We introduce the notion of a random partition of the stochastic interval [τ 0, τ ∞] as an analogy to the classical case and characterize the predictable processes associated with such partitions. Also we identify the operator algebras connected with the stochastic integrals of predictable processes and examine their mutual relations.
We look at analogues of the σ-algebras of events occurring up to a time and the events which are strictly prior to a time of the classical (commutative) theory. In the second case, we define the p-times and investigate the order structure of time projections associated with these times in an abstract set up.
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