In this paper, we develop a state space representation and Kalman filtering method for self-similar processes. Key components of our development are the concept of multivariate self-similarity and the mathematical framework of scale stationarity. We define multivariate self-similarity as joint self-similarity, in which the self-similarity is governed by a matrix valued parameter H. Such a generalization suits the nature of Multi-Input Multi-Output (MIMO) systems, since each channel is likely to be governed by a different self-similarity parameter. The system and measurement models for the proposed Kalman filter are defined as and y(t)=Cx(t)+Dv(t), respectively. Here, the derivative operator indicates that the memory of the process is stored in time scales, unlike the memory stored in time shifts for stationary processes. We exploit this fact in developing an insightful interpretation of the Riccati equation and the Kalman gain matrix, which lead to an efficient numerical implementation of the proposed Kalman filter via exponential sampling. Additionally, we include a discussion of network traffic modeling and communications applications of the proposed Kalman filter. This study demonstrates that the scale stationarity framework leads to mathematically tractable and physically intuitive formulation of Kalman filtering for self-similar processes.