The grey wolf optimization(GWO) algorithm is a stochastic optimization algorithm based on swarm intelligence which has been successfully used to solve many complex optimization problems. At present, there are many improved forms of the GWO algorithm, but the lack of rigorous convergence analysis leads to no theoretical support for the improved algorithm. In order to make up for this deficiency, the martingale theory is used to analyze its convergence for the first time. Firstly, the basic mathematical model is established according to the principle of the GWO algorithm. By defining the gray wolf state space and the gray wolf group state space, the Markov chain model of the GWO algorithm is established, and the Markov properties of the algorithm are analyzed. Secondly, the martingale theory is introduced, and a swarm evolution sequence with the supermartingale as the optimal fitness value is derived. Thirdly, the convergence of the GWO algorithm is analyzed using the supermartingale convergence theorem and its Markov properties. It is proved that the GWO algorithm can achieve global convergence with the possibility of 1. Finally, the convergence performance is verified by numerical experiments. The experimental results show that the GWO algorithm has strong global convergence, low computation time and high optimization accuracy.