The Mathematics of Financial Modelingand Investment Management

11-FinEcon-Time Series Page 290 Wednesday, February 4, 2004 12:58 PM

290 The Mathematics of Financial Modeling and Investment Management

AL()=

N

∑aiL

i

i = 1

Note that if the lag operator is applied to a series that starts from a

given point, initial conditions must be specified.

Within the domain of stationary series, infinite power series of the

lag operator can also be formed. In fact, as remarked above, given a sta-

tionary series, if the coefficients hi are absolutely summable, the series

∞

∑hiL

ix

t

i = 1

is well defined in the sense that it converges and defines another station-

ary series. It therefore makes sense to define the operator:

∞

∑hiL

AL()= i

i = 1

Now consider the operator I – λ L. If λ< 1 , this operator can be

inverted and its inverse is given by the infinite power series,

∞

∑

( I – λ L) – 

1 = λi Li

i = 1

λ

∞

∑

as can be seen by multiplying I – λL by the power series iLi :

i = 1

( I – λ L )

∞

∑λ^

iLi = L (^0) = I
i = 1
On the basis of this relationship, it can be demonstrated that any opera-
tor of the type
AL()=
N

∑aiL

i

i = 1

can be inverted provided that the solutions of the equation