We study state space equations within the white noise space setting. A commutative ring of power series in a countable number of variables plays an important role. Transfer functions are rational functions with coefficients in this commutative ring, and are characterized in a number of ways. A major feature in our approach is the observation that key characteristics of a linear, time invariant, stochastic system are determined by the corresponding characteristics associated with the deterministic part of the system, namely its average behavior.
D. Alpay, D. Levanony and A. Pinhas. Linear stochastic state space theory in the white noise space setting. SIAM Journal on Control and Optimization (SICON), Volume 48, Issue 8, pp. 5009-5027 (2010).
Society for Industrial and Applied Mathematics