What is common in most PhD application procedures in the UK?
Unfortunately the system in the UK is not uniform. You will really have to read each individual department's web site to know how to apply. There are, for example, Centres for Doctoral Training http://www.epsrc.ac.uk/skills/students/centres/ which you apply to like a US graduate program.  I don't know if there is a good one that covers stochastic analysis but it seems likely.

It sounds like the list you give is for people who want to apply for a specific supervisor.  For this route, your main issue will be funding as there is very little funding for non-UK students these days.  This will also vary wildly between different universities. There may be deadlines which you have missed already if you want to start in 2015.  However, simply for being accepted by a supervisor without funding the list you gave is fine.

Overall, read each department's web site on their PhD programmes, don't forget to look for any relevant Centres for Doctoral Training and good luck.