Statistical Theory for High- and
Infinite-dimensional Models
Ardjen Pengel
Exercise 2.0+
01-03-17
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Theorem: Ito isometry
Let f ∈ L2 [0, ∞) and Wt be a standard Brownian motion defined
on (Ω, F, P) then :
E[I (f )2 ] = ∫ f (t)2 dt
Consequently,
E[I (f )I (g )] = ∫ f (t)g (t)dt
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
Suppose first that f is simple i.e. f = ∑nj=0 aj 1(tj ,tj+1 ] ,where
0 = t0 < ⋯ < tn = t and aj ∈ R.
n
n
I (f ) = ∑ aj (Wtj+1 − Wtj ) = ∑ aj Dj
j=0
j=0
where Dj = Wtj+1 − Wtj .
n
I (f )2 = ∑ aj 2 Dj2 + 2 ∑ ai aj Di Dj
j=0
0≤i<j≤n
Lets first evaluate the expectation of the cross terms:
E[ai aj Di Dj ] = E[ai aj E[Di Dj ∣F(tj )]]
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
Suppose first that f is simple i.e. f = ∑nj=0 aj 1(tj ,tj+1 ] ,where
0 = t0 < ⋯ < tn = t and aj ∈ R.
n
n
I (f ) = ∑ aj (Wtj+1 − Wtj ) = ∑ aj Dj
j=0
j=0
where Dj = Wtj+1 − Wtj .
n
I (f )2 = ∑ aj 2 Dj2 + 2 ∑ ai aj Di Dj
j=0
0≤i<j≤n
Lets first evaluate the expectation of the cross terms:
E[ai aj Di Dj ] = E[ai aj E[Di Dj ∣F(tj )]]
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
Suppose first that f is simple i.e. f = ∑nj=0 aj 1(tj ,tj+1 ] ,where
0 = t0 < ⋯ < tn = t and aj ∈ R.
n
n
I (f ) = ∑ aj (Wtj+1 − Wtj ) = ∑ aj Dj
j=0
j=0
where Dj = Wtj+1 − Wtj .
n
I (f )2 = ∑ aj 2 Dj2 + 2 ∑ ai aj Di Dj
j=0
0≤i<j≤n
Lets first evaluate the expectation of the cross terms:
E[ai aj Di Dj ] = E[ai aj E[Di Dj ∣F(tj )]]
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
Suppose first that f is simple i.e. f = ∑nj=0 aj 1(tj ,tj+1 ] ,where
0 = t0 < ⋯ < tn = t and aj ∈ R.
n
n
I (f ) = ∑ aj (Wtj+1 − Wtj ) = ∑ aj Dj
j=0
j=0
where Dj = Wtj+1 − Wtj .
n
I (f )2 = ∑ aj 2 Dj2 + 2 ∑ ai aj Di Dj
j=0
0≤i<j≤n
Lets first evaluate the expectation of the cross terms:
E[ai aj Di Dj ] = E[ai aj E[Di Dj ∣F(tj )]]
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
Note that Di is F(tj ) measurable and Dj is independent of F(tj )
E[ai aj E[Di Dj ∣F(tj )]] = E[ai aj E[Dj ]] = 0
Thus we see that
n
n
j=0
j=0
E[I (f )2 ] = ∑ aj 2 E[Dj2 ] = ∑ aj 2 (tj+1 − tj )
∞ n
=∫
0
2
∑ aj 1(tj ,tj+1 ] dt = ∫
j=0
Ardjen Pengel
∞
f (t)2 dt
0
Statistical Theory for High- and Infinite-dimensional Models
Proof
Note that Di is F(tj ) measurable and Dj is independent of F(tj )
E[ai aj E[Di Dj ∣F(tj )]] = E[ai aj E[Dj ]] = 0
Thus we see that
n
n
j=0
j=0
E[I (f )2 ] = ∑ aj 2 E[Dj2 ] = ∑ aj 2 (tj+1 − tj )
∞ n
=∫
0
2
∑ aj 1(tj ,tj+1 ] dt = ∫
j=0
Ardjen Pengel
∞
f (t)2 dt
0
Statistical Theory for High- and Infinite-dimensional Models
Proof
Note that Di is F(tj ) measurable and Dj is independent of F(tj )
E[ai aj E[Di Dj ∣F(tj )]] = E[ai aj E[Dj ]] = 0
Thus we see that
n
n
j=0
j=0
E[I (f )2 ] = ∑ aj 2 E[Dj2 ] = ∑ aj 2 (tj+1 − tj )
∞ n
=∫
0
2
∑ aj 1(tj ,tj+1 ] dt = ∫
j=0
Ardjen Pengel
∞
f (t)2 dt
0
Statistical Theory for High- and Infinite-dimensional Models
Proof
We have shown that I ∶ S → L2 (Ω, F, P) preservers norms and thus
is a linear isometry.
2
1
2
2
1
2
∣∣I (f )∣∣L2 (Ω) = [E[I (f ) ]] = [ ∫ f (t) dt] = ∣∣f ∣∣L2 [0,∞)
Any linear isometric mapping also preserves the inner product:
∀f , g ∈ S ∶ ⟨I (f ), I (g )⟩L2 (Ω) = ⟨f , g ⟩L2 [0,∞)
This gives the required result:
E[I (f )I (g )] = ∫ f (t)g (t)dt
This can be extended to general functions f and g by the ”Cauchy
sequence an completeness”argument and an MCT for
interchanging the norm and the limit.
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
We have shown that I ∶ S → L2 (Ω, F, P) preservers norms and thus
is a linear isometry.
2
1
2
2
1
2
∣∣I (f )∣∣L2 (Ω) = [E[I (f ) ]] = [ ∫ f (t) dt] = ∣∣f ∣∣L2 [0,∞)
Any linear isometric mapping also preserves the inner product:
∀f , g ∈ S ∶ ⟨I (f ), I (g )⟩L2 (Ω) = ⟨f , g ⟩L2 [0,∞)
This gives the required result:
E[I (f )I (g )] = ∫ f (t)g (t)dt
This can be extended to general functions f and g by the ”Cauchy
sequence an completeness”argument and an MCT for
interchanging the norm and the limit.
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
We have shown that I ∶ S → L2 (Ω, F, P) preservers norms and thus
is a linear isometry.
2
1
2
2
1
2
∣∣I (f )∣∣L2 (Ω) = [E[I (f ) ]] = [ ∫ f (t) dt] = ∣∣f ∣∣L2 [0,∞)
Any linear isometric mapping also preserves the inner product:
∀f , g ∈ S ∶ ⟨I (f ), I (g )⟩L2 (Ω) = ⟨f , g ⟩L2 [0,∞)
This gives the required result:
E[I (f )I (g )] = ∫ f (t)g (t)dt
This can be extended to general functions f and g by the ”Cauchy
sequence an completeness”argument and an MCT for
interchanging the norm and the limit.
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
We have shown that I ∶ S → L2 (Ω, F, P) preservers norms and thus
is a linear isometry.
2
1
2
2
1
2
∣∣I (f )∣∣L2 (Ω) = [E[I (f ) ]] = [ ∫ f (t) dt] = ∣∣f ∣∣L2 [0,∞)
Any linear isometric mapping also preserves the inner product:
∀f , g ∈ S ∶ ⟨I (f ), I (g )⟩L2 (Ω) = ⟨f , g ⟩L2 [0,∞)
This gives the required result:
E[I (f )I (g )] = ∫ f (t)g (t)dt
This can be extended to general functions f and g by the ”Cauchy
sequence an completeness”argument and an MCT for
interchanging the norm and the limit.
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Theorem
The simple functions S are dense in L2 [0, ∞). Every measurable f
is the pointwise limit of simple funtions.
∃{fn }∞
n=1 ⊂ S s.t. f = sup fn
n
Proof: The idea is to divide the range of f .
First divide [0, 2n ] into 22n subintervals of width 2−n .
Define for every n:
Ak,n = f −1 [(k2−n , (k + 1)2−n )]], k = 0, 1, ⋯22n − 1)
and Bn = f −1 [(2n , ∞)].
This gives us the approximating simple functions:
22n −1
fn (x) ∶= ∑ k2−n 1Ak,n + 2n 1Bn
k=0
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Theorem
The simple functions S are dense in L2 [0, ∞). Every measurable f
is the pointwise limit of simple funtions.
∃{fn }∞
n=1 ⊂ S s.t. f = sup fn
n
Proof: The idea is to divide the range of f .
First divide [0, 2n ] into 22n subintervals of width 2−n .
Define for every n:
Ak,n = f −1 [(k2−n , (k + 1)2−n )]], k = 0, 1, ⋯22n − 1)
and Bn = f −1 [(2n , ∞)].
This gives us the approximating simple functions:
22n −1
fn (x) ∶= ∑ k2−n 1Ak,n + 2n 1Bn
k=0
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Theorem
The simple functions S are dense in L2 [0, ∞). Every measurable f
is the pointwise limit of simple funtions.
∃{fn }∞
n=1 ⊂ S s.t. f = sup fn
n
Proof: The idea is to divide the range of f .
First divide [0, 2n ] into 22n subintervals of width 2−n .
Define for every n:
Ak,n = f −1 [(k2−n , (k + 1)2−n )]], k = 0, 1, ⋯22n − 1)
and Bn = f −1 [(2n , ∞)].
This gives us the approximating simple functions:
22n −1
fn (x) ∶= ∑ k2−n 1Ak,n + 2n 1Bn
k=0
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
Note that indeed
(i) Ak,n and Bn are in F by the measurability of f
(ii) 0 ≤ fn ≤ f and fn ↑ f
These simple funtions are elements of L2 [0, ∞) and thus
∀f ∈ L2 [0, ∞) there is a sequence of simple functions {fn } s.t. by
the DCT
lim ∫ ∣f − fn ∣2 dt = 0
n→∞
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
Proof
Note that indeed
(i) Ak,n and Bn are in F by the measurability of f
(ii) 0 ≤ fn ≤ f and fn ↑ f
These simple funtions are elements of L2 [0, ∞) and thus
∀f ∈ L2 [0, ∞) there is a sequence of simple functions {fn } s.t. by
the DCT
lim ∫ ∣f − fn ∣2 dt = 0
n→∞
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models
It must be noted that the actual excercise was proving that the set
{f ∶ f = ∑nj=1 aj 1(tj ,tj+1 ] } was dense in L2 [0, ∞).
This can be done by using the fact that the set of compactly
supported continuous functions are dense in L2 [0, ∞). With similar
argumentation as the preceding proof it is enough to approximate
the L2 functions on a compact interval because by the integrability
assumption they must be bounded almost everywhere. And using
lower Riemann sums to approximate the continuous function from
below gives us the required result.
Ardjen Pengel
Statistical Theory for High- and Infinite-dimensional Models