Derive MA (Moving average) representation of a first-difference-process I have a non-stationary AR(1)-process. After taking the first difference, how can I derive the MA representation of the resulting „difference process“ Delta_xt?
As an example, consider 
xt = 1.5xt-1 + et 
<=>
(xt-xt-1=) Delta_xt = 0.5xt-1 + et.
How can I convert this process Delta_xt into an MA process?
 A: Your specific problem has a simple solution, which is that the AR(1) process you are describing is not stationary. Thus, it does not admit a moving average representation.
However, the more general problem can be relevant.
For some intuition, consider a stationary AR(1) model with an autoregressive parameter $\beta$, and assume it has been transformed into an MA($\infty$) model with moving average parameters $1,\beta,\beta^{2},...$. When considering the difference of that process, the coefficients are such that you need to difference the original MA($\infty$) model. So if the aforementioned coefficients were $1,\beta,\beta^{2},...$, then the differenced MA coefficients are $1,\beta-1,\beta^{2}-\beta,...$.
It's perhaps easiest to convert this into a state space form. We can combine together both the difference equation and the level equation into one form
$$
\left[\begin{array}{c}
\Delta y_{t}\\
y_{t}
\end{array}\right]=\left[\begin{array}{cc}
0 & \beta-1\\
0 & \beta
\end{array}\right]\left[\begin{array}{c}
\Delta y_{t-1}\\
y_{t-1}
\end{array}\right]+\left[\begin{array}{c}
1\\
1
\end{array}\right]\varepsilon_{t}
$$
Simplifying some of the notation, we can write this as
$$
X_{t}=AX_{t-1}+B\varepsilon_{t}
$$
Which through some standard time series techniques implies
$$
X_{t}=\left(1-AL\right)^{-1}B\varepsilon_{t}
$$
We can extract the difference series as
$$
\Delta y_{t}=\left[\begin{array}{cc}
1 & 0\end{array}\right]\left(1-AL\right)^{-1}B\varepsilon_{t}
$$
Note that $$\left(1-AL\right)^{-1}$$ is fairly standard in time series and expands to $$I+AL+\left(AL\right)^{2}+..$$
This can easily be extended to an AR(p) process and adding in constants, etc.
