Quantitative Convergence of Markov Equilibria for Time-Inconsistent Controlled Diffusions

Abstract

Time-inconsistent stochastic control admits a natural game-theoretic interpretation in which successive decision dates are represented by different selves. In the Markovian framework of Björk-Khapko-Murgoci, the corresponding continuous-time equilibrium HJB system can be motivated by discrete subgame-perfect games, but the general discrete-to-continuous passage is not automatic. We establish a quantitative convergence theorem for a smooth uniformly elliptic class in which the control may enter both drift and diffusion. Using the exact sample-and-hold transition law on each time step, we construct, near a verified continuous equilibrium branch, a unique regular discrete Markov subgame-perfect equilibrium and prove first-order convergence of the continuation objects in an anisotropic $C^{2+β}$ topology and of the equilibrium policies in $C^0$. The key stability mechanism is structural: the discrete first-order condition forces the differential feedback correction in the frozen linearization to be nilpotent, which yields mesh-uniform block estimates through a two-norm dyadic Schauder argument despite the second-order coupling caused by controlled volatility. We also formulate a conditional transfer principle for structured polynomial-growth classes, with error-space stability stated as a separate hypothesis.