Financial time series

L.Gerencsér Zs. Vágó B. Gerencsér

Lecture 4.Spectral theory, II.

The Fourier transform of (yn)Let us now reconsider the finite Fourier transform of (yn) itself:

N∑n=−N

yne−inω.

Assume that∑∞τ=0 r2(τ)

Integrating the Fourier transform of (yn)Rewrite the integral as

ζN(ω′) =

N∑n=−N

yn∫ ω′

0e−inω dω =

N∑n=−N

yncn.

Let us now compute E|ζN(ω′)|2. We get with some effort:

limN→∞

E|ζN(ω′)|2 = 2π∫ 2π

0χ[0,ω′)(ω)f (ω)dω = 2πF (ω′).

Thus the total energy content of ζN(ω′) is proportional to F (ω′).

Ultimate orthogonality of the incrementsThe mutual relationship of the increments: in general we have

limN→∞

E(ζN(a)− ζN(b))(ζN(d)− ζN(c)) = 2π∫ 2π

0χ[a,b)χ[c,d)(ω)f (ω)dω.

For non-overlapping (!) intervals [a, b) and [c, d) we get:

limN→∞

E(ζN(a)− ζN(b))(ζN(d)− ζN(c)) = 0.

Thus these increments are orthogonal.

The limit of the Fourier transformIt can be shown that ζN(ω′) has a limit ζ(ω′) in Lc2(Ω,F ,P):

limN→∞

ζN(ω′) = ζ(ω′).

Furthermore, for non overlapping intervals [a, b) and [c, d) we have

E(ζ(a)− ζ(b))(ζ(d)− ζ(c)) = 0.

Thus ζ(ω′) is a process of orthogonal increments!

Formally: dζ(ω) behaves like a random amplitude, see below.

The limit of the integrated Fourier transform

TheoremLet y = (yn) be a w.s.st. process with auto-covariance function such that∞∑τ=0

r2(τ)

The spectral representation process

DefinitionThe stochastic process ζ(ω′), with 0 ≤ ω′ ≤ 2π is called the spectralrepresentation process of y .

The reason for this terminology is that we can write yn as

yn =∫ 2π

0e iωndζ(ω),

with an appropriate interpretation of the integral to be given below.Recall that the energy content of ζ(ω′) is exactly F (ω′) :

E|ζ(ω′)|2 = 2πF (ω′).

Random orthogonal measures, I.An abstraction of ζ(ω′) is given in the following definition:

DefinitionA complex valued (!) stochastic process ζ(ω) in [0, 2π] is called a processwith orthogonal increments, if it is path-wise left continuous, ζ(0) = 0,

E|ζ(a)|2 =: G(a)

Random orthogonal measures, II.

DefinitionThe "measure dζ(.)" assigning the value ζ(b)− ζ(a) to an interval [a, b)is called a random orthogonal measure. The function G(.) is called thestructure function of dζ(.).

Exercise. Prove, that G(0) = 0, and for any 0 ≤ a < 2π we have

G(b)− G(a) = E |ζ(b)− ζ(a)|2 .

Thus G is monotone nondecreasing.

The representation problemFormally: dζ(ω) behaves a like random amplitude.

Question: can we represent a general w.s.st. process y as

yn =∫ 2π

0e inωdζ(ω)?

How to make sense of such a representation ?Recall the analogy with w.s.st. processes given by

yn =∑

k=1,mξke iωk n.

Integration w.r.t. to an orthogonal measure, I.Let dζ(ω) be a random orthogonal measure on [0, 2π).

Let f be a possibly complex valued step function of the form

f (ω) =∑k∈K

λkχ[ak ,bk ).

Here the intervals [ak , bk) are non-overlapping. Then define

I(f ) =∫ 2π

0f (ω)dζ(ω) =

∑k∈K

λk(ζ(bk)− ζ(ak)).

Thus I(f ) is a random variable which is obviously in Lc2(Ω,F ,P).

Integration w.r.t. to an orth. measure, II.

Exercise. Let f , g be two left continuous step functions on [0, 2π]. Then

EI(f )I(g) =∫ 2π

0f (ω)g(ω) dG(ω). (1)

Thus stochastic integration is a linear isometry:

From: the linear space of l.c. step-functions ⊂ Lc2([0, 2π], dG)

To: Lc2(Ω,F ,P).

Integration w.r.t. to an orth. measure, III.Extend this to a linear isometry:From: the closure, the full (!) orth. Lc2([0, 2π], dG) function spaceTo: Lc2(Ω,F ,P).

Thus for any f ∈ Lc2([0, 2π], dG) the stochastic integral

I(f ) =∫ 2π

0f (ω)dζ(ω)

is well-defined.

Isometry restated

TheoremLet f , g be two function from Lc2([0, 2π], dG). Then

E(I(f )I(g)

)=

∫ 2π0

f (ω)g(ω)dG(ω).

A benchmark example

Exercise. Let dζ(ω) be as above. Prove that

yn =∫ 2π

0e inωdζ(ω)

defines a w.s.st. process.

Formally: the increments dζ(ω) behave like random amplitudes.Surprise: all w.s.st, processes have the above form.

The spectral representation theoremThe most powerful tool in the theory of w.s.st. processes:

TheoremLet (yn) be a w.s.st. process. Then there exists a unique randomorthogonal measure dζ(ω), such that

yn =∫ 2π

0e inωdζ(ω).

The process ζ(.) is called the spectral representation process of (yn).

Change of measure, I.Let dζ(ω) be a random orthogonal measure on [0, 2π) with the structurefunction F (ω).

Let g ∈ Lc2(dF ), and define a new process

η(ω) =

∫ ω0

g(ω′)dζ(ω′) 0 ≤ ω < 2π.

Exercise. dη(ω) is a random orthogonal measure, with the structurefunction

dG(ω) = |g(ω|2dF (ω).

Change of measure, II.The corresponding random orthogonal measure will be written as

dη(ω) = g(ω)dζ(ω).

Let now h(ω) be a function in Lc2(dG) (!). Then∫ 2π0

h(ω)dη(ω)

is well-defined.

Change of measure, III.We have the following, intuitively obvious-looking result:

PropositionWe have ∫ 2π

0h(ω)dη(ω) =

∫ 2π0

h(ω)g(ω)dζ(ω).

Linear filters, I.The effect of linear filters on the spectral representation process:

Let (un) be a w.s.st. with spectral representation process dζu(ω):

un =∫ 2π

0e inωdζu(ω).

Define the w.s.st. process (yn) via an FIR filter as

yn =m∑

k=0hkun−k .

Linear filters, II.Define H(e−iω) =

∑mk=0 hke−ikω.

Exercise. Show that the spectral representation process of y is given by

dζy (ω) = H(e−iω)dζu(ω).

Linear filters, III.Let us now consider the infinite linear combination

yn =∞∑

k=0hkun−k .

We have seen that the r.h.s is well defined if the infinite series

H(e−iω) =∞∑

k=0hke−ikω

converges in Lc2(dF ).

Linear filters, IV.

PropositionThe spectral representation process of (yn) is given by

dζy (ω) = H(e−iω)dζu(ω).