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Stable Process
3. Stable Random Processes and Stochastic Intergrals
August, 2006
1. Stable stochastic processes.
2. Definition of stable integrals as a stochastic process.
3. α-stable random measures.
4. Constructive definition of stable integrals.
5. Properties of stable integrals.
6. Examples.
7. Sub-Gaussian process and sub-stable process.
8. Series representation for α-stable random measures.
9. condition S.
10. Poisson representation.
1
Stable stochastic processes
The finite dimensional distribution of a stochastic process {X(t), t ∈ T } are the distributions of the
vectors
X(t1 ), . . . , X(td ) , t1 , . . . , td ∈ T, d ≥ 1.
Definition A stochastic process {X(t), t ∈ T } is (strictly/symmetric) stable if all its finite dimentional distributions are (strictly/symmetric) stable. By consistancy, they must have the same index
of stability α, and we use the term α-stable stochastic process here.
Theorem 1.1 Let {X(t), t ∈ T } be a stochastic process.
(a) {X(t), t ∈ T } is strictly stable if and only if all linear combinations are strictly stable.
(b) {X(t), t ∈ T } is symmetric stable if and only if all linear combinations are symmetric stable.
(c) {X(t), t ∈ T } is α-stable if and only if all linear combinations are α-stable, given α ≥ 1.
Example α-stable Lev́y motion. A stochastic process {X(t), t ≥ 0} is called (standard) α-stable
Lev́y motion if
(1) X(0) = 0 a.s.
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(2) X has independent increments.
(3) X(t)−X(s) ∼ Sα (t−s)1/α , β, 0 for any 0 ≤ s < t < ∞. and for some 0 < α ≤ 2, −1 ≤ β ≤ 1.
NOTE: It is a Brownian motion when α = 2. The role that α-stble Lev́y motion plays among
α-stable process is similar to the role that Browian motion plays among Gaussian process.
2
Definition of stable integrals as a stochastic process
Suppose f is non-random function defined on the set E, I(·) is an operator, which can be exercised
onto f . The goal is to make the connection of the collections between {I(f ), f ∈ F } and a stochastic
process. That is to say, the element, such as I(f ), should be treated as a random variable, and the
stochastic process is indexed by f ∈ F .
Billingsley(1986) shows that, in order to define a stochastic process, one only need to define the
finite-dimensional distribution, and show that it is consistant according to Kolmogorov’s existance
theorem.
2.1
Construction
Suppose (E, E, m) is a measurable space, with m is σ-finite. β : E → [−1, 1] is a measurable funciton.
When α 6= 1, define F = Lα (E, E, m); when α = 1, define
Z
1
f (x)β(x) ln |f (x)| m(dx) < ∞ .
F = F(m, β) = f : f ∈ L (E, E, m) and
E
Note that F is a vector space.
For f1 , . . . , fd ∈ F , associate I(f1 ), . . . , I(fd ) with a probability measure Pf in Rd via characteristic function as follows:

o
n R P
Pd
πα
 exp − | d θj fj (x)|α 1 − iβ(x)sign
m(dx)
,
α 6= 1,
θ
f
(x)
tan
j=1 j j
2
n RE Pj=1
o
φf (θ) =
P
P
d
d θj fj (x) m(dx) , α = 1.
 exp − | d θj fj (x)| 1 + i 2 β(x)sign
j=1
j=1 θj fj (x) ln
j=1
E
π
One can show this is indeed a characteristic funciton of an α-stable vector. Notice that m is
not the spectrum measure, since E is not necessarily the sphere S d . Nevertheless, one can make
transformations
such that m becomes
Γ, and E becomes S d . For example, when α 6= 1, on space
n
o
P
E+ = x ∈ E : dj=1 fj (x)2 > 0 , transform vector f (x) to g(x) := f (x)/kf (x)k. Meanwhile, m is
transformed to m1 = kf (x)kα m. Notice that f ∈ Lα , then m1 is a finite measure.
Next, transform x to g(x), meanwhile, convert m 1 to Γ. This requires us to extract the intergrand
into two components, such as




d
d
X
X
πα  1 − β
πα  1 + β

1 − isign
θj gj (x) tan
+ 1 + isign
.
θj gj (x) tan
2
2
2
2
j=1
j=1
and make different transformations in different compoents.
The case of α = 1 is a bit more subtle, but similar.
To verify the consistancy of Pf , note that for any permutation π, we have
φfπ(1) ,...,fπ(d) (θπ(1) , . . . , θπ(d) ) = φf1 ,...,fd (θ1 , . . . , θd ),
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and for any n ≤ d,
φf1 ,...,fn ,...,fd (θ1 , . . . , θn ) = φf1 ,...,fn ,...,fd (θ1 , . . . , θn , 0, . . . , 0).
By Kolmogorov’s existance theorem, there is a stochastic process which we denote {I(f ); f ∈ F }
whose finite dimensional distributions are given in this section. Meanwhile, I(f ) is called α-stable
integral of f . We shall see why it is called intergral in latter sections. The measure m is called the
control measure, and the funciton β is called the skewness intensity.
2.2
Properties
Proposition 2.1 For any intergrable f 1 , . . . , fd , the integral I(f1 ), . . . , I(fd ) are jointly α-stable.
They are jointly SαS if the skewness intensity is zero.
Proposition 2.2 Let f be integrable, then I(f ) ∼ S α (σf , βf , µf ), where
σf
=
Z
βf
=
R
µf
=
E
α
E
|f (x)| m(dx)
1/α
,
f (x)<α> β(x)m(dx)
R
,
α
E |f (x)| m(dx)
0 R
− π2 E f (x)β(x) ln |f (x)|m(dx)
if α 6= 1;
if α = 1.
Proposition 2.3 I is a linear operator on F , i.e. if f 1 and f2 are integrable, then
I(a1 f1 + a2 f2 ) = a1 I(f1 ) + a2 I(f2 )
a.s.
for any real numbers a1 and a2 .
3
α-stable Random measures
In order to view I(f ) as an intergral, we first need to define the measure to serve as an itergrator.
A random measure is best viewed as a stochastic process M (·) indexed by
P sets A. Two major
characteristics are (M (A1 ), . . . , M (An )) is a random vector and M (∪Ai ) =
M (Ai ) a.s. if Ai are
disjoint.
Denote (Ω, F, P ) is the underling probability space, and L 0 (Ω) is the set of all real random
variables on this probability space. (E, E, m) is a measurable space, β : E → [−1, 1] is a measurable
funciton. Let E0 be the subset of E that contains all m-finite sets.
Definition An independent scattered σ-additive set function
M : E0 → L0 (Ω)
such that for each A ∈ E0 ,
R
1/α A β(x)m(dx)
M (A) ∼ Sα m(A)
,
,0
m(A)
is called an α-stable random measure on (E, E) with control measure m and skewness intensity β.
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Independent scattered means, random variables M (A i ) are independent for disjoint Ai in E0 .
σ-additive means, in addition, if ∪ ∞
j=1 Ai ∈ E0 , then


∞
∞
[
X


M
Ai =
M (Ai ) a.s.
j=1
j=1
Since the shift parameter is always 0, the random measure is characterized by m and β only.
The existance is easy to show from previous section. First, we have shown the existance of a
stochastic process {I(f ); f ∈ F }, setting M (A) = I(1 A ) for A ∈ E0 yeilds the existance of a stochastic
process {M (A); A ∈ E0 }. Its finite-dimensional distribution can be characterized via:
Proposition 3.1 For A1 , . . . , Ad ∈ E0 , and θ1 , . . . , θd ∈ R, we have
d
X
i=1
θj M (Aj ) ∼ Sα (σ, β, µ),
(1)
where
Z
σ =
E
|
d
X
θj 1Aj (x)|α m(dx)
i=1
!1/α
,
<α>
P
d
m(dx)
θ
1
(x)
β(x)
i=1 j Aj
E
,
β =
R Pd
α
m(dx)
θ
1
(x)
j
A
j
i=1
E
0
R P
P
µ =
− π2 E di=1 θj 1Aj (x)β(x) ln | di=1 θj 1Aj (x)|m(dx)
R
if α 6= 1,
if α = 1.
The rest is to show M is independently scattered(via characteristic functions), and σ-additive.
Proposition 3.2 M is a SαS random measure if the skewness intensity β = 0.
Example Let M be an α-stable random measure on [0, ∞), B with Lebsgue control measure and
constant skewness intensity β(x) = β, 0 ≤ x < ∞. Let
X(t) = M ([0, t]),
0 ≤ t < ∞.
M ([0, t]) is well defined since m([0, t] = t is finite for any t < ∞.
The stochastic process {X(t), 0 ≤ t < ∞} is α-stable Lev́y motion.
4
Constructive definition of stable integrals
After
definition of the random measure, we can show that I(f ) is bona
integral, which we denote
R
R fide
(n)
(x)M (dx), where f (n) are
E f (x)M (dx). First, we need to approximate this integral with E f
simple functions which approximates f (x); then show the converged limit is the desired integral.
Let M be an α-stable random measure on (E, E) with a control measure m and skewness intensity
β, and E0 is all the m-finte sets in E.
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4.1
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August, 2006
Simple functions
Write simple function f (x) =
I(f ) =
Z
f (x)M (dx) =
E
Pn
j=1 cj 1Aj (x)
n
X
with disjoint Aj ∈ E0 , define
cj M (Aj ).
(2)
j=1
From the definition of the random measure, we know that M (A j ) are independent α-stable
random variables with parameter
R
1/α A β(x)m(dx)
,
Sα m(A)
,0 .
m(A)
The linear combination of these random variables yield another α-stable random variable with parameter
Sα (σf , βf , µf ),
where
σf
βf
µf
=
Z
=
R
=
E
α
E
|f (x)| m(dx)
1/α
,
β(x) (f (x))<α> m(dx)
R
,
α
E |f (x)| m(dx)
0 R
− π2 E f (x)β(x) ln |f (x)|m(dx)
(3)
(4)
if α 6= 1,
if α = 1.
(5)
The operator I(·) is linear on the subspace of all simple functions.
4.2
Genearal functions
Genearlly, one can always find simple functions f (n) which approximates f (x), i.e.
f (n) (x) → f (x) for almost every x ∈ E,
|f (n) (x)|
≤ θ(x) for all n, x, and some θ ∈ F.
(6)
(7)
Then the sequence integral I(f (n) ) converges in probability, hence I(f ) is defined as
I(f ) = plimn→∞ I(f (n) ).
where plim denotes limite in probability. Also a check that the definition does not depend on the
choice of approximating sequence shows that the definition is well-stated.
Since convergence in probability infers convergence in distribution, we know the characteristic
function for I(f ) is

n R
o
 exp − |θf (x)|α 1 − iβ(x)sign θf (x) tan πα m(dx) ,
α 6= 1,
2
n RE
o
φf (θ) =
 exp − |θf (x)| 1 + i 2 β(x)sign θf (x) ln θf (x) m(dx) , α = 1.
π
E
After lineararity of I(·) is approved, the characteristic function of vector I(f1 ), . . . , I(fd ) could
be established. It is not difficult to see this construction is equivilent to the construction described
in section 2.
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Properties of stable integrals
R
R
Proposition 5.1 Let Xj = E fj (x)M (dx), j = 1, . . . , and X = E f (x)M (dx), where M is an
α-stable random measure with control measure m and skewness intensity β, then
plimj→∞ Xj = X
if and only if
Z
fj (x) − f (x)α m(dx) = 0,
lim
(8)
j→∞ E
and in the case of α = 1, if also
Z
lim
fj (x) − f (x) β(x) ln |fj (x) − f (x)|m(dx) = 0,
(9)
j→∞ E
R
Proposition 5.2 Let Xj = E fj (x)M (dx), j = 1, 2, where M is an SαS random measure with
1 < α ≤ 2 and control measure m, then
Z
f1 (x)f2 (x)<α−1> m(dx).
(10)
[X1 , X2 ]α =
E
R
Theorem 5.3 Let Xj = E fj (x)M (dx), j = 1, 2 be two α-stable random integrals with respect to an
α-stable random measure with control measure m and 0 < α < 2. Then X 1 and X2 are independent
if and only if
f1 (x)f2 (x) = 0
m − a.e. on E.
HINT: Using characteristic functions, independence is equivalent to say, for all θ i ,
Z
Z
Z
θ1 f1 (x) + θ2 f2 (x)α m(dx) = |θ1 |α
|f1 (x)|α m(dx) + |θ2 |α
|f2 (x)|α m(dx).
E
E
E
NOTE: This theorem is not true when α = 2. For example, let (R, B, dx) be the underling
probability space, and E = [−1, 1], E = B([−1, 1]), M is an 2-stable random measure with control
measure m equal to Lebesgue measure and zeroskewness intensity. f 1 (x) = x, and f2 (x) = 1 for
x ∈ [−1, 1]. The joint distribution of I(f1 ), I(f2 ) has characteristic function exp − 21 34 θ12 + 4θ22 ,
Hence I(f1 ) and I(f2 ) are independet(normal), but m {supp(f 1 ) ∩ supp(f2 )} = 2 > 0.
R
Corollary 5.4 Let Xj = E fj (x)M (dx), j = 1, . . . , d, be jointly α-stable. They are independet if
and only if they are pairwise independent, i.e., if and only if f k1 (x)fk2 (x) = 0 m-a.e. on E for any
subset {k1 , k2 } of {1, . . . , d}.
Proposition 5.5 (Change of variables) Let M m and Mν be either two α-stable random measure
with 0 < α ≤ 2 and α 6= 1, or two S1S random measures, with identical skewness intensity and with
control measures m and ν, respectively, satisfying
m(dx)
= (r(x))α ,
ν(dx)
x ∈ E,
where r(x) ≥ 0. Then
Z
Z
dist
f (x)Mm (dx) =
f (x)r(x)Mν (dx)
E
E
for any f ∈ Lα (E, m).
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HINT: Since they are all α-stable distributions, we only need to show their scale, skewness, shift
parameters are the same, recall (3).
NOTE: This change of variable will not hold if α = 1 and M m is not symmetric.
dist
Definition We say a process {X(t), t ∈ T } has the representation {Y (t), t ∈ T }, if X = Y , i.e. X
and Y has the same finite-dimensional distributions.
From α-stable random variable perspective, a spectra measure Γ on S d and a shift vector µ0
define the distribution. From α-stable random measure pespective, we need a measurable space
(E, E), on which defines a random measure M ; for d dimensional vector, we also need d funcions f j
in Lα (E, m) to define a strictly stable random variable I(f ), hence adding a shift vector η in R d
would resolve a α-stable random variable:
Z
Z
dist
fd (x)M (dx) + η.
(12)
f1 (x)M (dx), . . . ,
(X1 , . . . , Xd ) =
E
E
Theorem 5.6 (Representation theorem in R d )
Let X = (X1 , . . . , Xd ) be an α-stable random vector in Rd .
k·k
(i) Let k · k be an arbitrary norm on Rd , Sd = {s : ksk = 1} the corresponding unit sphere and let
Γk·k , µ0k·k be the corresponding spectra measure and shift parameter. The relation (12) always
holds with:
k·k
k·k
(E, E) = (Sd , Borel σ-algebra on Sd ),
M has control measure m = Γk·k and skewness intensity β(·) = 1,
k·k
fj : Sd → R is defined by fj (s1 , . . . , sd ) = sj , j = 1, . . . , d,
η = µ0k·k .
so that
dist
X = (X1 , . . . , Xd ) =
Z
k·k
Sd
s1 M (dx),
Z
k·k
s2 M (dx), . . . ,
Sd
Z
k·k
sd M (dx)
Sd
!
+ µ0k·k .
(13)
(ii) If X is SαS, then there exists bounded measurable functions f j such that
dist
X =
Z
0
1
f1 (x)M (dx),
Z
1
f2 (x)M (dx), . . . ,
0
Z
1
fd (x)M (dx) ,
0
where M is an SαS random measure with control measure m(dx) = dx.
(iii) If X is strictly α-stable with α =
6 1, then there exists bounded measurable functions f j such
that
Z 1
Z 1
Z 1
dist
X =
f1 (x)M (dx),
f2 (x)M (dx), . . . ,
fd (x)M (dx) ,
0
0
0
where M is an α-stable random measure with control measure m(dx) = dx and skewness
intensity β(x) = 1.
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(iv) If α = 1, then there exists bounded measurable functions f j such that
dist
X =
1
Z
f1 (x)M (dx),
0
Z
1
f2 (x)M (dx), . . . ,
0
Z
1
fd (x)M (dx) + η,
0
where M is an 1-stable random measure with control measure m(dx) = dx and skewness intensity β(x) = 1, and where η = (η1 , . . . , ηd ) with
1
ηj =
π
Z
1
fj (x) ln
0
d
X
fk (x)2
k=1
!
dx + µ0j ,
j = 1, . . . , d.
HINT: Recall equation (3) which proves (i). The key for the rest part can be viewed as sort
of transformation. This integral representaion holds not only for α-stable random variables, but
also for most α-stable
stochastic processes. In that case, one mey have to replace the commen
[0, 1], B [0, 1] , dx by some general measurable space (E, E, m).
NOTE : Proposition (3.2) shows that we can construct SαS random vector via symmetric αstable random measures. However, we can also contruct SαS random vector via non-symmetric
α-stable random measures. For example,
Z d
Xk =
1[k−1,k] (x) − 1[−k,−k+1](x) M (dx), , k = 1, . . . , d,
−d
where E = [−d, d], M is a α-stable random measure with Lebesgue control measure and skewness
intensity β = 1.
Corollary 5.7 Let X1 , . . . , Xd be jointly α-stable. If they are pairwise independent, then they are
independent.
6
Examples
Example SαS Lev́y motion.
R∞
Rt
Let X(t) = M ([0, t]) = 0 1{x ≤ t}M (dx) = 0 M (dx), t ≥ 0, where M is SαS on [0, ∞) with
control measure m(dx) = dx. Then
X(0) = 0,
a.s.
X(t) − X(s) =
Z
t
s
M (dx) = M ([s, t]) ∼ Sα (|t − s|1/α , 0, 0).
If 0 ≤ t1 < t2 < · · · < tn , then
X(t2 ) − X(t1 ), . . . , X(tn ) − X(tn−1 ) =
Z
t2
t1
M (dx), . . . ,
Z
tn
tn−1
M (dx) .
To summarize, {X(t), t ≥ 0} is a process starts at 0, has stationary independent increments and
SαS finite dimenstional ditribution. Hence it is SαS Lev́y motion.
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Example Moving average.
R∞
Let f be a measurable function in R1 satifying −∞ |f (x)|α dx < ∞, 0 < α ≤ 2, and define
Z ∞
X(t) =
f (t − x)M (dx), t ∈ R1 ,
(14)
−∞
where M is SαS with Lebesgue control measure. A process X(·) of the form 14 is called an SαS
moving average process. X(·) is stationary, since for any t 1 , . . . , td , h ∈ R, and real θ1 , . . . , θd ,
k
d
X
θk X(tk + h)kαα
k=1
=
Z
∞
|
d
X
−∞ k=1
α
θk f (tk + h − x)| dx =
Z
∞
|
d
X
−∞ k=1
α
θk f (tk − x)| dx = k
d
X
θk X(tk )kαα
k=1
Example Ornstein-Uhlenbeck process.
Let λ > 0 and M be a SαS random measure, 0 < α ≤ 2 with Lebesgue control measure. The
process
Z t
e−λ(t−x) M (dx), t ∈ R1 ,
X(t)
−∞
is called an Ornstein-Uhlembeck process. Note that it is a moving average process with
f (x) = e−λx 1[0,∞) (x),
x ∈ R1 ,
Hence it is well-defined, and stationary.
Note for any fixed s < t, we have
Z t
−λ(t−s)
X(t) − e
X(s) =
e−λ(t−x) M (dx)
(15)
s
P
d
is a SαS random varaible independent of any linear combination
k = 1 bk X(u
k ), uk ≤ s,
−λ(t−s)
k = 1, . . . , d. Therefore, X(t) − e
X(s) is independent of σ X(u), u ≤ s , implying that
the Ornstein-Uhlenbeck process is Markovian.
Example Reverse Ornstein-Uhlenbeck process.
Let λ > 0 and M be a SαS random measure, 0 < α ≤ 2 with Lebesgue control measure. The
process
Z ∞
X(t)
e−λ(x−t) M (dx), t ∈ R1 ,
t
is called a reverse Ornstein-Uhlembeck process. Note for any fixed s > t, we have
Z s
−λ(s−t)
X(t) − e
X(s) =
e−λ(t−x) M (dx)
(16)
t
Hence the reverse Ornstein-Uhlenbeck process is also Markovian.
Recall the Gausssian case α = 2, all stationary Markov process are Ornstein-Uhlenbeck, meaning
when α = 2, the Ornstein-Uhlenbeck process and the reverse Ornstein-Uhlenbeck process are the
same process. However, they are different when 0 < α < 2.
Example Well-balanced linear fractional stable motion.
Let M be SαS, 0 < α ≤ 2 with Lebesgue control measure and
Z ∞
H−1/α
H−1/α
X(t) =
|t − x|
− |x|
M (dx), t ∈ R1 ,
−∞
where 0 < H < 1, and H 6= 1/α. The process is called Well-balanced linear fractional stable motion,
with two major properties.
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• Self-similar with index H, or H − ss, i.e.
X(ct1 ), . . . , X(ctd )
dist
=
cH X(t1 ), . . . , cH X(td ) .
• Stationary increment, i.e.
dist
{X(t) − X(0), t ∈ R} = {X(t + τ ) − X(τ ), t ∈ R}.
Example Log-fractional stable motion
It looks like the well-balanced linear fractional stable motion, with the limit H goes to 1/α, the
power function changed to the log function. Assuming 0 < α < 1, the process
Z ∞
ln |t − x| − ln |x| M (dx), t ∈ R1 ,
X(t) =
−∞
is called the (symmetric) log-fractional stable motion. It is self similar with index 1/α, and has
stationary increment.
NOTE: When α = 2, the log-fractional stable motion and the well-balanced linear fractional
stable motion becomes one stochastic process, it is called fractional Brownian motion.
7
7.1
Sub-stable process
Sub-Gaussian processes
Recall that if {G(t), t ∈ T } is a Gaussian process, and let A ∼ S α/2
cos πα
4
2/α
, 1, 0 , where α < 2,
and A is independent of {G(t), t ∈ T }. Then {X(t) = A 1/2 G(t), t ∈ T } is a sub-Gaussian process
with underling Gaussian process {G(t), t ∈ T }. {X(t)} is SαS, and has finte sub-Gaussian SαS
distributions.
Let {Ω, F, P } be the underling probability space on which {G(t)} is defined. Note in this case,
we have E = Ω, and E = F as in previous notations. Write {G(t), t ∈ T } = {G(t, ω), t ∈ T, ω ∈ Ω}
to make the dependence on Ω explicit. To avoid confusion, use x instead of ω. Let M be a SαS
random measure on {Ω, F} with control measure P .
Proposition 7.1 The sub-Gaussian process {X(t) = A 1/2 G(t), t ∈ T } has the representation
Z
1
dist
√
G(t, x)M (dx), t ∈ T ,
{X(t), t ∈ T } =
dα 2 Ω
where dα = (E|Z|α )1/α , with Z ∼ N (0, 1).
10
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Stable Process 3
Zhi Ouyang
August, 2006
Proof For any d ≥ 1, t1 , . . . , td ∈ T , θ1 , . . . , θd ∈ R,




Z
Z d
d



 X
X
α
1
1
G(tj , x)M (dx)
= exp −d−α
θj G(tj , x) P (dx)
E exp i
θj √
√
α




dα 2 Ω
2 j=1
Ω
j=1
 
α/2 


Z d


X
2
1
[ notice that G is Gaussian ] = exp −  √
θj G(tj , x) P (dx)


2 j=1
Ω




d
d X

 1 X
θj θk G(tj , x)G(tk , x)
[ expand ] = exp − 
 2
j=1 k=1


d
 X

1/2
[ notice X = A G ] = E i
θj Xj .


j=1
7.2
Sub-α-stable processes
To generalize the result, let {Y (t), t ∈ T } be a Sα 0 S process with α0 < 2. For fixed 0 < α < α0 , let
πα α0 /α A ∼ Sα/α0 cos 0
,
2α
Definition The process
0
X(t) = A1/α Y (t),
t ∈ T,
(18)
is SαS, can it is called sub-stable process or sub-α-stable process with underling stable process
{Y (t), t ∈ T }.
Similarly, let M be a SαS random measure on {Ω, F} with control measure P .
0
Proposition 7.2 The sub-stable process {X(t) = A 1/α Y (t), , t ∈ T } has the representation
Z
dist
−1
{X(t), t ∈ T } = dα,α0
Y (t, x)M (dx), t ∈ T ,
(19)
Ω
where dp,α0 = (E|Z|p )1/p , Z ∼ Sα0 (1, 0, 0).
R
NOTE: When the random measure M is not symmetric, the process {X(t) = Ω W (t)M (dx), t ∈
T } is still well-defined when corresponding W (t) has finte α th -moment. However, X(t) no longer
bears the simple form as Ap W (t).
8
Series representation for α-stable random measures
To generalize the result on the one-dimensional α-stable random variable, it indicates that α-stable
random measures have essentially a “discrete” structure.
Let M be an α-stable random measure on (E, E) with finte control measure m and skewness
intensity β(·). Denote m̂ as the standerdized control measure m/m(E). Let {Γ 1 , Γ2 , . . . } be a
sequence of arrival times of a Poission process with unit arrival rate, and let {(V 1 , γ1 ), (V2 , γ2 ), . . . }
be a sequence of i.i.d. random vectors, independent of {Γ i }, such that Vi has distribution m̂ on E,
and
1 + β(Vi )
P (γi = 1|Vi ) = 1 − P (γi = 0|Vi ) =
2
11
Stable Process 3
Zhi Ouyang
August, 2006
Theorem 8.1 The series representation for M is
)
(
Z
∞ h
i
1/α X
dist
−1/α
(α)
Cα m(E)
γi Γi
1(Vi ∈ A) − bi
β(x)m̂(dx) + ηA , A ∈ E ,(20)
{M (A), A ∈ E} =
A
i=1
where Cα is the constant about the tail convergence, i.e.
Z ∞
−1 1−α
if α 6= 1;
−α
Γ(2−α) cos(πα/2) ,
Cα =
=
x sin x dx
2/π,
if α = 1.
0
and
(α)
bi
=
ηA =


 R0
1/(i−1)


x−2
1/i
α
α
α−1
α−1 (i
0
2
π
ln
if 0 < α < 1,
if α = 1,
sin xdx
α
− (i − 1) α−1
2
π m(E)
R
if 1 < α < 2,
A β(x)m(dx)
if α 6= 1,
if α = 1.
Equation (20) shows that M is decomposed into two components: a random component consisting
−1/α
a signed random point mass γi Γi
placed at random points Vi ; and a non-random component
proportional to the signed measure m s (dx) = β(x)m(dx). The non-random component is absent
when 0 < α < 1, and it is equal to zero when M is SαS.
dist
Theorem 8.2 For any integrable function f ∈ F , we have I(f ) = S(f ), where S(f ) is an a.s.
convergent random series defined as follows:
(i) When 0 < α < 1,
S(f ) = Cα m(E)
∞
1/α X
−1/α
γi Γi
f (Vi ),
i=1
(ii) When α = 1,
∞
S(f ) =
Xh
2
(1)
γi Γ−1
m(E)
i f (Vi ) − bi
π
i=1
Z
E
i
f (x)β(x)m̂(dx) + ηf ,
where
Z
2 2
ηf = ln
m(E)
f (x)β(x)m(dx).
π
π
E
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(iii) When 1 < α < 2,
S(f ) = Cα m(E)
∞ h
1/α X
−1/α
γi Γi
f (Vi )
i=1
−
(α)
bi
Z
E
f (x)β(x)m̂(dx)
i
We refer S(f ) as the series representation of the stable stochastic integral I(f ). This definition is
equivalent to previous definitions about α-stable integral of f .
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Stable Process 3
Zhi Ouyang
August, 2006
Corollary 8.3 If fi ∈ F , then
dist
I(f1 ), . . . , I(fd ) = S(f1 ), . . . , S(fd ) .
Example Lev́y α-stable motion.
Let {X(t), 0 ≤ t ≤ 1} be standard Lev́y α-stable motion on [0, 1], with X(1) ∼ S α (1, β, 0). We
have shown that
Z 1
dist
{X(t), 0 ≤ t ≤ 1} =
1[0,t] M (dx), 0 ≤≤ 1 ,
0
where M is an α-stable random measure on [0, 1], B with Lebesgue control measure and constant
skewness intensity β(x) = β.
Hence, for 0 < α < 1,
)
(
∞
X
1/α
dist
−1/α
γi Γi
1(Ui ≤ t), 0 ≤ t ≤ 1 ,
{X(t), 0 ≤ t ≤ 1} =
Cα
i=1
for α = 1,
dist
{X(t), 0 ≤ t ≤ 1} =
(
)
∞
i
2 X h −1
2 2
(1)
γi Γi 1(Ui ≤ t) − βtbi + βt ln , 0 ≤ t ≤ 1 ,
π
π π
i=1
for 1 < α < 2
dist
{X(t), 0 ≤ t ≤ 1} =
(
Cα
∞ h
1/α X
i=1
(1)
γi Γ−1
i 1(Ui ≤ t) − βtbi
i
)
,0≤t≤1 ,
where {γi , . . . } is a sequence of i.i.d random variables satisfying
P (γi = 1) = 1 − P (γi = 0) =
1+β
,
2
{Γ1 , . . . } is a sequence of arrival times of a Poisson process with unite arrival rate, and {U 1 , . . . } is
a sequence of i.i.d random variables uniformluy distributed on [0, 1]. They are independent.
Meaning, when 0 < α < 1, the α-stable Lev́y motion, and when 1 < α < 2, the SαS Lev́y motion
can be regarded as pure jump process. The instants U i s of the jumps are uniformly distributed in
[0, 1], the directions of the jumps are up by (1 + β)/2, and down by (1 − β)/2, with jump heights
following a decreasing order distributed as −1/α power of arrival times of Poisson process with unit
rate.
Non-symetric α-stable Lev́y motion with 1 ≤ α ≤ 2 can be regarded as a pure jump process with
a linear deterministic trend.
Corollary 8.4 In the symmetric case, if M is SαS, 0 < α < 2, then for any f ∈ F ,
dist
I(f ) =
Cα m(E)
∞
1/α X
−1/α
i Γi
f (Vi ),
i=1
where {i , . . . } is the fair toss bernulli sequence independent of {Γ} and {V i }.
13
Stable Process 3
9
Zhi Ouyang
August, 2006
Condition S
To replace the Rademacher sequence { i } in Corollary 8.4 by sequence {d−1
α Gi }, where Gi are i.i.d.
1/α
α
−1
α
standard normal random variable, and d α = E|G1 |
, or E|dα G1 | = 1. Notice that
α + 1
,
2
dαα = 2α/2 π −1/2 Γ
then denote
Cα0
=
d−α
α Cα
=2
−α/2 1/2
π
Then
dist
I(f ) =
Cα0 m(E)
∞
1/α X
α + 1
Γ
2
−1/α
Gi Γi
−1
Cα ,
(22)
f (Vi ),
(23)
i=1
For a SαS stochastic process represented as
Z
X(t) =
ft (x)M (dx), t ∈ T ,
E
by Corollary 8.3, we obtain
(
dist
{X(t), t ∈ T } =
Cα0 m(E)
∞
1/α X
−1/α
Gi Γi
i=1
ft (Vi ), t ∈ T
)
.
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View this series on the right-hand side as the product of two probability space: (Ω 1 , F1 , P1 ) on which
the Gaussian sequence Gi is defined, (Ω2 , F2 , P2 ) on which Γi and Vi are defined. Hence conditional
on Γi and Vi , it is a mixture of zero mean Gaussian processes on the probability space (Ω 1 , F1 , P1 ).
Proposition 9.1 Let M be a SαS random measure
R on (E, E) with a finte control measure m.
Let ft ∈ F , then SαS stochastic process {X(t) = E ft (x)M (dx), t ∈ T } is conditionally centered
Gaussian, i.e., it is a probability mixture of zero mean Gaussian processes.
This indicates, besides sub-Gaussian processes, they are many more SαS processes are conditionally centered Gaussian.
Definition Let {X(t), t ∈ T } be a stochastic process. It satisfies condition S, if there is a countable
subset To ⊆ T , such that for every t ∈ T , there exists t k ∈ To , and X(t) = plimk→∞ X(tk ).
NOTE: Condition S, the letter “S” stands for seperability in probability. All SαS processes that
satisfies condition S bears the representation equation
Z
dist
ft (x)M (dx), t ∈ T , m finite.
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{X(t), t ∈ T } =
E
Morever, we have following relationship:
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Stable Process 3
Zhi Ouyang
August, 2006
Conditionally centered Gaussian
KS
R
E
ft (x)M (dx) with m finite ks
KS
R
E gt (x)M
0 (dx)
with m0 σ-finite
Condition S
KS
(T, d) sep., ctb. T1 ⊆ T , s.t. {X(t), t ∈ T \T1 } conti. i.p.
Example Counterexample: Consider X(t), t ∈ [0, 1] be i.i.d. (non-degenerate) SαS random variables,
R then it certainly not satisfy condition S, nor it has a finte m(or σ-finite m) representation
as E ft M (dx). But it is conditionally centered Gaussion, since we can write X(t) = A(t) 1/2 G(t),
where A(t) are i.i.d. totally skewed α/2-stable, and G(t) is i.i.d. standard normal random variables
independent of A(t).
10
Poisson representation
Let M be an α-stable random measure on (E, E) with control measure m and skewness intensity
β(·), let f ∈ F . The goal of this section is to write I(f ) in terms ofa integral with respect to a
Poisson random measure N on E × R0 , where R0 = R\{0}.
condition here. In fact, since
R f ∈ αF implies that m is σ-finte, which−i is a very important
−i+1
< |f (x)| ≤ 2
} has finte m-measure for every
E |f (x)| M (dx) < ∞, write Ei = {x : 2
integer i, and the union of them is whole space E.
To define a Poisson random measure, recall the conterpart of the definition of α-stable random
measure
Definition Let (S, S, n) be a measurable space, with S 0 = {A ∈ S : n(A) < ∞}. (Ω, F, P ) is an
underlining probability space. A Poisson random measure N on (S, S, n) is an independent scattered
σ-additive set function
N : S0 −→ L0 (Ω)
A 7−→ N (A) ∼ P o n(A)
That is to say, N (A) has the discrete distribution that
k
−n(A) n(A)
, k = 0, 1, 2, . . . .
P n(A) = k = e
k!
n is called the control measure of N .
In this context, set S = E × R0 , and
(
1 + β(x) m(dx) |u|du
α+1
n(dx, du) = EN (dx, du) =
1 − β(x) m(dx) |u|du
α+1
15
u>0
u<0
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