Path Regularity and Regularity in Mean of 1st Order Processes

The study of path properties of stochastic processes is a very active area of probabilistic research, where it is of considerable interest to find necessary conditions and sufficient conditions for the regularity of the trajectories of a stochastic process. (1-4,5-10) In Ref. 4, we proved that every random function holomorphic in mean on an open subset D of the complex field is equivalent to a random function whose paths are holomorphic on D. It is of considerable interest in the stochastic analysis to know whether tj> has a version with regular paths whenever is a 1st order process which is regular in mean on a region TcR . We prove that if the process $ is n times differentiable in mean on an interval T and its n th derivative in mean is continuous in mean on T, then d has a version whose paths are in C n 1 ( T ) . Thus, if in addition d is separable, then with probability one the paths of d lie in C n 1 ( T ) .