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Gauge Institute Journal,
H. Vic Dannon
Ito Integral
H. Vic Dannon
[email protected]
March, 2013
t =b
Abstract
We show that Ito’s integral
∫
X (ζ, t )dB(ζ , t ) ,
t =a
where X (ζ , t ) is a Random Process, and B(ζ, t ) is Random
Walk is ill defined because it
ƒ violates
Riemann’s
Oscillation
Condition
for
Integrability: The Oscillation Integral is not an
infinitesimal.
ƒ violates the Fundamental Theorem of the Integral
Calculus: Reversing the integration returns a function
different from the original integrand.
ƒ depends on the partitions of [a, b ] , and on the choice of
the intermediate points in them. Thus, the Random
Variable has an undetermined Expectation.
Consequently, the Ito Integral does not exist, and formulas
that include that integral do not hold.
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H. Vic Dannon
Keywords: Ito Integral, Ito Process, Ito Formula, Stochastic
Integration, Infinitesimal, Infinite-Hyper-real, Hyper-real,
Calculus, Limit, Continuity, Derivative, Integral, Delta
Function, Random Variable, Random Process, Random
Signal, Stochastic Process, Stochastic Calculus, Probability
Distribution,
Bernoulli
Random
Variables,
Binomial
Distribution, Gaussian, Normal, Expectation, Variance,
Random Walk, Poisson Process,
2000 Mathematics Subject Classification 26E35; 26E30;
26E15; 26E20; 26A06; 26A12; 03E10; 03E55; 03E17; 03H15;
46S20; 97I40; 97I30.
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Contents
Introduction
1. Hyper-real Line
2. Hyper-real Function
3. Integral of a Hyper-real Function
4. Delta Function
5. Random Walk B(ζ, t )
6. Integration sums of B(ζ, t ) with respect to B(ζ, t ) .
t =b
7.
∑ B(ζ, t )dB(ζ, t ) is ill-defined.
t =a
t =b
8.
∑ B(ζ, t )dB(ζ, t ) depends on the choice of the
t =a
intermediate points in the partition of [B(ζ, a ), B(ζ, b)]
9.
The Myth that Lebesgue Integral generalizes Riemann’s
t =b
10.
∑ B(ζ, t )dB(ζ, t ) violates the Fundamental Theorem of
t =a
the Integral Calculus.
11. Ito’s Process and Ito’s Formula
References
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Introduction
In [Dan5], we defined in Infinitesimal Calculus the Wiener
t =b
Integral
∫
f (t )dB(ζ, t ) , where f (t ) is integrable hyper-real
t =a
function, and B(ζ, t ) is a Random Walk.
In [Ito], Ito argued that f (t ) can be replaced with a Hyperreal Random Process X (ζ , t ) .
We argue here that his claim fails for X (ζ, t ) = B(ζ, t ) .
We show that Ito’s approach through Lebesgue Integration,
ignores the basics of Integration set by Riemann.
The Ito integration sums do not converge to an Integral.
t =b
The Ito integral
∫
B(ζ, t )dB(ζ , t ) is ill defined because it
t =a
ƒ violates
Riemann’s
Oscillation
Condition
for
Integrability: The Oscillation Integral is not an
infinitesimal.
ƒ violates the Fundamental Theorem of the Integral
Calculus: Reversing the integration returns a function
different from the original integrand.
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Gauge Institute Journal,
H. Vic Dannon
ƒ depends on the partitions of [a, b ] , and on the choice of
the intermediate points in them. Thus, the Random
Variable has an undetermined Expectation.
Any integration is an infinite summation. Then, we cannot
tell the difference between
t =b
∑ X (ζ, t )[X (ζ, t + dt ) − X (ζ, t )] ,
t =a
t =b
∑ X (ζ, t + dt )[X (ζ, t + dt ) − X (ζ, t )] ,
t =a
and any summation in between them
t =b
∑ ⎡⎣ λX (ζ, t ) + (1 − λ)X (ζ, t + dt ) ⎤⎦ [X (ζ, t + dt ) − X (ζ, t )]
t =a
It is not up to us to choose the sum with λ = 12 , and it is
meaningless to say that we did that.
The infinite summation yields a definite value, only if all
these –impossible to distinguish between- results, are all the
same.
Furthermore, when we encounter uncertain outcomes, our
first measure of certainty is the average. The average is
what we expect, give or take some deviation.
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Thus, our first measure is the Expectation of the Random
Variable, and our second measure is the Variance, which
over time will limit the Expectation within a band.
If we cannot tell the Average of our outcomes, our Random
Variable is undefined, and should be discarded.
Thus, a Random Process X (ζ , t ) may not be integrated with
respect to a Random Walk B(ζ, t ) , and an Integral in which
f (t ) is replaced by X (ζ , t ) does not exist.
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1.
Hyper-real Line
The minimal domain and range, needed for the definition
and analysis of a hyper-real function, is the hyper-real line.
Each real number α can be represented by a Cauchy
sequence of rational numbers, (r1, r2 , r3 ,...) so that rn → α .
The constant sequence (α, α, α,...) is a constant hyper-real.
In [Dan2] we established that,
1. Any
totally ordered set of positive, monotonically
decreasing to zero sequences (ι1, ι2 , ι3 ,...) constitutes a
family of infinitesimal hyper-reals.
2. The infinitesimals are smaller than any real number,
yet strictly greater than zero.
3. Their reciprocals
(
1 1 1
, ,
ι1 ι2 ι3
,...
)
are the infinite hyper-
reals.
4. The infinite hyper-reals are greater than any real
number, yet strictly smaller than infinity.
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H. Vic Dannon
5. The infinite hyper-reals with negative signs are
smaller than any real number, yet strictly greater than
−∞ .
6. The sum of a real number with an infinitesimal is a
non-constant hyper-real.
7. The Hyper-reals are the totality of constant hyperreals,
a
family
of
infinitesimals,
a
family
of
infinitesimals with negative sign, a family of infinite
hyper-reals, a family of infinite hyper-reals with
negative sign, and non-constant hyper-reals.
8. The hyper-reals are totally ordered, and aligned along
a line: the Hyper-real Line.
9. That line includes the real numbers separated by the
non-constant hyper-reals. Each real number is the
center of an interval of hyper-reals, that includes no
other real number.
10.
In particular, zero is separated from any positive
real by the infinitesimals, and from any negative real
by the infinitesimals with negative signs, −dx .
11.
Zero is not an infinitesimal, because zero is not
strictly greater than zero.
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H. Vic Dannon
12.
We do not add infinity to the hyper-real line.
13.
The
infinitesimals,
the
infinitesimals
with
negative signs, the infinite hyper-reals, and the infinite
hyper-reals with negative signs are semi-groups with
respect to addition. Neither set includes zero.
14.
The hyper-real line is embedded in \∞ , and is
not homeomorphic to the real line. There is no bicontinuous one-one mapping from the hyper-real onto
the real line.
15.
In particular, there are no points on the real line
that can be assigned uniquely to the infinitesimal
hyper-reals, or to the infinite hyper-reals, or to the nonconstant hyper-reals.
16.
No
neighbourhood
of
homeomorphic to an \n ball.
a
hyper-real
is
Therefore, the hyper-
real line is not a manifold.
17.
The hyper-real line is totally ordered like a line,
but it is not spanned by one element, and it is not onedimensional.
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2.
Hyper-real Function
2.1 Definition of a hyper-real function
f (x ) is a hyper-real function, iff it is from the hyper-reals
into the hyper-reals.
This means that any number in the domain, or in the range
of a hyper-real f (x ) is either one of the following
real
real + infinitesimal
real – infinitesimal
infinitesimal
infinitesimal with negative sign
infinite hyper-real
infinite hyper-real with negative sign
Clearly,
2.2
Every function from the reals into the reals is a hyper-
real function.
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3.
Integral of Hyper-real Function
In [Dan3], we defined the integral of a Hyper-real Function.
Let f (x ) be a hyper-real function on the interval [a, b ] .
The interval may not be bounded.
f (x ) may take infinite hyper-real values, and need not be
bounded.
At each
a ≤ x ≤b,
there is a rectangle with base [x − dx2 , x + dx2 ] , height f (x ) ,
and area
f (x )dx .
We form the Integration Sum of all the areas for the x ’s
that start at x = a , and end at x = b ,
∑
f (x )dx .
x ∈[a ,b ]
If for any infinitesimal dx , the Integration Sum has the
same hyper-real value, then f (x ) is integrable over the
interval [a, b ] .
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Then, we call the Integration Sum the integral of f (x ) from
x = a , to x = b , and denote it by
x =b
∫
f (x )dx .
x =a
If the hyper-real is infinite, then it is the integral over [a, b ] ,
If the hyper-real is finite,
x =b
∫
f (x )dx = real part of the hyper-real . ,
x =a
3.1 The countability of the Integration Sum
In [Dan1], we established the equality of all positive
infinities:
We proved that the number of the Natural Numbers,
Card ` ,
equals
the
number
of
Real
Numbers,
Card \ = 2Card ` , and we have
Card `
Card ` = (Card `)2 = .... = 2Card ` = 22
= ... ≡ ∞ .
In particular, we demonstrated that the real numbers may
be well-ordered.
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Consequently, there are countably many real numbers in the
interval [a, b ] , and the Integration Sum has countably many
terms.
While we do not sequence the real numbers in the interval,
the summation takes place over countably many f (x )dx .
The Lower Integral is the Integration Sum where f (x ) is
replaced
by its lowest value on each interval [x − dx2 , x + dx2 ]
3.2
∑
x ∈[a ,b ]
⎛
⎞
⎜⎜
inf
f (t ) ⎟⎟⎟dx
⎜⎝ x −dx ≤t ≤x + dx
⎠⎟
2
2
The Upper Integral is the Integration Sum where f (x ) is
replaced by its largest value on each interval [x − dx2 , x + dx2 ]
3.3
⎛
⎞⎟
⎜⎜
f (t ) ⎟⎟dx
∑ ⎜⎜ x −dxsup
⎟
dx
≤t ≤x +
⎠⎟
x ∈[a ,b ] ⎝
2
2
If the integral is a finite hyper-real, we have
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3.4 A hyper-real function has a finite integral if and only if
its upper integral and its lower integral are finite, and differ
by an infinitesimal.
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4.
Delta Function
In [Dan4], we defined the Delta Function, and established its
properties
1. The Delta Function is a hyper-real function defined
from the hyper-real line into the set of two hyper-reals
⎧
1 ⎫⎪
⎪
⎪
⎪
0,
⎬ . The hyper-real 0 is the sequence
⎨
⎪
dx
⎪
⎪⎪
⎩
⎭
The infinite hyper-real
0, 0, 0,... .
1
depends on our choice of
dx
dx .
2. We will usually choose the family of infinitesimals that
is spanned by the sequences
1
1
1
,
,
,… It is a
n
n2
n3
semigroup with respect to vector addition, and includes
all the scalar multiples of the generating sequences
that are non-zero. That is, the family includes
infinitesimals with negative sign. Therefore,
1
will
dx
mean the sequence n . Alternatively, we may choose
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the
1
2n
H. Vic Dannon
family
,
2n .
1
3n
,
spanned
1
4n
,… Then,
by
the
sequences
1
will mean the sequence
dx
Once we determined the basic infinitesimal dx ,
we will use it in the Infinite Riemann Sum that defines
an Integral in Infinitesimal Calculus.
3. The Delta Function is strictly smaller than ∞
δ(x ) ≡
4. We define,
where
1
dx
χ
⎡ −dx , dx ⎤ (x ) ,
⎢⎣ 2 2 ⎥⎦
χ
⎧1, x ∈ ⎡ − dx , dx ⎤
⎪
⎢⎣ 2 2 ⎥⎦ .
⎪
⎡ −dx , dx ⎤ (x ) = ⎨
⎪
⎣⎢ 2 2 ⎦⎥
0, otherwise
⎪
⎩
5. Hence,
™ for x < 0 , δ(x ) = 0
™ at x = −
™ for
dx
1
, δ(x ) jumps from 0 to
,
2
dx
1
.
x ∈ ⎡⎢⎣ − dx2 , dx2 ⎤⎦⎥ , δ(x ) =
dx
™ at x = 0 ,
™ at x =
δ(0) =
1
dx
1
dx
, δ(x ) drops from
to 0 .
dx
2
™ for x > 0 , δ(x ) = 0 .
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Gauge Institute Journal,
H. Vic Dannon
™ x δ(x ) = 0
6. If dx =
7. If dx =
8. If dx =
1
n
2
n
1
n
χ
, δ(x ) =
χ
(x ), 2
[− 1 , 1 ]
2 2
1
, δ(x ) =
,
∫
2
,
(x )...
[− 1 , 1 ]
6 6
3
,...
, δ(x ) = e−x χ[0,∞), 2e−2x χ[0,∞), 3e−3x χ[0,∞),...
δ(x )dx = 1 .
x =−∞
10.
4 4
2 cosh2 x 2 cosh2 2x 2 cosh2 3x
x =∞
9.
χ
(x ), 3
[− 1 , 1 ]
1
δ(ξ − x ) =
2π
k =∞
∫
e −ik (ξ −x )dk
k =−∞
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Gauge Institute Journal,
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5.
Random Walk B(ζ, t )
The Random Walk of small particles in fluid is named after
Brown, who first observed it, Brownian Motion. It models
other processes, such as the fluctuations of a stock price.
In a volume of fluid, the path of a particle is in any direction
in the volume, and of variable size
5.1 Bernoulli Random Variables of the Walk
We restrict the Walk here to the line, in uniform
infinitesimal size steps dx :
To the left, with probability
p = 12 ,
or to the right, with probability
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Gauge Institute Journal,
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q = 1 − p = 12 .
At time t , after
N infinitesimal time intervals dt ,
N =
t
dt
, is an infinite hyper-real,
the particle is at the point
x.
At the i th step we define the Bernoulli Random Variable,
Bi (right step) = dx ,
ζ1 = right step .
Bi (left step) = −dx ,
ζ2 = left step .
where i = 1, 2,..., N .
Pr(Bi = dx ) = 12 ,
Pr(Bi = −dx ) = 12 ,
E [Bi ] = dx ⋅ 12 + (−dx ) ⋅ 12 = 0 ,
E [Bi 2 ] = (dx )2 ⋅ 12 + (−dx )2 ⋅ 12 = (dx )2
Var[Bi ] = E [Bi 2 ] − (E [Bi ])2 = 0
N
(dx )2
0
5.2 The Random Walk
B(ζ, t ) = B1 + B2 + ... + BN
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Gauge Institute Journal,
H. Vic Dannon
is a Random Process with
E [B(ζ, t )] = 0 ,
Var[B(ζ, t )] = N (dx )2 .
Proof: Since the Bi are independent,
E [B(ζ, t )] = E [B1 ] + ... + E [BN ] = 0
N
0
0
Var[B(ζ, t )] = Var[B1 ] + ... + Var[BN ] = N (dx )2 . ,
(dx )2
5.3
(dx )2
B(ζ, t + dt ) − B(ζ, t ) is a Bernoulli Random Variable Bi
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Gauge Institute Journal,
H. Vic Dannon
6.
Integration Sums of B(ζ , t ) with
respect to B(ζ , t ).
t =b
In [Dan5], we defined the Integral
∫
f (t )dB(ζ , t ) :
t =a
6.1 Integration Sums of f (t ) with respect to B(ζ, t ) .
Let f (t ) be a hyper-real function on the bounded time
interval [a, b ] . f (t ) need not be bounded.
At each a ≤ t ≤ b , there is a Bernoulli Random Variable
dB(ζ , t ) = B(ζ , t + dt ) − B(ζ, t ) = Bi (ζ, t ) = B (ζ, t )dt .
We form the Integration Sum
t =b
t =b
t =a
t =a
∑ f (t )dB(ζ, t ) = ∑ f (t )Bi (ζ, t ).
For any dt ,
(1) the First Moment of the Integration Sum is
⎡ t =b
⎤
⎢
E ⎢ ∑ f (t )Bi (ζ, t ) ⎥⎥ =
⎢⎣ t =a
⎥⎦
t =b
[Bi (ζ , t )] = 0 .
∑ f (t ) E
t =a
0
(2) the Second Moment of the Integration sum is
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Gauge Institute Journal,
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⎡
τ =b
⎞⎤
⎞⎟2 ⎥⎤
⎡⎛ t =b
⎞⎛
⎢ ⎜⎛ t =b
⎟
⎟⎟⎜
⎜
⎢
⎟
E ⎢ ⎜⎜ ∑ f (t )Bi (ζ, t ) ⎟ ⎥ = E ⎢ ⎜⎜ ∑ f (t )Bi (ζ, t ) ⎟⎜⎜ ∑ f (τ )B j (ζ , τ ) ⎟⎟ ⎥⎥
⎟
⎟
⎟⎟⎜
⎢ ⎝⎜ t =a
⎠⎟ ⎦⎥
⎠⎟ ⎥⎥
⎠⎝
τ =a
⎢ ⎝⎜ t =a
⎣
⎢⎣
⎦
t =b τ =b
=
∑ ∑ f (t )f (τ )E[B j (ζ, τ )Bi (ζ, t )]
t =a τ =a
Since the Bernoulli Random Variables are independent,
E [B j (ζ , τ )Bi (ζ , t )] = E [Bi 2 (ζ , t )] = (dx )2
only for t = τ . Then,
⎡ ⎛ t =b
⎞⎟2 ⎤⎥
⎢⎜
E ⎢ ⎜⎜ ∑ f (t )Bi (ζ , t ) ⎟⎟ ⎥ =
⎟
⎢ ⎜⎝ t =a
⎠⎟ ⎥⎥
⎢⎣
⎦
t =b
dx )2 ,
∑ f 2(t )(N
t =a
(2D )dt
t =b
= 2D ∑ f 2 (t )dt ,
t =a
t =b
= 2D ∫ f 2 (t )dt ,
t =a
assuming (dx )2 = (2D )dt , and f (t ) integrable.
Thus, for any dt , the Integration Sum is a unique welldefined hyper-real Random Variable I (ζ ) .
We call I (ζ ) the integral of f (t ) , with respect to B(ζ, t ) from
t =b
x = a , to x = b , and denote it by
∫
t =a
22
f (t )dB(ζ , t ) .
Gauge Institute Journal,
H. Vic Dannon
6.2 Integration Suns of B(ζ, t ) , with respect to B(ζ, t )
t =b
To define
∫
B(ζ, t )dB(ζ , t ) , we form the Integration Sum
t =a
t =b
t =b
t =a
t =a
∑ B(ζ, t )dB(ζ, t ) = ∑ B(ζ, t )Bi (ζ, t ) .
(1) the First Moment of the Integration Sum is
⎡ t =b
⎤
⎢
E ⎢ ∑ B(ζ, t )Bi (ζ , t ) ⎥⎥ =
⎣⎢ t =a
⎦⎥
t =b
∑ E[B(ζ, t )Bi (ζ, t )]
t =a
t =b
=
[Bi 2 (ζ, t )] = 2D(b − a ) .
∑ E
t =a
(dx )2 =(2D )dt
(2) the Second Moment of the Integration sum is
⎡ ⎛ t =b
τ =b
⎞⎤
⎞⎟2 ⎤⎥
⎡⎛ t =b
⎞⎛
⎢⎜
⎟
⎟⎟⎜
⎜
⎢
⎟
E ⎢ ⎜⎜ ∑ B(ζ, t )Bi (ζ , t ) ⎟ ⎥ = E ⎢ ⎜⎜ ∑ B(ζ, t )Bi (ζ , t ) ⎟⎜⎜ ∑ B(ζ , τ )B j (ζ , τ ) ⎟⎟ ⎥⎥
⎟
⎟
⎟⎟⎜
⎢ ⎜⎝ t =a
⎠⎟ ⎥⎦
⎠⎟ ⎥⎥
⎠⎝
τ =a
⎢⎣ ⎝⎜ t =a
⎣⎢
⎦
This expression cannot be easily resolved, and we are not
able to conclude that for any dt , the Integration Sum is a
unique well-defined hyper-real Random Variable I (ζ ) .
We shall see that for a variety of reasons the integral of
B(ζ, t ) , with respect to B(ζ, t ) from x = a , to x = b , cannot
be defined.
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Gauge Institute Journal,
H. Vic Dannon
7.
t =b
∑ B(ζ, t )dB(ζ, t ) is ill-defined
t =a
Proof:
The Expectation of the Oscillation Integral [Dan3,
p.46], of B(ζ, t ) with respect to B(ζ, t ) is
⎡ t =b
⎤
E ⎢⎢ ∑ ⎡⎣ B(ζ, t + dt ) − B(ζ, t ) ⎤⎦ dB(ζ, t ) ⎥⎥ .
⎢⎣ t =a
⎥⎦
Since B(ζ, t + dt ) − B(ζ, t ) is a Bernoulli Random Variable Bi ,
the Oscillation Integral of B(ζ, t ) with respect to B(ζ, t )
equals
t =b
∑
t =a
E [Bi 2 ]
= (2D )(b − a ) ≠ infinitesimal ,
(dx )2 =(2D )dt
violating Riemann’s Oscillation Condition for Integrability of
⎡ t −b
⎤
E ⎢⎢ ∑ B(ζ, t )dB(ζ, t ) ⎥⎥ .
⎢⎣ t =a
⎥⎦
⎡ t −b
⎤
⎢
⎥
Thus, E ⎢ ∑ B(ζ, t )dB(ζ, t ) ⎥ diverges, and is not an Integral.
⎣⎢ t =a
⎦⎥
That is, the first Moment of the Random Variable
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Gauge Institute Journal,
H. Vic Dannon
t −b
∑ B(ζ, t )dB(ζ, t )
t =a
does not exist, and the Random Variable is ill-defined.
Consequently, the Random Variable diverges, and is not an
integral.
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Gauge Institute Journal,
H. Vic Dannon
8.
t =b
∑ B(ζ, t )dB(ζ, t )
depends on the
t =a
choice
of
the
intermediate
points in the partition
Proof:
To converge to an Integral, the Integration sum
⎡ t =b
⎤
⎡ t =b
⎤
⎢
⎥
⎢
E ⎢ ∑ B(ζ, t )dB(ζ , t ) ⎥ = E ⎢ ∑ B(ζ, t )( B(ζ, t + dt ) − B(ζ , t ) ) ⎥⎥
⎢⎣ t =a
⎥⎦
⎢⎣ t =a
⎥⎦
should remain unchanged for any choice of the intermediate
points between B(ζ, t ) , and B(ζ, t + dt ) .
We check three different sets of intermediate points
(1) Intermediate points at
1
2
( B(ζ, t + dt ) + B(ζ, t )) .
The Integration Sum becomes
t =b −dt
∑ 12 ( B(ζ, t + dt ) + B(ζ, t ))( B(ζ, t + dt ) − B(ζ, t )) =
t =a
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Gauge Institute Journal,
H. Vic Dannon
t =b −dt
∑ 12 ( B 2(ζ, t + dt ) − B 2(ζ, t )) ,
=
t =a
=
1
2
( B 2(ζ, b) − B 2(ζ, a )) .
(2) Intermediate points at B(ζ, t )
The Integration Sum is,
t =b
∑ B(ζ, t )( B(ζ, t + dt ) − B(ζ, t ) ) =
t =a
t =b −dt
∑ 12 ( B(ζ, t + dt ) + B(ζ, t ))( B(ζ, t + dt ) − B(ζ, t ) )
t =a
1
2
( B 2 (ζ ,b )−B 2 (ζ ,a ) )
t =b −dt
− 12
B(ζ, t + dt ) − B(ζ , t ) )( B(ζ , t + dt ) − B(ζ, t ) )
∑ (
t =a
Bi 2
Then,
⎡ t =b −dt
⎤
⎢
E ⎢ ∑ B(ζ , t )( B(ζ, t + dt ) − B(ζ , t ) ) ⎥⎥ =
⎣⎢ t =a
⎦⎥
t =b
=
1
E
2
⎡ B 2 (ζ, b ) − B 2 (ζ, a ) ⎤ − 1
E [B 2 ]
⎣⎢
⎦⎥ 2 ∑ i t =a
(dx )2 =(2D )dt
= 12 E ⎡⎢ B 2(ζ, b ) − B 2(ζ, a ) ⎤⎥ − D(b − a ) .
⎣
⎦
(3) Intermediate points at B(ζ, t + dt ) ,
27
Gauge Institute Journal,
H. Vic Dannon
the Integration Sum is,
t =b −dt
∑
B(ζ , t + dt )( B(ζ , t + dt ) − B(ζ , t ) ) =
t =a
t =b −dt
∑ 12 ( B(ζ, t + dt ) + B(ζ, t ))( B(ζ, t + dt ) − B(ζ, t ) )
t =a
1
2
( B 2 (ζ ,b )−B 2 (ζ ,a ) )
t =b −dt
B(ζ, t + dt ) − B(ζ , t ) )( B(ζ, t + dt ) − B(ζ, t ) )
∑ (
+ 12
t =a
Bi 2
Then,
⎡ t =b−dt
⎤
⎢
E ⎢ ∑ B(ζ, t + dt )( B(ζ, t + dt ) − B(ζ , t ) ) ⎥⎥ =
⎣⎢ t =a
⎦⎥
=
1
E
2
⎡ B 2 (ζ, b) − B 2 (ζ, a ) ⎤ +
⎣⎢
⎦⎥
t =b
1
2
∑
t =a
E [Bi 2 ]
(dx )2 =(2D )dt
= 12 E ⎡⎢ B 2 (ζ, b ) − B 2(ζ, a ) ⎤⎥ + D(b − a ) .
⎣
⎦
⎡ t =b
⎤
Each of the three cases, has E ⎢⎢ ∑ B(ζ, t )dB(ζ, t ) ⎥⎥ that differs
⎢⎣ t =a
⎥⎦
from the others by non-infinitesimal number.
Thus,
the
First
Moment
of
t =b
∑ B(ζ, t )dB(ζ, t ) is undefined.
t =a
28
the
Random
Variable
Gauge Institute Journal,
H. Vic Dannon
That is, the Integration Sum is ill-defined, and does not
exist.
Indeed, for any intermediate point
λB(ζ, t + dt ) + (1 − λ)B(ζ, t ) ,
0 ≤ λ ≤ 1,
the integration sum is different
⎡ t =b
⎤
⎢
⎡
⎤
E ⎢ ∑ ⎣ λB(ζ, t + dt ) + (1 − λ)B(ζ, t ) ⎦ ( B(ζ, t + dt ) − B(ζ, t ) ) ⎥⎥
⎢⎣ t =a
⎥⎦
⎡ t =b
⎤
⎢
= λ E ⎢ ∑ B(ζ, t + dt )( B(ζ, t + dt ) − B(ζ , t ) ) ⎥⎥ +
⎢⎣ t =a
⎥⎦
1 E ⎡ B 2 (ζ ,b )−B 2 (ζ ,a ) ⎤ + D (b −a )
2 ⎣⎢
⎦⎥
⎡ t =b
⎤
+(1 − λ) E ⎢⎢ ∑ B(ζ, t )( B(ζ , t + dt ) − B(ζ , t ) ) ⎥⎥ .
⎢⎣ t =a
⎥⎦
1 E ⎡ B 2 (ζ ,b )−B 2 (ζ ,a ) ⎤ −D (b −a )
⎥⎦
2 ⎢⎣
= 12 E ⎡⎢ B 2 (ζ, b) − B 2 (ζ, a ) ⎤⎥ + (λ − 12 )D(b − a ) . ,
⎣
⎦
29
Gauge Institute Journal,
H. Vic Dannon
9.
The
Myth
that
Lebesgue’s
Integral generalizes Riemann’s
Ito’s use of Lebesgue’s Integration, is consistent with his
ignorance of the basics of Integration, and suggests a belief
in the Myth that Lebesgue’s Integration generalizes
Riemann’s . But in [Dan6] we observed that
ƒ Riemann’s Function [Dan6] is Riemann-Integrable over
a Non-Measurable set of Discontinuities,
ƒ the Lebesgue integral cannot be defined over the nonmeasurable
rationals, while the Riemann Integral
may be defined over the rationals,
Thus, we concluded that
Riemann Integral generalizes Lebesgue’s, and the Lebesgueintegrable functions are a subset of the Riemann-integrable
functions.
Riemann’s requirements that
o the oscillation integral must be infinitesimal, and
30
Gauge Institute Journal,
H. Vic Dannon
o the integral may not depend on the choice of the
intermediate points,
are the foundations of any treatment of integration.
The misinterpretation of the ill-defined Random Variable
t =b
∑ B(ζ, t )dB(ζ, t )
t =a
as different Integrals, named after Ito, for λ = 1 , and after
Stratonovich for λ = 12 ,.., misses the point that these
different values prove that there is no integral.
Ito never comprehended why f (t ) of the Wiener Integral
cannot be replaced by a Random Process f (ζ, t ) , and believed
that this can be resolved by applying the “more general”
Lebesgue Integral.
As demonstrated in [Dan7], Lebesgue’s theories of measure
and integration are conducive to delusionary claims.
31
Gauge Institute Journal,
H. Vic Dannon
10.
t =b
∑ B(ζ, t )dB(ζ, t )
violates
the
t =a
Fundamental Theorem of the
Integral Calculus.
The Fundamental Theorem of Calculus guarantees that
Integration and Differentiation are well defined inverse
operations, that when applied consecutively yield the
original function.
It is well known to hold in the Calculus of Limits under
given
conditions.
In
the
Infinitesimal
Calculus,
Fundamental Theorem holds with almost no conditions.
In [Dan5], we established
10.1 Let f (x ) be Hyper-real Integrable on[a, b ]
u =x
p.v.D ∫ f (u )du = f (x )
Then, for any x ∈ [a, b ] ,
u =a
32
the
Gauge Institute Journal,
H. Vic Dannon
t =x
Consequently, if F (x ) =
∫
f (t )dt ,
t =a
t =b
∫
t =a
t =b
f (t )dt = ∫ dF (t ) =F (b) − F (a )
t =a
Thus, the Fundamental Theorem of Calculus requires that
we have
t =b
∫
t =a
t =b
B(ζ, t )dB(ζ , t ) =
∫ d ( 12 B 2(ζ, t ) ) = 12 B 2(ζ, b) − 12 B 2(ζ, a ) ,
t =a
and
⎡ t =b
⎤
⎢
⎥
E ⎢ ∫ B(ζ, t )dB(ζ , t ) ⎥ = 12 E ⎡⎢ B 2 (ζ, b) − B 2 (ζ , a ) ⎤⎥ .
⎣
⎦
⎢ t =a
⎥
⎣
⎦
Instead, Ito has an additional λ -dependent term
⎡ t =b
⎤
⎢
⎥
E ⎢ ∫ B(ζ, t )dB(ζ, t ) ⎥ = 12 E ⎡⎢ B 2(ζ, b) − B 2 (ζ, a ) ⎤⎥ + (λ − 12 )D(b − a )
⎣
⎦
⎢ t =a
⎥
⎣
⎦
Thus, Ito’s ill-defined Integral violates the Fundamental
Theorem of the integral Calculus.
33
Gauge Institute Journal,
H. Vic Dannon
11.
Ito’s Process and Ito’s Formula
11.1 Ito’s Process
is defined as the solution X (ζ, t ) of the differential Equations
dX (ζ, t ) = f (ζ, t )dB(ζ, t ) + g(ζ, t )dt .
Then,
E ⎡⎣ dX (ζ, t ) − f (ζ, t )dB(ζ, t ) − g(ζ, t )dt ⎤⎦ = 0 ,
and
2⎤
⎡
E ⎢ dX (ζ , t ) − f (ζ , t )dB(ζ , t ) − g(ζ , t )dt ⎥ = 0
⎣
⎦
Summation over time of the first Moment equation yields
the meaningless,
⎡
⎤
⎢
⎥
τ
=
t
τ =t
⎢
⎥
E ⎢ X (ζ , t ) − X (ζ , 0) − ∑ f (ζ , τ )dB(ζ , τ ) − ∫ g(ζ , τ )d τ ⎥ ≈ 0 .
⎢
⎥
τ
=0
τ =0
⎢
⎥
⎢⎣
⎥⎦
ill-defined Ito Integral
11.2 Ito’s Formula
Let B = B(ζ , t ) be a Random Walk
X = X (ζ, t ) be an Ito Process,
34
Gauge Institute Journal,
H. Vic Dannon
and F = F (X , t ) .
Then,
dX = f (ζ, t )dB(ζ, t ) + g(ζ, t )dt ,
(dX )2 = f 2 (dB )2 + 2 fg(dB )dt + g 2 (dt )2 ,
E [(dX )2 ] = E ⎡⎢ f 2(dB )2 + 2 fg(dB )dt + g 2(dt )2 ⎤⎥
⎣
⎦
= E [ f 2 ] E [(dB )2 ] + 2E [ fg ] E [dB ]dt + E [g 2 ](dt )2
N
(dx )2 =(2D )dt
0
≈ E [ f 2 ](2D )dt
Substituting into
⎡
∂F
∂F ⎤⎥
1 ∂2F
⎢
E dF −
dX −
(dX )2 −
dt ≈ 0 ,
⎢
⎥
2
∂
∂
2
X
t
∂
X
⎣
⎦
we have, Ito’s Formula
⎡
⎪⎧ ∂F
∂F
∂2F 2
∂F ⎪⎪⎫ ⎥⎤
E ⎢ dF −
fdB − ⎪⎨
g+
f D+
⎬ dt ⎥ ≈ 0 .
⎢
2
⎪
⎪
∂
∂
∂
X
X
t
∂
X
⎪⎩
⎪⎭ ⎦
⎣
The Formula is stated loosely, ([Oksendal], [Kuo]), dropping
the Expectation operator.
In particular, the Variance of the Random Process has to be
infinitesimal. That is,
⎡
⎧ ∂F
⎫
⎪
∂F
∂2F 2
∂F ⎪
⎢
⎪
⎪
E ⎢ dF −
fdB − ⎨
g+
f D+
⎬ dt
2
⎪
⎪⎪
X
t
∂X
∂
∂
⎢
X
∂
⎪
⎩
⎭
⎣
35
2
⎤
⎥
⎥ ≈ 0.
⎥
⎦
Gauge Institute Journal,
H. Vic Dannon
So far, the Formula was applied after a summation over time
that generated an Integral Formula with the meaningless
Ito Integral,
∂F
∑ ∂X f (ζ, t )dB(ζ, t ) .
t
That Integral Formula is referred to as “The Fundamental
Theorem of the Stochastic Integral Calculus”…
11.2 Example of the Ito Formula
F (X (ζ, t ), t ) = B 2(ζ, t ) ,
Taking
X (ζ, t ) = B(ζ, t ) ,
f (ζ, t ) = 1 ;
g(ζ, t ) = 0
⎫ ⎤
⎡
⎧
⎪
⎪
⎪
⎪
⎢
2
⎪
⎪
∂F
∂F
∂F 2
∂F ⎪ ⎥⎥
2
⎢
⎪
E ⎢ d (B ) −
f D+
⎬ dt ⎥ ≈ 0
Nf dB − ⎨⎪ ∂X gN + ∂X 2 N
X
t
∂
∂
N1
N ⎪⎪ ⎥
⎢
⎪
N
0
1
⎪
⎪⎭ ⎥⎦
⎢⎣
2B
0 ⎪
⎪⎩
2
E ⎡⎢ d (B 2 ) − 2BdB − 2Ddt ⎤⎥ ≈ 0 .
⎣
⎦
This leads the books to d(B 2 ) = 2BdB + 2Ddt
which is summed up over time to yield the “Fundamental
2
etc.”, B (ζ , t ) =
τ =t
∑ B(ζ, τ )dB(ζ, τ ) + 2(D )t .
τ
=0
ill-defined Ito Integral
36
Gauge Institute Journal,
H. Vic Dannon
References
[Dan1] Dannon, H. Vic, “Well-Ordering of the Reals, Equality of all
Infinities, and the Continuum Hypothesis” in Gauge Institute Journal
Vol.6 No 2, May 2010;
[Dan2] Dannon, H. Vic, “Infinitesimals” in Gauge Institute Journal
Vol.6 No 4, November 2010;
[Dan3] Dannon, H. Vic, “Infinitesimal Calculus” in Gauge Institute
Journal Vol.7 No 4, November 2011;
[Dan4]
Dannon, H. Vic,
“The Delta Function” in Gauge Institute
Journal Vol.8 No 1, February 2012;
[Dan5] Dannon, H. Vic, “Infinitesimal Calculus of Random Processes”
posted to www.gauge-institute.org
[Dan6] Dannon, H. Vic, “Lebesgue Integration” in Gauge Institute
Journal Vol.7 No 1, February 2011;
[Dan7]
Dannon, H. Vic,
“Khintchine’s Constant and Lebesgue’s
Measure” posted to www.gauge-institute.org
[Gard]
Thomas
Gard,
“Introduction
to
Stochastic
Differential
Equations” Dekker, 1988
[Gardiner] Crispin Gardiner “Stochastic Methods” fourth Edition,
Springer, 2009.
[Gnedenko] B. V. Gnedenko, “The Theory of Probability”, Second
Edition, Chelsea, 1963.
[Grimmett/Welsh]
Geoffrey
Grimmett
“Probability, an introduction”, Oxford, 1986.
37
and
Dominic
Welsh,
Gauge Institute Journal,
[Hsu]
H. Vic Dannon
Hwei Hsu, “Probability, Random Variables, & Random
Processes”, Schaum’s Outlines, McGraw-Hill, 1997.
[Ito] Kiyosi Ito, “On Stochastic Differential Equations” Memoires of
the American Mathematical Society, No 4. American Mathematical
Society, 1951
[Karlin/Taylor] Howard Taylor, Samuel Karlin, “An Introduction to
Stochastic Modeling”, Academic Press, 1984.
[Kuo] Hui Hsiung Kuo, “Introduction
to Stochastic Integration”,
Springer, 2006.
[Larson/Shubert] Harold Larson, Bruno Shubert, “Probabilistic Models
in Engineering Sciences, Volume II, Random Noise, Signals, and
Dynamic Systems”, Wiley, 1979.
[Oksendal] Bernt Oksendal, “Stochastic Differential Equations, An
Introduction with Applications”, Fourth Corrected Printing of the Sixth
Edition, Springer, 2007.
[Stratonovich] R. L. Stratonovich, “Conditional Markov Processes and
their Applications to the Theory of Optimal Control” Elsevier, 1968
38