Example of stochastic process that is adapted but not predictable

Question

Can someone give me an example of stochastic process that is adapted but not predictable?

I know that the process must be non-left-continuous, but I don't know how to come up with an example by myself.

Thank you

Answer 1

In discrete time:

Consider independent $(Y_n)_{n\in\Bbb{N}}$ with $$P(Y_n = 0) = P(Y_n = 1) = \frac{1}{2}$$

And define $$X_n := \sum_{k=1}^n Y_k$$ being the sum with the filtration given by $$F_n = \sigma(Y_1,\ldots,Y_n)$$

Then $X_n$ is adapted but not predictable

In continuous time:

Consider a poisson point process $(X_t)_{t\ge 0}$ and the related natural filtration given by $$F_t = \sigma(X_s, s\le t)$$

Then $X_t$ is adapted but not predictable.

You can visualize this by the following: $(X_t)_{t\ge 0}$ is a jump process where the time till the next jump is independent from the elapsed time. So if you just have information about the time $s<t$ you won't know if the process will jump at time $t$ hence it's not predictable. But at time $t$ you know if you had jumped or not so it's adapted.