A novel continuous-time stochastic differential equation (SDE) for spacecraft attitude quaternion kinematics with state-multiplicative noise and a novel continuous-time exact optimal quaternion filter are developed in the framework of Itô (mean-square) calculus. The quaternion Itô SDE contains dissipative terms that ensures the mean-square stability of the process. A closed-form deterministic propagation equation is developed for the second-order moment of the quaternion. Its dynamical properties are analyzed and a closed-form solution is found for the constant angular velocity case. The quaternion filter produces the linear minimum-variance unbiased estimate of the quaternion from continuous observations with additive white noise. The filter gain computations include coupled Riccati equations of the estimation error matrix and of the quaternion second-order moment. These computations are not estimate-dependent and can therefore be performed off-line. The special case of gyro error white noise with independent identically distributed components is considered. The case of correlated components can be addressed straightforwardly. Additive gyro biases are easily handled via state augmentation. Extensive Monte-Carlo simulations are performed in order to validate the Itô model and to illustrate the proposed filter accuracy. Comparative simulations of the novel filter and of a standard additive extended Kalman filter are performed both for attitude-only estimation and for attitude-bias estimation. For low process noise levels, the novel modeling and filtering approach shows results that are similar to the standard approach. For high process noise levels however, the numerical study suggests that the novel filter can increase the accuracy of a conventional Kalman filter by orders of magnitudes.