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 ).
 Supremum of a class of functions
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.
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])
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 will recall the definition of semicontinuity of a function and some of related properties. (pdf) Continue reading