Stochastic Processes and Financial Mathematics
(part two)
Chapter 10 The transition to continuous time
Up until now, we have always indexed time by the integers,
As before, we work over some probability space
As in discrete time, we will usual write
In words, we should think of a continuous stochastic process as a random continuous function. For example, if
In continuous time, our standard setup is that we will work over a filtered space
Lastly, we upgrade our definition of a martingale into continuous time.
There are continuous time equivalents of the results (e.g. the martingale convergence theorem) that we proved for discrete time martingales, but they are outside of the scope of this course.
Organization
Whilst studying these notes you will also start thinking about revision for the summer exam. Some advice on how to structure your revision for this course is included in Appendix D. We’ll discuss this in lectures at a suitable point during the semester.
Those of you taking MAS61023 have a chapter of independent reading in Chapter 19, marked with a (). This material relies only on the previous semester and is already already accessible to you. You can study this chapter whenever you like, during semester two. Section 13.3 is also independent reading for you, but you’ll have to wait until we reach up to that point within the semester two lectures to study it.
You can find general information on the organization of this course in Section 0.1, within part one of these lecture notes.