Hi Jan, personally I believe it is still called Kalman filter and you don't need any special name for it. You may simply say that your dynamic model is a random walk model. This does not add a specialty to the filter. If one works on parameter estimation problems, the dynamical model could be even simpler, e.g., in the form of x_(k+1)=x_(k) (which will then be called a static model).

I agree, it is called a random walk model.

I understand your question. Truly speaking there is no name of the system you defined. It is still the Kalman filter, where A, B, C, D matrix are different. Which you mentioned in question is right that the Kalman filter is an optimal filter. When the process or measurement or both becomes nonlinear then neither kalman filter could be applied nor any optimal filter exists. Fortunately  there are many sub optimal algorithm exist in literature. For detailed literature survey on nonlinear estimators based on sampling points and weights you could read http://www.tutorialpoint.org/Filter/nonlinear-estimation-literature-review-page1.html. 

http://www.tutorialpoint.org/Filter/nonlinear-estimation-literature-review-page1.html

Apparently what you are modeling can be written as 

x_(k+1)=Ax_(k)+w_(k)

y_(k+1)=x_(k)

y_(k+2)=Ay_(k+1)+0*y_(k) + w_(k) 

which is an AR(2) process. However, |A| must be smaller than 1. If you measure y_(k) = x_(k), then you would have y_(k+1)=Ay_(k) + w_(k), which is an AR(1) process. Still you need |A|

(1)This is Standard Kalman Filter, which A=1. In general vectorized form:

A= [0,1,0

0,0,1

0,0,0]

In this Standard Kalman Filter, the assumption about the estimated new signal follows the Gaussian Distribution.

(2) Compared to the Extended Kalman Filter, A is evaluated at each projection estimation.