This paper aims to design an asynchronous state feedback controller for Markov jump time-delay systems. The highlight of this work lies in that the state feedback is quantized by a logarithmic quantizer, and both the controller and quantizer are asynchronous with the controlled systems. By Lyapunov-Krasovskii functional, a sufficient condition is presented to ensure that the resulting closed-loop system is stochastically mean square stable with a prescribed H-infinity performance index. Finally, an example is presented to illustrate the effectiveness and new features of proposed design method. (C) 2018 Elsevier Ltd. All rights reserved.