# Integral of a Brownian motion

We are going to show that ${M_{t} = \int_{0}^{t} B_{s} ds}$ is a normal distribution ${\mathcal N(0, \frac 1 3 t^{3})}$ for each fixed ${t\ge 0}$, where ${B}$ 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.

# 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 ${t\in (0, \infty)}$, we denote by ${\mathbb D^{d}_{t}}$ the collection of RCLL processes on ${[0, t]}$ taking values in ${\mathbb R^{d}}$. We also use ${\mathbb D^{d}_{\infty}}$ to denote the collection of RCLL processes on ${[0, \infty)}$.

# 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