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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).