Non-anticipative Integral Transformations of Stochastic Processes

Ljiljana Petrović

Let $X$ be a stochastic process, defined on the interval $[0;1]$, and $Y$ its non-anticipative integral transformation defined by $$ Y(t)=\int\limits^t_0 g(t, u) X(u)du $$ In this paper we shall investigate conditions related to the family $$ G=\{g(t,u), t\in[0;1],u \leq t\} $$ under which the process $Y\!:\!1^\circ$ generates the spaces $H(Y;t)$ equal to the corresponding spaces $H(X;t)$ of the process $\bold X; 2\circ$ belongs to the same class as the process $X; 3^\circ$ is continuous, provided $X$ is continuous.