Parameter estimation in stochastic systems: some recent results and applications

Consider a stochastic process {X,},,, where JI is either Iw + = [0, cc) or N (the set of natural numbers). For some compact metric space D, let PO, 8 ED, be a family of probability measures on the trajectory (‘canonical’) space of ( X,},EJ, equipped with the appropriate topology and the corresponding Bore1 u-field. Assume that the ‘true’ probability measure is PO,, for some 19, ED. Pe7-, T E ,II, will denote the restriction of PO to the u-field generated by {X,, tG T). For each T E J, we assume that Per, 8 ED, is a family of mutually absolutely continuous probability measures. Then for t E J, B E D, the Radon-Nikodym derivative A,(e) = d P,,/d PO,,, is a martingale with respect to the progressive u-fields f, generated by [X,, s G t}. By Jensen’s inequality, it follows that L,( 0) = In A,( 0) is an f,-supermartingale. Under suitable conditions, we can apply the Doob-Meyer decomposition theorem to obtain the unique decomposition