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Nowadays, control-dependent stochastic noise is ruled out in adaptive control, whereas such noise can be encountered in financial, aerospace, and biomechanical models. The status is due to the presence of adaptive control in the diffusion term. The diffusion coefficient, heavily dependent on the adaptive signal (with unexpected size), undermines stochastic stability via its quadratic form which cannot be suppressed merely by the control itself in the drift term. This compels us to exploit the positive role of the diffusion term, entailing new sophisticated analysis in the nonlinear and adaptive context. This article aims to enlarge the applicability of adaptive control, and especially, seek adaptive stabilization via dynamic gain in the context of control-dependent stochastic noise. Specifically, basic theorems on stochastic convergence are proposed, particularly revealing the intrinsic relation between the system convergence and the gain evolution by exploiting the underlying positive role of the diffusion term. Then, a distinctive martingale-based analysis pattern for adaptive control is established, recognizing the inapplicability of classical Lyapunov/LaSalle theorems. In this way, for a specific class of uncertain nonlinear systems, global stabilization with almost sure asymptotic/exponential convergence is achieved by incorporating typical dynamic gains in the presence of control-dependent stochastic noise.