# Measurability of the infimum of a class of functions

Ref:

In the earlier post [1], we discussed measurability of the infimum of a class of measurable functions. In particular, for the infimum of a class of measurable functions as a function, we can show that it may not be measurable. Therefore, we shall need additional conditions to have the infimum function measurable. In this post, we show that the infimum function is

• measurable if the class size is countable;
• lower semicontinuous (thus measurable) if each function in the class is lower semicontinuous.

# Two representations of the fractional Laplacian operator

There are a few equivalent definitions for the fractional Laplacian operator ${(-\Delta)^{\alpha}}$ for a constant ${\alpha \in (0, 2)}$. One is given by

$\displaystyle - (- \Delta)^{\alpha} \phi(x) = C' \int_{\mathbb R^{d}} (\phi(x+y) - \phi(x) - D \phi(x) \cdot y I_{B_{1}}(y)) \nu(y) dy,$

and the other one is given by

$\displaystyle - (- \Delta)^{\alpha} \phi(x) = C \int_{\mathbb R^{d}} (\phi(x+y) + \phi(x-y) - 2\phi(x)) \nu(y) dy,$

where ${C}$ and ${C'}$ are normalizing constants and ${\nu(y) = \frac{1}{|y|^{d + \alpha}}}$ is a symmetric Levy measure on ${\mathbb R^{d}}$. In this below, we will show they are actually given equivalent with ${2 C = C'}$. Continue reading

# Skorhod metric on a finite time interval

The notion of Skorohod metric provides a very useful topology in a discontinuous curve spaces, which is often arising as a sample space in stochastic analysis. Understanding Skorohod metric is also an interesting process for an undergraduate student, as a concrete example of a metric space but not a normed space. This note below is written by an undergraduate student with plenty of original examples.

PDF: here

Reference: [Bil99] ([Patrick Billingsley 1999])

# 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]).