It is a note on the proof of unique solvability on graphon MFG.

# When does American and European call have the same price?

We will prove the following fact arising from finance.

- American call and European call with the same strike and maturity has the same price given that their underlying assets are the same non-dividend stock.

Of course, this fact shall not be generalized to put or some other scenarios.

Continue reading

# Two slides on Dirichlet problems

Today, I’ve presented my recent work on the generalized solution of the Dirichlet problem at IMA, here is the slide ( pdf ). There is also a video available for the presentation ( link ). Unfortunately, it does not show my board work during the talk, but it’s still a lot fun to look back at my own performance.

One may compare this work with my previous work on strong solutions of the Dirichlet problem, see the paper here ( link ), and slides here ( pdf ).

# Measurability of the infimum of a class of functions

Ref:

[1] Supremum of a class of functions

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 for a constant . One is given by

and the other one is given by

where and are normalizing constants and is a symmetric Levy measure on . In this below, we will show they are actually given equivalent with . 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 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]).