Final_1.pdf

Now the state equation is given as

The observation equation is given as

The variance of Ytis calculated as described in the discussion of the obser-
vation equation. We now define

gt= 1 – Kt, where Ktis the Kalman gain as describedin the standard predictor-
corrector framework. The Kalman equations providing the minimum vari-
ance linear estimate are as follows:

We now proceed to obtain the recursive relation for. The
Kalman equation for subscript t– 1 is as follows:

Substituting for Xˆtt−− 12 | , we have

XgXtt||−−−− 1112 =+−t ˆt t () 1 gYt t− 1

cov(XXˆtt−− 12 ,ˆ )

var ˆ

var ˆ var

var ˆ var |

|

X

XY

tt

tt t

()=

()()

()+ ()

−

1

ˆˆ

XgXtt||=+−t tt− 1 () 1 gYt t

g

Y

YX

t

t ttt

=

()

()+ ()−

var var var ˆ|1

YXttt=+ˆ η

ˆˆ ˆˆ ˆˆ

ˆˆˆˆ

var ˆ var ˆ

|

XXrXX rXX

XrXrXrX

XrX

tt t t t t t t t

tt tt tt tt

tt t t

−− −− −−

−− −−

−−

=+ −()+−()()−

=+()+−()−−()

()=+()

11 12 23

11 23

1

2 1

1

1121

(^1) (()+−()()+−()()
++()()− ()
−+()()− ()
−−()()−
−−
−−
−−

12 1

21 1 2

21 1

21 2 1

2 2

2 3

12

13

rX rX

rrXX

rr XX

rr

tt tt

tt

var ˆ var ˆ

cov ˆ ,ˆ

cov()XXXˆtt−− 13 ,ˆ

202 RISK ARBITRAGE PAIRS