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 Topic Title: 2-dimensional Kalman Bucy filter Created On Fri May 17, 13 09:30 PM Topic View: Branch View Threaded (All Messages) Threaded (Single Messages) Linear

frame
Member

Posts: 60
Joined: Sep 2009

Fri May 17, 13 09:30 PM

There are 2 unobservable states:
dx = a (xx-x) dt + b dBx
dy = c (yy-y) dt + d dBy

And 3 observable signals:
dZ/Z = (z0 + zx x + zy y) dt + e dB + f dBy
ds1 = x dt + g dB1
ds2 = y dt + h dB2

The drift of Z depends on both the states. However Z has non-zero exposition to the shock of one of the states (f dBy).
Instead s1 and s2 are just noisy versions of the states.

My problem is how to set the kalman bucy filter, i.e. the differential for filtered x and y, the Riccati for the posterior variances and the differential of dZ/Z given the filtered states.

textbooks usually look at the case of only one state but with f=0.

Suggestions?

ps: the interpretation of dZ/Z is that of the process defined as Z=W exp(m y) where dW/W = (w0 + w1 x) dt + e dB. Hence, we can interpret x and y respectively as permanent and transitory shocks to Z.

frame
Member

Posts: 60
Joined: Sep 2009

Mon May 20, 13 01:52 PM

Nobody has hints?

However, even a simpler thing would be very useful to me.

How should I adjust the usual kalman (bucy) filter when the unobservable state is correlated with the observable signal?

e.g. I want to filter x with dx = -k x dt + s dB and I observe Y with dY = x dt + b dW and dB and dW are correlated.

Alan
Senior Member

Posts: 7200
Joined: Dec 2001

Mon May 20, 13 03:32 PM

Quote

Originally posted by: frame
Nobody has hints?

However, even a simpler thing would be very useful to me.

How should I adjust the usual kalman (bucy) filter when the unobservable state is correlated with the observable signal?

e.g. I want to filter x with dx = -k x dt + s dB and I observe y with dy = x dt + b dW and dB and dW are correlated.

I don't know anything about the Kalman filter, but I see a discussion in Oksendahl's SDE book.
He says that

dX = (f1 X + f2 Y) dt + C dB1
dY = (g1 X + g2 Y) dt + D dB2

with X unobserved and Y observed, is amenable to such a filter and gives refs.
Here (B1, B2) are uncorrelated Brownians. I am using capital letters for his variables and small letters for yours.

Now your last eqn pair (with small case variable) could be transformed to Oksendahl's form by keeping
the observed Y = y and defining a new unobservable X = x - A y and choosing A appropriately.

Edited: Mon May 20, 13 at 04:18 PM by Alan

frame
Member

Posts: 60
Joined: Sep 2009

Mon May 20, 13 06:12 PM

Thank you Alan. I'll work on this.