This paper deals with channel estimation over flat fading Rayleigh channel with Jakes' Doppler Spectrum. Many estimation algorithms exploit the time-domain correlation of the channel by employing a Kalman filter based on a first-order (or sometimes second-order) approximation of the time-varying channel with a criterion based on correlation matching (CM), or on the Minimization of Asymptotic Variance (MAV). In this paper, we first consider a reduced complexity approach based on Least Mean Square (LMS) algorithm, for which we provide closed-form expressions of the optimal step-size coefficient versus the channel state statistic (additive noise power and Doppler frequency) and of corresponding asymptotic mean-squared-error (MSE). However, the optimal tuning of the step-size coefficient requires knowledge of the channel's statistic. This knowledge was also a requirement for the aforementioned Kalman-based methods. As a second contribution, we propose a self-adaptive estimation method based on a stochastic gradient which does not need a priori knowledge. We show that the asymptotic MSE of the self-adaptive algorithm is almost the same as the first order Kalman filter optimized with the MAV criterion and is better than the latter optimized with the conventional CM criterion. We finally improve the speed and reactivity of the algorithm by computing an adaptive speed process leading to a fast algorithm with very good asymptotic performance.