The estimation of order and parameters in a process of stochastic differential equations with uncertain observations

In this paper we consider a continuous-time stochastic process which is described by a differential equation with unknown order and unknown coefficients. The input to the process is unobservable and assumed to be white; the output can be measured but is corrupted by noise. The problem is to estimate the order and coefficients of the differential equation solely from the output data. The approach taken here is to form sample correlations from output data and then derive the spectral density function from these correlations. The order can be determined from observing the degree of dependency among sample correlations and the coefficients are calculated from the spectral density function. The concept of optimally spacing the sample correlations for the purpose of order and coefficient estimation is introduced and discussed in detail.