I tried to understand intuitively what a predictable stochastic process is (in particular, what is "predictable" about it), but found the definition via the measurability with respect to a certain sigma-algebra highly unintuitive.
Can someone give me an enlightning example of a predictable and a non-predictable stochastic process in the situation that our probability space $\Omega$ is equal to the space of continuous paths starting at $0 \in \mathbb{R}^n$ and the probability measure is the Wiener measure?
Best Answer
Hi to highlight saz' comment you have to realize that every continuous process is predictable. To find non predictable process you have to come up with bigger space $\Omega$ for this take the space of càdlàg processes.
A very classical process that is not predictable is the Poisson process $N$ as it is càg right continuous with jumps of size one, intuition tells you that you won't be able to predict the process at jump time with any sequence of càg processes, in a way, $N$ at jump times is total surprise (unpredictable with knowledge of immediate past).
For more on this you have to work out all the properties of stopping times and their classification, associated representation of predictable sigma algebra (and others) with stopping time, representation of jumps of processes etc... a good reference is Dellacherie Meyer, or more reasonably Protter's book or the book of He, Wang, Yan "Semimartingale theory and Stochastic Calculus" or the blog of George Lowther ( which is regretfully not active anymore)
Best regards