Stochastic Processes

Let X be a non empty set, and ℝ X be the set of all real valued functions defined on X. For every x ∈ X let Π{x} : ℝ X ↦ ℝ{x} = ℝ be the canonical projection onto the xth coordinate. The standard σ-algebra of subsets of ℝ X is the coarsest σ-algebra M0(X) such that all projections Π{x} are Borel measurable. It is well known that a stochastic process with parameter set X is nothing but a probability measure defined on M0(X) . If (X, ϱ) is an ordinary metric space, then the set UC(X) of all uniformly continuous, real valued functions is not necessarily a measurable subset of ℝ X — i.e. in general the set UC(X) is not an element of M0(X) . Hence the characterization of those stochastic processes having uniformly continuous trajectories is a well known non trivial problem. The aim of this chapter is to look at this problem from the view point of many valued topology. We will give a topological characterization of stochastic processes with uniformly continuous trajectories which is based on many valued continuity. In Section 8.4 we will extend this result to countable inductive limits E of separable Fréchet spaces and show that a stochastic process π with parameter set E has continuous linear trajectories iff the induced random function Θ>π associated with π is linear and B(ℝE)-valued continuous where B(ℝE) denotes the probability σ-algebra determined by the probability space (ℝ E , M0(E, π). Finally, in Subsection 8.4.1 we explain the role of Minlos’ Theorem in the range of Boolean valued topologies.