We are going to show that is a normal distribution for each fixed , where is a Brownian motion.

# Ornstein-Uhlenbeck process

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.

# Semicontinuity

We will recall the definition of semicontinuity of a function and some of related properties. (pdf) Continue reading

# Portmanteau Theorem

We recall definition of convergence in distribution and its related Portmanteau Theorem here. This is based on [Bil99] ([Patrick Billingsley 1999]).

# Skorohod metric

PDF: here

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 .

# The definition of the viscosity solution of Dirichlet problem

We have discussed the definition of viscosity property (here). In this note, we will discuss the definition of the viscosity solution of Dirichlet problem. (PDF) Continue reading

# The definition of the viscosity solution property

In this note, we will discuss the definition of the viscosity solution property. (PDF) Continue reading