We are going to show that is a normal distribution for each fixed , where is a Brownian motion.
An important Ito process in fiance is Ornstein-Uhlenbeck process (or Mean-reverting process). In fiance, It’s called as Hull-White model to describe a stochastic interest rate.
We recall definition of convergence in distribution and its related Portmanteau Theorem here. This is based on [Bil99] ([Patrick Billingsley 1999]).
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 .