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Walking randomly in the realm of partial
differential equations and probability theory
The Laplace operator and Brownian motion
Department of mathematical sciences
12 March 2014
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2
Introduction
We study the following Cauchy problem (heat or diffusion equation)
(
∂t u = 12 ∆u
(x, t) ∈ Rd × (0, ∞)
,
(1)
u(x, 0) = δ0 (x) x ∈ Rd
where δ0 is the Dirac delta centered at the origin. A solution of (1)
is called a fundamental solution.
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Fundamental solution
Let
φ̂(ξ) := F(φ)(ξ) =
ˆ
1
d
(2π) 2
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e−iξ·x φ(x) dx,
Rd
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Fundamental solution
Let
φ̂(ξ) := F(φ)(ξ) =
ˆ
1
d
(2π) 2
e−iξ·x φ(x) dx,
Rd
then F(∆φ)(ξ) = −|ξ|2 F(φ)(ξ) (integration by parts).
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Fundamental solution
Let
φ̂(ξ) := F(φ)(ξ) =
ˆ
1
d
(2π) 2
e−iξ·x φ(x) dx,
Rd
then F(∆φ)(ξ) = −|ξ|2 F(φ)(ξ) (integration by parts). Taking the
Fourier transform of (1) yields
∂ û
1
= − |ξ|2 û
∂t
2
1
2
û(ξ, t) = Ce− 2 t|ξ| ,
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Fundamental solution
Let
φ̂(ξ) := F(φ)(ξ) =
ˆ
1
e−iξ·x φ(x) dx,
d
(2π) 2
Rd
then F(∆φ)(ξ) = −|ξ|2 F(φ)(ξ) (integration by parts). Taking the
Fourier transform of (1) yields
∂ û
1
= − |ξ|2 û
∂t
2
1
2
û(ξ, t) = Ce− 2 t|ξ| ,
d
and using that F(δ0 ) = (2π)− 2 we get
d
1
2
û(ξ, t) = (2π)− 2 e− 2 t|ξ| .
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(2)
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Fundamental solution
Now, take the Fourier inverse of (2)
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Fundamental solution
Now, take the Fourier inverse of (2) (don’t worry û(·, t) ∈ S(Rd ))
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Fundamental solution
Now, take the Fourier inverse of (2) (don’t worry û(·, t) ∈ S(Rd ))
d
1
2
u(x, t) =F −1 ((2π)− 2 e− 2 t|·| )(x)
ˆ
1
d
2
− d2
=(2π)
eiξ·x (2π)− 2 e− 2 t|ξ| dξ
Rd
=
1
d
|x|2
− 2t
e
(3)
.
(2πt) 2
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Fundamental solution
Now, take the Fourier inverse of (2) (don’t worry û(·, t) ∈ S(Rd ))
d
1
2
u(x, t) =F −1 ((2π)− 2 e− 2 t|·| )(x)
ˆ
1
d
2
− d2
=(2π)
eiξ·x (2π)− 2 e− 2 t|ξ| dξ
Rd
=
1
d
|x|2
− 2t
e
(3)
.
(2πt) 2
Hopefully this is a well-known function!
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Fundamental solution
Now, take the Fourier inverse of (2) (don’t worry û(·, t) ∈ S(Rd ))
d
1
2
u(x, t) =F −1 ((2π)− 2 e− 2 t|·| )(x)
ˆ
1
d
2
− d2
=(2π)
eiξ·x (2π)− 2 e− 2 t|ξ| dξ
Rd
=
1
d
|x|2
− 2t
e
(3)
.
(2πt) 2
Hopefully this is a well-known function! It is e.g. the probability
density function of the normal distribution (with µ = 0 and Σ = tI).
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Normal distribution (or Gaussian
distribution) with µ = 0 and t = 1
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Family of probability measures
Consider the family {Qt , t ≥ 0} where "Qt ( dx) = qt (x) dx".
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Family of probability measures
Consider the family {Qt , t ≥ 0} where "Qt ( dx) = qt (x) dx". Then
this family is a family of probability measures if qt (x) := u(x, t)
satisfies
i) qt (x) is non-negative; and
´
ii) Rd qt (x) dx = 1,
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Family of probability measures
Consider the family {Qt , t ≥ 0} where "Qt ( dx) = qt (x) dx". Then
this family is a family of probability measures if qt (x) := u(x, t)
satisfies
i) qt (x) is non-negative; and
´
ii) Rd qt (x) dx = 1,
and it does.
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Family of probability measures
Consider the family {Qt , t ≥ 0} where "Qt ( dx) = qt (x) dx". Then
this family is a family of probability measures if qt (x) := u(x, t)
satisfies
i) qt (x) is non-negative; and
´
ii) Rd qt (x) dx = 1,
and it does. In addition, qt (x) satisfies
iii) qt+s (x) = (qt ∗ qs )(x); and
iv) Qt is weakly convergent to δ0 .
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Family of probability measures
Consider the family {Qt , t ≥ 0} where "Qt ( dx) = qt (x) dx". Then
this family is a family of probability measures if qt (x) := u(x, t)
satisfies
i) qt (x) is non-negative; and
´
ii) Rd qt (x) dx = 1,
and it does. In addition, qt (x) satisfies
iii) qt+s (x) = (qt ∗ qs )(x); and
iv) Qt is weakly convergent to δ0 .
(All of these properties can be proved using (3).)
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Random variable
Definition
(Ω, F, P) is called a probability space.
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Random variable
Definition
(Ω, F, P) is called a probability space.
´ −x 2 /2
If Ω = R, F = B(R) and P(A) = A e√
dm, then we have the
2π
normal distribution with µ = 0 and σ 2 = 1.
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Random variable
Definition
(Ω, F, P) is called a probability space.
´ −x 2 /2
If Ω = R, F = B(R) and P(A) = A e√
dm, then we have the
2π
normal distribution with µ = 0 and σ 2 = 1.
Definition
X : Ω → Rd is a random variable if X is F measurable (i.e.,
(F, B(Rd )) measurable).
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Random variable
Definition
(Ω, F, P) is called a probability space.
´ −x 2 /2
If Ω = R, F = B(R) and P(A) = A e√
dm, then we have the
2π
normal distribution with µ = 0 and σ 2 = 1.
Definition
X : Ω → Rd is a random variable if X is F measurable (i.e.,
(F, B(Rd )) measurable).
Think of this as assigning a number to each outcome of an
experiment.
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Stochastic process
Definition
A stochastic process is parametrized collection of random
variables
{Xt }t∈T
defined on a probability space (Ω, F, P) and assuming values in
Rd .
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Stochastic process
Definition
A stochastic process is parametrized collection of random
variables
{Xt }t∈T
defined on a probability space (Ω, F, P) and assuming values in
Rd .
Note that Xt = Xt (ω) where ω ∈ Ω.
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Stochastic process
Definition
A stochastic process is parametrized collection of random
variables
{Xt }t∈T
defined on a probability space (Ω, F, P) and assuming values in
Rd .
Note that Xt = Xt (ω) where ω ∈ Ω. Here it is useful to think of t as
time, and ω as a particle (or an experiment). Then t 7→ Xt (ω) would
represent the position (or the result) as a function of time t of the
particle (experiment) ω.
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Transition probabilities
Definition
For each 0 ≤ s ≤ t < ∞, B ∈ B(Rd ), x ∈ Rd , we define
Ps,t (x, B) = P(Xt ∈ B|Xs = x)
as the transition probability.
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Transition probabilities
Definition
For each 0 ≤ s ≤ t < ∞, B ∈ B(Rd ), x ∈ Rd , we define
Ps,t (x, B) = P(Xt ∈ B|Xs = x)
as the transition probability.
Note that Ps,t gives the probability of going from the point x at time
s to the set B at time t.
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Chapman-Kolmogorov equations
If we let
d
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|y −x|2
− 2(t−s)
Ps,t (x, dy ) = qt−s (y − x) dy = (2π(t − s))− 2 e
dy ,
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Chapman-Kolmogorov equations
If we let
d
or equivalently
|y −x|2
− 2(t−s)
Ps,t (x, dy ) = qt−s (y − x) dy = (2π(t − s))− 2 e
dy ,
ˆ
qt−s (y − x) dy ,
Ps,t (x, B) =
B
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Chapman-Kolmogorov equations
If we let
d
or equivalently
|y −x|2
− 2(t−s)
Ps,t (x, dy ) = qt−s (y − x) dy = (2π(t − s))− 2 e
dy ,
ˆ
qt−s (y − x) dy ,
Ps,t (x, B) =
B
then the transition probabilities satisfies (remember that
qt+s = qt ∗ qs )
ˆ
Ps,t (y , B)Pr ,s (x, dy )
Pr ,t (x, B) =
(4)
Rd
for all 0 ≤ r ≤ s ≤ t < ∞, x ∈ Rd , and B ∈ B(Rd ).
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Chapman-Kolmogorov equations
If we let
d
or equivalently
|y −x|2
− 2(t−s)
Ps,t (x, dy ) = qt−s (y − x) dy = (2π(t − s))− 2 e
dy ,
ˆ
qt−s (y − x) dy ,
Ps,t (x, B) =
B
then the transition probabilities satisfies (remember that
qt+s = qt ∗ qs )
ˆ
Ps,t (y , B)Pr ,s (x, dy )
Pr ,t (x, B) =
(4)
Rd
for all 0 ≤ r ≤ s ≤ t < ∞, x ∈ Rd , and B ∈ B(Rd ). Note that (4)
says that the probability of going from x to dy is independent of
going from dy to B.
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Finite-dimensional distribution
For 0 ≤ t1 ≤ . . . ≤ tn we define a measure νt1 ,...,tn by
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Finite-dimensional distribution
For 0 ≤ t1 ≤ . . . ≤ tn we define a measure νt1 ,...,tn by
νt1 ,...,tn (B1 × · · · × Bn )
ˆ
=
qt1 (x1 ) · · · qtn −tn−1 (xn − xn−1 ) dx1 . . . dxn
B1 ×...×Bn
ˆ
− d2
=
(2π(t1 ))
e
(5)
|x |2
1)
− 2(t1
B1 ×...×Bn
d
· · · (2π(tn − tn−1 ))− 2 e
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−
|xn −xn−1 |2
2(tn −tn−1 )
dx1 . . . dxn .
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Finite-dimensional distribution
For 0 ≤ t1 ≤ . . . ≤ tn we define a measure νt1 ,...,tn by
νt1 ,...,tn (B1 × · · · × Bn )
ˆ
=
qt1 (x1 ) · · · qtn −tn−1 (xn − xn−1 ) dx1 . . . dxn
B1 ×...×Bn
ˆ
− d2
=
(2π(t1 ))
e
(5)
|x |2
1)
− 2(t1
B1 ×...×Bn
d
· · · (2π(tn − tn−1 ))− 2 e
−
|xn −xn−1 |2
2(tn −tn−1 )
dx1 . . . dxn .
Note that νt1 ,...,tn answers the question "what is the probability of
Xt1 ∈ B1 , and, . . . , and Xtn ∈ Bn ?". Hence, it is closely related to
transition probabilities.
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Kolmogorov’s existence theorem
Theorem
Given a family of probability measures {νt1 ,...,tn , ti ∈ R+ and n ∈ N}
satisfying the Kolmogorov consistency criteria. Then there exists a
probability space (Ω, F, P) and a stochastic process {Xt }t≥0 on Ω;
Xt : Ω → Rd such that
νt1 ,...,tn (B1 × · · · × Bn ) = P(Xt1 ∈ B1 , . . . , Xtn ∈ Bn ),
for all ti ∈ R+ , k ∈ N and all Borel sets Bi .
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Existence of a stochastic process
We "know" that νt1 ,...,tn defined by (5) satisfies Kolmogorov’s
consistency criteria. Hence, there exists a stochastic process
{Xt }t≥0 on Ω such that
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Existence of a stochastic process
We "know" that νt1 ,...,tn defined by (5) satisfies Kolmogorov’s
consistency criteria. Hence, there exists a stochastic process
{Xt }t≥0 on Ω such that
νt1 ,...,tn (B1 × · · · × Bn ) = P(Xt1 ∈ B1 , . . . , Xtn ∈ Bn ),
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Existence of a stochastic process
We "know" that νt1 ,...,tn defined by (5) satisfies Kolmogorov’s
consistency criteria. Hence, there exists a stochastic process
{Xt }t≥0 on Ω such that
νt1 ,...,tn (B1 × · · · × Bn ) = P(Xt1 ∈ B1 , . . . , Xtn ∈ Bn ),
that is, the stochastic process Xt has νt1 ,...,tn as its finite
dimensional distribution.
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Brownian motion
Definition
A stochastic process Bt : Ω × [0, ∞) → Rd is called Brownian
motion if
i) B0 = 0
ii) Btn − Btn−1 is N (0, (tn − tn−1 )I)
iii) Bt1 , . . . , Btn − Btn−1 is independent
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Brownian motion
Definition
A stochastic process Bt : Ω × [0, ∞) → Rd is called Brownian
motion if
i) B0 = 0
ii) Btn − Btn−1 is N (0, (tn − tn−1 )I)
iii) Bt1 , . . . , Btn − Btn−1 is independent
The process Xt with finite dimensional distribution given by (5)
satisfies all of these axioms!
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Brownian motion
Definition
A stochastic process Bt : Ω × [0, ∞) → Rd is called Brownian
motion if
i) B0 = 0
ii) Btn − Btn−1 is N (0, (tn − tn−1 )I)
iii) Bt1 , . . . , Btn − Btn−1 is independent
The process Xt with finite dimensional distribution given by (5)
satisfies all of these axioms! That is, we have constructed
Brownian motion using the fundamental solution of (1).
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Why??
Let us write down the finite dimensional distribution
ˆ
|x |2
d − 1
(2π(t1 ))− 2 e 2(t1 )
B1 ×...×Bn
− d2
· · · (2π(tn − tn−1 ))
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e
−
|xn −xn−1 |2
2(tn −tn−1 )
dx1 . . . dxn .
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Why??
Let us write down the finite dimensional distribution
ˆ
|x |2
d − 1
(2π(t1 ))− 2 e 2(t1 )
B1 ×...×Bn
− d2
· · · (2π(tn − tn−1 ))
e
−
|xn −xn−1 |2
2(tn −tn−1 )
dx1 . . . dxn .
Consider the first axiom; B0 = 0.
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Why??
Let us write down the finite dimensional distribution
ˆ
|x |2
d − 1
(2π(t1 ))− 2 e 2(t1 )
B1 ×...×Bn
− d2
· · · (2π(tn − tn−1 ))
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e
−
|xn −xn−1 |2
2(tn −tn−1 )
dx1 . . . dxn .
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Why??
Let us write down the finite dimensional distribution
ˆ
|x |2
d − 1
(2π(t1 ))− 2 e 2(t1 )
B1 ×...×Bn
− d2
· · · (2π(tn − tn−1 ))
e
−
|xn −xn−1 |2
2(tn −tn−1 )
dx1 . . . dxn .
Consider the second axiom; Btn − Btn−1 is N (0, (tn − tn−1 )I).
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Why??
Let us write down the finite dimensional distribution
ˆ
|x |2
d − 1
(2π(t1 ))− 2 e 2(t1 )
B1 ×...×Bn
− d2
· · · (2π(tn − tn−1 ))
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e
−
|xn −xn−1 |2
2(tn −tn−1 )
dx1 . . . dxn .
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Why??
Let us write down the finite dimensional distribution
ˆ
|x |2
d − 1
(2π(t1 ))− 2 e 2(t1 )
B1 ×...×Bn
− d2
· · · (2π(tn − tn−1 ))
e
−
|xn −xn−1 |2
2(tn −tn−1 )
dx1 . . . dxn .
Consider the third axiom; Bt1 , . . . , Btn − Btn−1 is independent.
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Figure of Brownian motion in R1
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Properties of Brownian motion
Brownian motion (or a version of Brownian motion) has these
properties
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Properties of Brownian motion
Brownian motion (or a version of Brownian motion) has these
properties
i) for all 0 ≤ t1 , . . . , ≤ tn the random variable Z = (Bt1 , . . . , Btn ) is
multinormal distributed;
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Properties of Brownian motion
Brownian motion (or a version of Brownian motion) has these
properties
i) for all 0 ≤ t1 , . . . , ≤ tn the random variable Z = (Bt1 , . . . , Btn ) is
multinormal distributed;
ii) the function t 7→ Bt (ω) is continuous;
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Properties of Brownian motion
Brownian motion (or a version of Brownian motion) has these
properties
i) for all 0 ≤ t1 , . . . , ≤ tn the random variable Z = (Bt1 , . . . , Btn ) is
multinormal distributed;
ii) the function t 7→ Bt (ω) is continuous;
iii) the function t 7→ Bt (ω) continuous is nowhere differentiable;
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Properties of Brownian motion
Brownian motion (or a version of Brownian motion) has these
properties
i) for all 0 ≤ t1 , . . . , ≤ tn the random variable Z = (Bt1 , . . . , Btn ) is
multinormal distributed;
ii) the function t 7→ Bt (ω) is continuous;
iii) the function t 7→ Bt (ω) continuous is nowhere differentiable;
iv) the function t 7→ Bt (ω) continuous has infinite variation on each
interval (of t); and
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Properties of Brownian motion
Brownian motion (or a version of Brownian motion) has these
properties
i) for all 0 ≤ t1 , . . . , ≤ tn the random variable Z = (Bt1 , . . . , Btn ) is
multinormal distributed;
ii) the function t 7→ Bt (ω) is continuous;
iii) the function t 7→ Bt (ω) continuous is nowhere differentiable;
iv) the function t 7→ Bt (ω) continuous has infinite variation on each
interval (of t); and
v)
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1
c Bc 2 t
is also a Brownian motion; it is scalar invariant.
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Pollen particles
In 1827, Robert Brown studied pollen grains suspended in water
under a microscope.
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Pollen particles
He noticed that the path created by a single pollen particle was
very irregular.
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Pollen particles
He noticed that the path created by a single pollen particle was
very irregular. To exclude life-like motion, the experiment was
repeated with non-organic material. The result was, indeed, the
same.
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Pollen particles
He noticed that the path created by a single pollen particle was
very irregular. To exclude life-like motion, the experiment was
repeated with non-organic material. The result was, indeed, the
same.
In 1905, Albert Einstein considered the density of Brownian
particles. He showed that the density satisfies (1) up to some
constant.
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Pollen particles
He noticed that the path created by a single pollen particle was
very irregular. To exclude life-like motion, the experiment was
repeated with non-organic material. The result was, indeed, the
same.
In 1905, Albert Einstein considered the density of Brownian
particles. He showed that the density satisfies (1) up to some
constant.
We then know that the path of one of these particles ω will be given
by (B1 (t, ω), B2 (t, ω)); two dimensional Brownian motion.
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Pollen particles
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Kolmogorov’s forward equation
(Also called the Fokker-Planck equation.)
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Kolmogorov’s forward equation
(Also called the Fokker-Planck equation.)
Assume that Bt has a nice, smooth transition probability density
ps,t (x, y ), that is,
ˆ
P(Bt ∈ B|B0 = 0) =
ps,t (0, y ) dy
∀B ∈ B(Rd ).
B
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Kolmogorov’s forward equation
(Also called the Fokker-Planck equation.)
Assume that Bt has a nice, smooth transition probability density
ps,t (x, y ), that is,
ˆ
P(Bt ∈ B|B0 = 0) =
ps,t (0, y ) dy
∀B ∈ B(Rd ).
B
Then this density will satisfy
(
∂t p = 12 ∆y p
(y , t) ∈ Rd × (0, ∞)
.
p0 (0, y ) = δ0 (y ) (y , t) ∈ Rd × {0}
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Fractional Laplace
We turn our attention to the following Cauchy problem (heat or
diffusion equation)
(
α
∂t u = −(−∆) 2 u (x, t) ∈ Rd × (0, ∞)
,
(6)
u(x, 0) = δ0 (x)
x ∈ Rd
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Fractional Laplace
We turn our attention to the following Cauchy problem (heat or
diffusion equation)
(
α
∂t u = −(−∆) 2 u (x, t) ∈ Rd × (0, ∞)
,
(6)
u(x, 0) = δ0 (x)
x ∈ Rd
where δ0 is the Dirac delta centered at the origin,
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24
Fractional Laplace
We turn our attention to the following Cauchy problem (heat or
diffusion equation)
(
α
∂t u = −(−∆) 2 u (x, t) ∈ Rd × (0, ∞)
,
(6)
u(x, 0) = δ0 (x)
x ∈ Rd
α
where δ0 is the Dirac delta centered at the origin, and −(−∆) 2 is
defined by the Fourier transform
α
F(−(−∆) 2 φ)(ξ) = −|ξ|α F(φ)(ξ)
(7)
for all φ ∈ S(Rd ).
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24
Fractional Laplace
We turn our attention to the following Cauchy problem (heat or
diffusion equation)
(
α
∂t u = −(−∆) 2 u (x, t) ∈ Rd × (0, ∞)
,
(6)
u(x, 0) = δ0 (x)
x ∈ Rd
α
where δ0 is the Dirac delta centered at the origin, and −(−∆) 2 is
defined by the Fourier transform
α
F(−(−∆) 2 φ)(ξ) = −|ξ|α F(φ)(ξ)
(7)
for all φ ∈ S(Rd ). Note that the (7) is consistent with the Fourier
transform of ∆.
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25
Lévy processes
By more theoretically advanced theory, we can do similar
computations as in the case of the Laplace operator.
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25
Lévy processes
By more theoretically advanced theory, we can do similar
computations as in the case of the Laplace operator. These
computations will show that we can obtain α-stable isotropic
(rotationally invariant) Lévy processes from the fundamental
solution of (6),
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25
Lévy processes
By more theoretically advanced theory, we can do similar
computations as in the case of the Laplace operator. These
computations will show that we can obtain α-stable isotropic
(rotationally invariant) Lévy processes from the fundamental
solution of (6), where the fundamental solution is given by
d
α
u(x, t) = F −1 ((2π)− 2 e−t|ξ| )(x)
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α ∈ (0, 2).
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25
Lévy processes
By more theoretically advanced theory, we can do similar
computations as in the case of the Laplace operator. These
computations will show that we can obtain α-stable isotropic
(rotationally invariant) Lévy processes from the fundamental
solution of (6), where the fundamental solution is given by
d
α
u(x, t) = F −1 ((2π)− 2 e−t|ξ| )(x)
α ∈ (0, 2).
Observe that if we take α = 2 in the above equation, we get
Brownian motion up to some constant (there is a one-half missing).
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26
α-stable distributions with µ = 0
Symmetric α−stable densities, β = 0, γ = 1, δ = 0
α = 0.5
α = 1.0
α = 1.5
α= 2.0
0.6
0.5
0.4
0.3
0.2
0.1
0
−5
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−4
−3
−2
−1
0
1
2
3
4
5
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27
Figures of Lévy processes
Picture due to A. Meucci (2009).
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28
Figures of Lévy processes
Picture due to A. Meucci (2009).
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29
Figures of Lévy processes
Picture due to A. Meucci (2009).
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30
Figures of Lévy processes
Picture due to A. Meucci (2009).
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31
The Black-Scholes option pricing model
Let us look at an European call option. That is, the right to buy one
stock for K NOK at a fixed time T > t. We call K the strike price
(the agreed price) and St the spot price (the price of the stock at
time t). If we are lucky we earn ST − K NOK, so the pay-off is
max{ST − K , 0}.
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31
The Black-Scholes option pricing model
Let us look at an European call option. That is, the right to buy one
stock for K NOK at a fixed time T > t. We call K the strike price
(the agreed price) and St the spot price (the price of the stock at
time t). If we are lucky we earn ST − K NOK, so the pay-off is
max{ST − K , 0}.
The problem is how much does this European call option cost? Or,
how do we get a good estimate on St ?
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20
03
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The Black-Scholes option pricing model
350 Stock price NAS 300 250 200 150 Stock price NAS 100 50 0 www.ntnu.no
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33
The Black-Scholes option pricing model
In 1997, Fischer Black and Myron Scholes won the Nobel Prize in
Economics for the equation
(
σ2 2 ∂ 2 V
∂V
∂V
∂t + 2 S ∂S 2 + rS ∂S − rV = 0 t ∈ [0, T ) ,
V (S, T ) = max{S − K , 0}
t =T
where V = V (S, t) is the price of the option, r is the risk-free
interest rate, and σ is the volatility of the stock.
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33
The Black-Scholes option pricing model
In 1997, Fischer Black and Myron Scholes won the Nobel Prize in
Economics for the equation
(
σ2 2 ∂ 2 V
∂V
∂V
∂t + 2 S ∂S 2 + rS ∂S − rV = 0 t ∈ [0, T ) ,
V (S, T ) = max{S − K , 0}
t =T
where V = V (S, t) is the price of the option, r is the risk-free
interest rate, and σ is the volatility of the stock.
The solution of this problem is given by the Feynman-Kac formula
h
i
V (S, t) = E e−r (T −t) max{STt,x − K , 0} .
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34
The Black-Scholes option pricing model
In the previous slide, St was actually modelled as geometric
Brownian motion:
dSt = rSt dt + σSt dBt .
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34
The Black-Scholes option pricing model
In the previous slide, St was actually modelled as geometric
Brownian motion:
dSt = rSt dt + σSt dBt .
This model has a lot of weaknesses. One crucial example is that
Browninan motion has a continuous version; so, this model cannot
model sudden jumps in price
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34
The Black-Scholes option pricing model
In the previous slide, St was actually modelled as geometric
Brownian motion:
dSt = rSt dt + σSt dBt .
This model has a lot of weaknesses. One crucial example is that
Browninan motion has a continuous version; so, this model cannot
model sudden jumps in price (which we know occurs in real life).
Furthermore, since the tail of a gaussian distribution is very thin,
the probability of extreme events is very low.
This is where Lévy processes enter. We allow sudden
discontinuous jumps in such processes, and this is a very useful
tool when modelling stock prices.
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35
The Black-Scholes option pricing model
As a consequence of this fact, it is common nowadays (at least in
finance) to add
ˆ
u(x + y , t) − u(x, t) − (ey − 1)∂x u(x, t)ν( dy )
|y |>0
to the Black-Scholes equation.
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35
The Black-Scholes option pricing model
As a consequence of this fact, it is common nowadays (at least in
finance) to add
ˆ
u(x + y , t) − u(x, t) − (ey − 1)∂x u(x, t)ν( dy )
|y |>0
to the Black-Scholes equation. Note that ν is a positive Radon
measure at least satisfying
ˆ
min{|y |2 , 1}ν( dy ).
R\{0}
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35
The Black-Scholes option pricing model
As a consequence of this fact, it is common nowadays (at least in
finance) to add
ˆ
u(x + y , t) − u(x, t) − (ey − 1)∂x u(x, t)ν( dy )
|y |>0
to the Black-Scholes equation. Note that ν is a positive Radon
measure at least satisfying
ˆ
min{|y |2 , 1}ν( dy ).
R\{0}
The solution is still given by a formula similar to Feynman-Kac’, but
St is now a Lévy process.
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