When is a Markov process independent-increment?

An independent-increment stochastic process must be Markov. I am now wondering about the reverse case. Why do some Markov processes fail to be independent-increment?

1. What are some examples of Markov processes that are not independent-increment?

2. Is it possible to have some sufficient and necessary condition for a Markov process to be independent-increment?

Thanks and regards!

Solution 1:

First, observe that an independent-increment process depends on the fact that the sequence is defined on $R$. A Markov Chain can be defined in any set $S$. If $S \neq R$, you might have trouble to even define what independent-increment would be.

You can also consider a process in $R$ defined in the following way: $X_{t+1} = X_{t} + Z$, where $Z|(X_{t},X_{t-1},\ldots,X_{0}) \sim N(-X_{t},1)$. This an example of a process such that the increment depends only on the last step and, therefore, is not independent but the process is Markovian.

Regarding 2, I don't think I have much to add. You can always write $X_{t} = X_{t-1} + Z_{t}$. since $X_{t}$ is Markovian, $Z_{t} = f(X_{t},U_{t})$, where $U_{t}$ is independent of $(X_{t-1},X_{t-2}, \ldots, X_{0})$. I guess you have to prove that $f(x,u)$ is constant on $x$.