- #1

Stephen Tashi

Science Advisor

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## Main Question or Discussion Point

The explanation of a continuous Markov process [itex] X(t) [/itex] defines an indexed collection of sigma algebras by [itex] \mathcal{F}_t = \sigma\{ X(s): s < t\} [/itex] and this collection is said to be

I'm trying to understand why the notation used for set inclusion is used to express the relation of "increasing" for a collection of sigma algebras.

A straightforward approach is to think of a set of sigma algebras that are each a collection of subsets of the same set and to define the concept of sub-sigma algebra in terms of one collection of sets being a subset of another collection of sets.

However, don't [itex] \mathcal{F}_t [/itex] and [itex] \mathcal{F}_s [/itex] denote sigma algebras defined on different sets when [itex] s \ne t [/itex] ? I think of [itex] \mathcal{F}_t [/itex] as being a sigma algebra of subsets of (only) the set of all trajectories of the process up to time [itex] t [/itex]. [itex] \ [/itex] If [itex] s > t [/itex] then isn't [itex] \mathcal{F}_s [/itex] a sigma algebra of subsets of a different set of trajectories?

*increasing*with respect to the index [itex] t [/itex].I'm trying to understand why the notation used for set inclusion is used to express the relation of "increasing" for a collection of sigma algebras.

A straightforward approach is to think of a set of sigma algebras that are each a collection of subsets of the same set and to define the concept of sub-sigma algebra in terms of one collection of sets being a subset of another collection of sets.

However, don't [itex] \mathcal{F}_t [/itex] and [itex] \mathcal{F}_s [/itex] denote sigma algebras defined on different sets when [itex] s \ne t [/itex] ? I think of [itex] \mathcal{F}_t [/itex] as being a sigma algebra of subsets of (only) the set of all trajectories of the process up to time [itex] t [/itex]. [itex] \ [/itex] If [itex] s > t [/itex] then isn't [itex] \mathcal{F}_s [/itex] a sigma algebra of subsets of a different set of trajectories?