A Games-in-Games Paradigm for Strategic Hybrid Jump-Diffusions: Hamilton-Jacobi-Isaacs Hierarchy and Spectral Structure

This paper develops a hierarchical games-in-games control architecture for hybrid stochastic systems governed by regime-switching jump-diffusions. We model the interplay between continuous state dynamics and discrete mode transitions as a bilevel differential game: an inner layer solves a robust stochastic control problem within each regime, while a strategic outer layer modulates the transition intensities of the underlying Markov chain. A Dynkin-based analysis yields a system of coupled Hamilton-Jacobi-Isaacs (HJI) equations. We prove that for the class of Linear-Quadratic games and Exponential-Affine games, this hierarchy admits tractable semi-closed form solutions via coupled matrix differential equations. We prove that for the class of Linear-Quadratic games and Exponential-Affine games, this hierarchy admits tractable semi-closed form solutions via coupled matrix differential equations. The framework is demonstrated through a case study on adversarial market microstructure, showing how the outer layer’s strategic switching pre-emptively adjusts inventory spreads against latent regime risks, which leads to a hyper-alert equilibrium.

💡 Research Summary

The paper introduces a novel hierarchical “games‑in‑games” framework for controlling hybrid stochastic systems that combine continuous dynamics with discrete regime‑switching jump‑diffusions. The authors model the interaction between the two layers of decision‑making as a bilevel zero‑sum differential game. The inner layer operates on the fast time‑scale: for each regime (mode) it solves a robust stochastic control game between a controller and a disturbance, yielding a value function V_i that satisfies a Hamilton‑Jacobi‑Isaacs (HJI) partial‑integro‑differential equation (PIDE). The outer layer works on the slower time‑scale: two macro‑players choose mixed actions that directly influence the transition intensities μ_{ij} of the underlying continuous‑time Markov chain. Their objective depends on the inner‑layer value functions, leading to a second HJI system that couples the transition rates with the inner‑layer outcomes.

Key technical contributions are: (1) a unified Dynkin‑formula based decomposition that separates the continuous‑layer Isaacs equation from the strategic switching equation without circular dependencies; (2) a rigorous viscosity‑solution analysis proving existence, uniqueness, and a comparison principle for the coupled HJI system under standard Lipschitz and quadratic growth assumptions; (3) closed‑form tractability for two important classes—Linear‑Quadratic (LQ) and exponential‑affine (CARA) games—where the coupled HJI reduces to a set of matrix differential Riccati‑type equations whose coefficients include the controllable transition rates; (4) a spectral analysis that characterizes stability and optimality in terms of eigenvalues of the combined system matrix, revealing how the outer‑layer can shape the spectrum by adjusting μ_{ij}.

The authors validate the theory with a case study on adversarial market microstructure. Here a market maker (inner‑layer controller) manages inventory spreads while a high‑frequency trader (inner‑layer disturbance) attempts to exploit price movements. The outer‑layer players represent the exchange’s policy and a regulator, who can modulate the intensity of regime switches between “liquid” and “illiquid” market states. By solving the coupled Riccati system, the authors show that pre‑emptive adjustment of transition intensities leads to a “hyper‑alert” equilibrium: the market maker’s spread is dynamically tightened before a regime shift, mitigating adverse selection risk. Numerical simulations illustrate the reduction in expected loss and the emergence of a robust equilibrium that persists under model perturbations.

Overall, the paper fills a gap in the literature where hybrid systems have been treated either with exogenous switching or with a single optimal‑switching controller. By treating the switching mechanism itself as a strategic game, the work opens new avenues for resilient control, cyber‑physical security, and financial engineering where adversaries can manipulate mode transitions. Limitations include the assumption of full observability of the state for the outer‑layer policies and the restriction to LQ/CARA cost structures for tractable solutions. Future research directions suggested are partial‑information extensions, non‑quadratic cost functions, multi‑player generalizations, and data‑driven learning of optimal switching policies.