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 C2+β 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.
Related
- Global Sobolev Convergence of Howard Iteration for Finite-Horizon Controlled Diffusions
- Equilibria for Time-inconsistent Regular-singular Control Problems
- Local W2,1p convergence of penalization for time-inconsistent stopping problems
- Long-term behaviour of the CAO-system: Optimal polynomial H1-convergence to degenerate equilibria
- Exponential rate of convergence of relative value iteration algorithms for ergodic controls of diffusions
BibTeX
@misc{desmettre2026quantitative,
title = {{Quantitative Convergence of Markov Equilibria for Time-Inconsistent Controlled Diffusions}},
author = {Sascha Desmettre},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18376},
eprint = {2610.18376},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18376},
}
RIS
TY - UNPB
TI - Quantitative Convergence of Markov Equilibria for Time-Inconsistent Controlled Diffusions
AU - Sascha Desmettre
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.18376
UR - https://arxiv.org/abs/2610.18376
AB - 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+β 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.
ER -