Strategic Arms Accumulation as a Stochastic Differential Game: Feedback Equilibria, Regime Shocks, and Deterrence Dynamics

We develop a dynamic differential game model of strategic arms accumulation between two rival states, reflecting the security dilemma of major geopolitical adversaries such as the United States and China or the United States and the former Soviet Union. The model features both determinis tic and stochastic formulations, where each country maximizes intertem poral utility over its own military stock while negatively internalizing the rival's arms buildup. We derive Markov-Perfect Nash equilibrium strate gies in both open-loop and feedback forms, with special emphasis on the characterization and solution of the coupled Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck-Kolmogorov (FPK) partial diferential equa tions. In the linear-quadratic case, we solve the time-consistent Riccati equations and analyze the elasticity of optimal strategies with respect to changes in strategic preferences, investment costs, and external threats. The model is extended to include regime shocks-discontinuous jumps in threat perception or strategic aggressiveness-which may destabilize de terrence and trigger explosive arms escalation. Comparative statics from Riccati dynamics reveal thresholds beyond which deterrence collapses and feedback control solutions blow up in finite time. We also explore welfare trade-offs, deterrence sustainability, and policy implications of dynamic treaties. The mathematical framework developed here provides a foun dation for modeling arms races as nonlinear stochastic systems under en- dogenous expectations, belief dynamics, and political instability.