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 with strongly concave Hamiltonians and interior equilibrium controls, allowing control in both drift and diffusion. Using the exact sample-and-hold transition law on each time step, we construct, near a given verified continuous-time BKM weak equilibrium, a unique regular discrete Markov subgame-perfect equilibrium and prove first-order convergence of continuation objects in $C^{2+\beta }$ Hölder norms with mixed regularity and of policies and values uniformly. The key stability mechanism is structural: the discrete first-order condition makes the frozen differential feedback correction nilpotent. A two-norm dyadic Schauder argument, local Gaussian comparison, and spatial localization turn this cancellation into mesh-uniform block stability despite the second-order coupling caused by controlled volatility.

arXiv comment30 pages