Oct 20, 2018
I have spent much time on Probability Theory, Chapter 14. Now, it is taking shape given my research in the previous chapters. The definition of stochastic process has taken a significant update, now I require a stochastic process to be measurable in the product measure space of the time index measure space and the underlying probability measure space. In Professor Burkholder’s notes, these processes are called progressively measurable. For discrete-time processes, this updated version of definition is equivalent to the old definition. For continuous-time processes, this updated version adds significant constraints on what can be called a stochastic process. My past research had been a bit misguided since I never go into the depth of defining a stochastic process, and just take other researchers’ work at their face value. Now, I thought deep and hard into this problem and decided that this definition is the right way to go forward. I hope that this hiccup is going to be forgiven by all my readers. Currently, I am thinking deep and hard into the definition of Itô integral. Hope that I can make something out of it.