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Apologies if this is not the right place for this question.

I have been digging quite a bit around Kalman Filtering recently and I understand how the base case works. However, I am trying to solve quite a complicated problem using the Kalman filter, but am not sure whether the Kalman Filter can be applied in this context.

I am doing work on UK interest rates and for the data inputs for the analysis I am using Government Spot Rates (government bonds) = C (below), SWAP rates = A (below) and a War Bond Perpetuity = B (below). Now, I have the history of Government bonds (for example) from 01/01/1980 until 31/12/2017. I have the history of the perpetuity from 01/01/1932 until 31/12/2017. I have the history of SWAPs from 01/01/2002 until 31/12/2017. What I am trying to do is to use a Kalman Filter to backfill the SWAP history, so say from 01/01/2002 back to 01/01/1980 using inputs and relationships from the available data that I have described above.

The problem can be described in more simple terms imagining the following:

We have three cars – A, B and C. A is driving on a road in the middle, B is driving on a road above A and C is driving on a road below A (as graph below):

B--------------------------------------------------------------------

A----------------------------------------------------X

C--------------------------------------------------------------------

A simple model to calculate the position of A would be to take the difference between B and C and the difference between A and C so that A = (B-C) – (A-C)

We also know that there is correlation between the movements of the cars (i.e. B is correlated to A, A is correlated to C and C is correlated to B).

I am going to use the observed values I have as my measurement inputs (so A, B and C) which feed into the Kalman Filter algorithm to get my estimated state based on my measurements (so by saying measurements I mean the observed data I have) and my simple equation above (theoretical state) – also introducing some noise around these.

The question I have is what happens when I run out of observations for A and I also loose the relationship between A and C (indicated by the X in the above chart) as the Kalman Filter relies on receiving observations. I have thought of setting the measurement of A to 0 and giving it infinitely big variance, so it gets ignored. So the big questions is - is there a way to use the Kalman Filter to “predicted” where my A will go, given I have lost measurement updates for A at point X (if I was using a GPS for the measurement the GPS has stopped working) but I still know where my B and C are, and I also know the distance between B and C but I have lost the distance between A and C (so my model starts relying on less information)? I was thinking something along the lines of Markov Chain Monte Carlo applying the known correlation structure from before but not sure how this can feed into a Kalman Filter process.

Any thoughts will be much appreciated.

Thanks in advance,

Todor