents, building upon the model of Aid et al. (2022). The model features two large populations—producers and consumers—whose individual production and consumption rates evolve according to controlled Itô dynamics with quadratic costs. A market signal, interpreted as a price index, aggregates total production and consumption and feeds back into agents’ instantaneous profits, thereby coupling micro-level decisions with macro-level variables. Each agent optimizes expected cumulative profit over a finite horizon, leading to Hamilton–Jacobi–Bellman equations exhibiting quadratic growth. The objective of this work is to reconstruct optimal feedback strategies and value functions from observed market signals and empirical agent dynamics. We derive explicit formulas in the mean-field limit, where the system converges to coupled McKean–Vlasov dynamics and nonlinear Kolmogorov equations. These formulas provide a tractable characterization of optimal controls and enable efficient calibration of model parameters. We will propose numerical experiments to validate the theoretical results and illustrate how the reconstructed strategies reproduce key features of the observed market evolution. Our framework combines stochastic control, inverse problems, and mean-field analysis to offer a systematic approach for calibrating large-scale economic systems driven by production–consumption interactions.
Reference: R. Aïd, O. Bonesi, G. Callegaro, L. Campi. A McKean-Vlasov game of commodity production, consumption and trading. Applied Mathematics and Optimization, 86(40), 2022