Minimax estimation in systems of observation with Markovian chains by integral criterion

A problem of estimation of states and parameters in stochastic dynamic systems of observation with discrete time containing a Markovian chain is studied. Matrices of transient probabilities and observation plans are random with unknown distribution with a given compact carrier. Observations, on the basis of which the estimation is made, are available at a fixed interval of time $[0,T]$. As a loss function, we have a conditional mathematical expectation with respect to the available observations of $\ell_2$-norm of the estimation error of a signal process on $[0,T]$. The problem is in constructing an estimate minimizing losses correspondent to the worst distribution of the pair “a matrix of transient probabilities – a matrix of observation plan” form a set of allowable distributions. For a correspondent minimax problem is demonstrated the existence of a saddle point and is obtained a form of the wanted minimax estimation. The applicability of the obtained results is illustrated by a numerical example of the estimation of a state of TCP under the conditions of uncertainty of communication channel parameters.