Section: New Results

Separability of Set-Indexed Processes

Participant : Alexandre Richard.

A classical result states that any (multiparameter) stochastic process has a separable modification, thus ensuring the measurability property of the sample paths. We extend this result to set-indexed processes.

Let (T,𝒪) be a topological space. We assume that this space is second-countable, ie there exists a countable subset 𝒪 ˜𝒪 such that any open set of 𝒪 can be expressed as a union of elements of 𝒪 ˜.

A process {X t ,tT} is separable if there exists an at most countable set ST and a null set Λ such that for all closed sets F𝐑 and all open set O𝒪,

ω:X s (ω)F for all sOSω:X s (ω)F for all sOΛ.

This definition is different of the one found in [57] , where the space is “linear", in that this author considers the previous equation only when O is an interval. It happens that this notion needs not be defined in a general topological space. However when restricted to a vector space, our definition implies the previous one.

Theorem 0.2 (Doob's separability theorem) Any T-indexed stochastic process X={X t ;tT} has a separable modification.

If T is an indexing collection in the sense of [62] , the topology induced by the distance d T has to be second-countable. This happens for instance when (T,d T ) is totally bounded, which is the case in