Convergence of Generalized Supermartingales with Continous Parameter

We consider stochastic processes fX(t; ) : t 2 Tg with continuous parameter t 2 T = [0;1[. These processes are so called generalized supermartingales, where the generalization comes from a modification of the right side of the supermartingale inequality. The assumptions on the process and the probability space are the same as for the classical convergence theorem of DOOB (see KOPP [2]). The aim of the paper are statements on generalized supermartingales specially the behaviour of such processes with regard to P-almost sure convergence. Earlier papers have enhanced the interest in the use of a martingale-type approach to the study of the convergence properties, in a stochastic enviroment, of recursive identification and control schemes (see LJUNG [4], GOODWIN et. al. [1], SOLO [10], SIN and GOODWIN [9], LANDAU [3]). Further the application to financial market models (see RÖDLER [6,7,8]).