Kalman Filtering 61
proximations of the equilibrium price for the time chunk, and the notion of
observation error begins to make sense. It is therefore definitely reasonable
to apply Kalman-filtering ideas to stock price series with the logarithm of
prices modeled as a random walk.
Summarizing the discussions so far, we have assigned ourselves the task
of watching a random walk and making observations at regular time inter-
vals. Each of the observations has a measure of error associated with it. The
purpose of the exercise is to come up with a plausible set of system states.
Keeping with the notational conventions already established, let us de-
note the sequence of observations starting from time t= 0, the beginning of
our watching exercise as yt, and the true states as xt. The observation equa-
tion may be written yt=xt+et; that is, the observation is the true state plus
some error. Now, according to the definition of the random walk, we also
havext=xt–1+et(the current state is the previous state plus an innovation).
Writing this as a sequence of predictions and observations, we have
y 0 =x 0 +e 0 (observation)
x 1 =x 0 +e 1 (prediction)
y 1 =x 1 +e 1 (observation)
x 2 =x 1 +e 2 (prediction)
y 2 =x 2 +e 2 (observation)
...Writing the equations in matrix form, we have
For notational convenience let us denote the system of equations as
Y 2 =H 2 X 2 +h 2The 2 in the subscript is the index of the state and is indicative of the num-
ber of state estimates used in forming the equations. The above is a set of five
equations with three unknown values x 0 ,x 1 ,x 2. Given that there are more
equations than unknowns, the system is also referred to as an overdeter-
mined set. If the errors at each stage are drawn from identical, independent
yyyx
x
xeee0120
1
20
1
1
2
20
0
100
110
010
011
001
=
−
−
+
−
−
εε