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

Abstract: 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.