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
We recall definition of convergence in distribution and its related Portmanteau Theorem here. This is based on [Bil99] ([Patrick Billingsley 1999]).