I have a question concerning page 15 in Chapter 2, where the notes state that "the process X has the Markov property". I don't understand how the process X can have the Markov property if X_n = X_(n-1) + Y_n and Y_n does not have the Markov property. If Y_n does not have the Markov property, doesn't that automatically mean that X_n then also does not have the Markov property?
I think it's because Yn only depends on the value of Xn-1 and therefore Xn = Xn-1 + Yn still only depends on Xn-1 and not any earlier terms of the series Xn. If it were not Markov, then knowing the value of some of Xn-2, xn-3 etc would give us 'more information' about the value of Xn. But since Xn only depends on Xn-1 and Y, where Yn also only depends on the value of Xn-1, it is Markov. Hopefully this is along the right lines?
Hi George97, Yes, maybe you are right. I just checked Chapter 1 and page 18 states that even when a process "does not have independent increments, then it may still be Markov if it satisfies the Markov definition". Best regards, Stephan
