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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 .

For , most straightforward metric can be induced by in , where is the sup norm

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 –