It is a note on the proof of unique solvability on graphon MFG.
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.
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.
In the earlier post , 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.
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
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.
Reference: [Bil99] ([Patrick Billingsley 1999])
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]).