In the paper the basics of passification based adaptive control theory for stochastic continuous-time systems are introduced. Conditions for the mean square dissipativity of adaptive stabilization systems for a linear plant under coordinate-parametric white noise disturbances are obtained. A linear adaptive regulator with adaptation algorithm designed by the passification approach is proposed. The number of plant inputs may differ from that of outputs. The proof is based on the construction of a quadratic stochastic Lyapunov function. (In the case of purely parametric perturbations, the obtained conditions are known to be necessary and sufficient for the existence of a Lyapunov function with these properties.) Ultimate mean square boundedness (Levinson dissipativity) conditions for the designed closed loop system are obtained; it is shown that, in some special cases, the dissipativity of the closed loop system is preserved under white noise perturbations of arbitrary intensity. Stochastic G-passivity for nonsquare systems is introduced and necessary and sufficient conditions for strict stochastic G-passifiability are formulated.