Reference: [Bil99] ([Patrick Billingsley 1999])
Goal: RCLL space is the collection of all right continuous processes with left limit exists. In this below, we will give a definition of Skorohod metric on this space. More precisely, for a , we denote by the collection of RCLL processes on taking values in . We also use to denote the collection of RCLL processes on .
Obviously, we have , and this motivates the Skorhod metric below.
- For , we denote by by the class of strictly increasing continuous mappings of onto itself. In particular, and for all . The identity on also belongs to . We can define a functional in by
Note that may not be necessarily finite in .
- For , define the distance function in by
Show that for and given by (2).
– QED –
By using the above definition , we are ready for the definition of :
- We define the distance function in by
where for all with a continuous function given by
The projection mapping is continuous at if is continuous at .
Proof: It’s a consequence of Theorem 12.5 of [Bil99].
Consider . Let .
- is not upper semicontinuous at given by
since where .
- is not lower semicontinuous at given by
In fact, setting , we have
– QED –