Download Change of Measure formula and the Hellinger Distance of two Lévy

Change of Measure formula and the Hellinger
Distance of two Lévy Processes
Erika Hausenblas
University of Salzburg, Austria
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.1
Outline
Hellinger distances
Poisson Random Measures
The Main Result
The Change of Measure formula
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.2
The Kakutani–Hellinger Distance
Let (Θ, B) be a measurable space.
Definition 1 Let α ∈ (0, 1). For two σ–finite measures P1 and P2 on
(Θ, B) we define
Hα (P1 , P2 ) =
1−α
dP1
dP
α Z
dHα (P1 , P2 ),
dP2
dP
,
and
hα (P1 , P2 ) =
Θ
where P is a σ–finite measure such that P1 , P2 ≪ P . We call Hα (P1 , P2 )
the Hellinger-Kakutani inner product of order α of P1 and P2 . The total
mass of Hα (P1 , P2 ) is written as
Z
hα (P1 , P2 ) =
dHα (P1 , P2 ).
Θ
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.3
The Kakutani–Hellinger Distance
Remark 1 The definition of the Kakutani–Hellinger affinity H is
independent from the choice of P , as long as P1 , P2 ≪ P holds.
Definition 2 Let α ∈ (0, 1). For two σ –finite measures P1 and P2 on
(Θ, B) we define
Kα (P1 , P2 ) = αP1 + (1 − α)P2 − Hα (P1 , P2 ),
and
kα (P1 , P2 ) =
Z
dKα (P1 , P2 ),
Θ
The latter we call the Kakutani-Hellinger distance of order α between
P1 and P2 .
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.4
The Kakutani–Hellinger Distance
Some Applications:
Statistics of Random processes
(Jacod and Shirayev, Griegelionis (2003))
Application to Contiguity
( Shirayev and Greenwood (1985))
Application to the Likelihood Ratio, Information theory
(Vajda (2006), Liese and Vajda (1987));
A measure of Bayes estimator
(Vajda, Liese and Vajda)
Application to Risk Minimization (Vostrikova)
Application in Martingale measures
(Keller, 1997).
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.5
The Kakutani–Hellinger Distance
Some Properties:
hα (P1 , P2 ) = 0 ⇐⇒ P1 and P2 are singular;
Suppose that (Ωi , Fi ), 1 ≤ i ≤ n, are measurable spaces and Pi1
and Pi2 , probability measures on (Ωi , Fi ), 1 ≤ i ≤ n.
Then
h1
2
2
⊗ni=1 Pi1 , ⊗ni=1 Pi
=
n
Y
i=1
h1
2
Pi1 , Pi2
.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.6
The Lévy Process L
Assume that L = {L(t), 0 ≤ t < ∞} is a Rd –valued Lévy process over
(Ω; F ; P). Then L has the following properties:
L(0) = 0;
L has independent and stationary increments;
for φ bounded, the function t 7→ Eφ(L(t)) is continuous on R+ ;
L has a.s. cádlág paths;
the law of L(1) is infinitely divisible;
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.7
The Lévy Process L
The Fourier Transform of L is given by the Lévy - Hinchin - Formula:
Z E eihL(1),ai = exp ihy, aiλ +
eiλhy,ai − 1 − iλy1{|y|≤1} ν(dy) ,
Rd
where a ∈ Rd , y ∈ E and ν : B(Rd ) → R+ is a Lévy measure.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.8
The Lévy Process L
Definition 3 (see Linde (1986), Section 5.4) A σ–finite symmetric
Borel-measure ν : B(Rd ) → R+ is called a Lévy measure if ν({0}) = 0
and the function
Z
E ′ ∋ a 7→ exp
(cos(hx, ai) − 1) ν(dx) ∈ C
a
Rd
is a characteristic function of a certain Radon measure on Rd . An
arbitrary σ-finite Borel measure ν is a Lèvy measure if its symmetrization
ν + ν − is a symmetric Lévy measure.
a
ν(A) = ν(−A) for all A ∈ B(Rd )
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.9
Poisson Random Measure
Let L be a Lévy process on R with Lévy measure ν over (Ω, F , P).
Remark 2 Defining the counting measure
B(R) ∋ A 7→ N (t, A) = ♯ {s ∈ (0, t] : ∆L(s) = L(s) − L(s−) ∈ A}
one can show, that
N (t, A) is a random variable over (Ω, F , P);
N (t, A) ∼ Poisson (tν(A)) and N (t, ∅) = 0;
For any disjoint sets A1 , . . . , An , the random variables
N (t, A1 ), . . . , N (t, An ) are pairwise independent;
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.10
Poisson Random Measure
Definition 4 Let (S, S) be a measurable space and (Ω, A, P) a probability
space. A random measure on (S, S) is a family
η = {η(ω, ), ω ∈ Ω}
of non-negative measures η(ω, ) : S → R+ , such that
η(, ∅) = 0 a.s.
η is a.s. σ–additive.
η is independently scattered, i.e. for any finite family of disjoint sets
A1 , . . . , An ∈ S, the random variables
η(·, A1 ), . . . , η(·, An )
are independent.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.11
Poisson Random Measure
A random measure η on (S, S) is called Poisson random measure iff
for each A ∈ S such that E η(·, A) is finite, η(·, A) is a Poisson random
variable with parameter E η(·, A).
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.12
Poisson Random Measure
A random measure η on (S, S) is called Poisson random measure iff
for each A ∈ S such that E η(·, A) is finite, η(·, A) is a Poisson random
variable with parameter E η(·, A).
Remark 3 The mapping
S ∋ A 7→ ν(A) := EP η(·, A) ∈ R
is a measure on (S, S).
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.12
Poisson Random Measure
+
ˆ
Let (Z, Z) be a measurable space. If S = Z × R+ , S = Z ×B(R
), then a
Poisson random measure on (S, S) is called Poisson point process.
Remark 4 Let ν be a Lévy measure on a Banach space E and
• S = Z × R+
+)
ˆ
• S = Z ×B(R
• ν ′ = ν × λ (λ is the Lebesgue measure).
Then there exists a time homogeneous Poisson random measure
η : Ω × Z × B(R+ ) → R+
such that E η( , A, I) = ν(A)λ(I),
ν is called the intensity of η.
A ∈ Z, I ∈ B(R+ ),
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.13
Poisson Random Measure
Definition 5 Let E be a topological vector space and
η : Ω × B(E) × B(R+ ) → R+
be a Poisson random measure over (Ω; F; P) and {Ft , 0 ≤ t < ∞} the filtration
induced by η. Then the predictable measure
γ : Ω × B(E) × B(R+ ) → R+
is called compensator of η, if for any A ∈ B(E) the process
η(A, (0, t]) − γ(A, [0, t])
is a local martingale over (Ω; F; P).
Remark 5 The compensator is unique up to a P-zero set and in case of a time
homogeneous Poisson random measure given by
γ(A, [0, t]) = t ν(A),
A ∈ B(E).
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.14
Poisson random measures
Example 1 Let x0 ∈ R, x0 6= 0 and set ν = δx0 .
Let η be the time homogeneous Poisson random measure on R with
intensity ν.
Then
t 7→ L(t) :=
Z tZ
0
x η(dx, ds),
R
and
P(L(t) = kx0 ) = exp(−t)
Since
R
R x γ(dx, dt)
tk
k! ,
k ∈ IN.
= x0 dt, the compensated process is given by
RtR
0 R x η̃(dx, ds) = L(t) − x0 t.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.15
Poisson random measures
Example 2 Let α ∈ (0, 1) and ν(dx) = xα−1 .
Let η be the time homogeneous Poisson random measure on R with
intensity ν.
Then
t 7→ L(t) :=
is an α stable process and
RtR
0
R x η̃(dx, ds),
E(e−λL(t) ) = exp(−λtα ).
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.16
Newman’s Result (1976)
By means of the following formula
•
h1
2
2
⊗ni=1 Pi1 , ⊗ni=1 Pi
=
n
Y
i=1
h1
2
Pi1 , Pi2
,
where above (Ωi , Fi ), 1 ≤ i ≤ n, are different probability spaces and Pi1
and Pi2 two probability measures, and,
•
since the counting measure of a Lévy process is
independently scattered,
Newman was able to show the following:
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.17
Newman’s Result (1976)
Let L1 and L2 be two Lévy processes with Lévy measures ν1 and ν2 .
Let
ν 1 := H 1 (ν1 , ν2 );
2
2
R
ai (t) := R z νi − ν 1 (dz),
2
i = 1, 2;
Pi : B(ID(R+ ; R)) ∋ A 7→ P (Li + ai ∈ A) ,
i = 1, 2;
Then
H 1 (P1 , P2 ) := exp −t k 1 (ν1 , ν2 ) P 1 ,
2
2
2
where P 1 is the probability measure on ID(R+ ; R) of the process L 1
2
2
given by the Lévy measure ν 1 .
2
Inoue (1996) extended the result to non time homogeneous, but
deterministic Lévy processes. See also Liese (1987).
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.18
Jacod’s and Shiryaev’s Approach
• L1 and L2 be two semimartingales (here, of pure jump type);
• Pi , i = 1, 2, be the probability measures induced from L1 and L2 on ID(R+ ; R);
• Q be a measure such that P1 ≪ Q and P2 ≪ Q.
Let
z1 :=
dP1
dQ ,
z1 :=
dP2
dQ ,
and for t ≥ 0 let z1 (t) and z2 (t) be the restriction of z1 and z2 on Ft .
Let
1
2
1
2
Y (t) := (z1 (t)) (z1 (t)) ,
t > 0.
Then there exists a predictable increasing process h 12 a , called Hellinger process,
such that h 21 (0) = 0 and
Rt
t 7→ Y (t) + 0 Y (s−)dh 21 (s),
is a Q-martingale. (see also Jacod (1989), Grigelionis (1994))
a
In terms of Newman, h should be k .
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.19
Prm - Bismut’s Setting
Let (Ω; F; P) a probability space and
µ : B(R) × B(R+ ) −→ IN0
a Poisson random measure over (Ω; F; P) with compensator γ given by
B(R) × B(R+ ) ∋ (A, I) 7→ γ(A, I) := λ(A)λ(I).
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.20
Prm - Bismut’s Setting
Let (Ω; F; P) a probability space and
µ : B(R) × B(R+ ) −→ IN0
a Poisson random measure over (Ω; F; P) with compensator γ given by
B(R) × B(R+ ) ∋ (A, I) 7→ γ(A, I) := λ(A)λ(I).
———–
Let c : R → R be a given mapping such that c(0) = 0 and
R
2
|c(z)|
dz < ∞.
R
Then, the measure ν defined by
B(R) ∋ A 7→ ν(A) :=
R
R
1A (c(z)) dz
is a Lévy measure and the process L given by
Z tZ
t 7→ L(t) :=
c(z) (µ − γ)(dz, ds),
0
R
is a (time homogeneous) Lévy process.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.20
Poisson random measures
Example 2 Let α ∈ (0, 1) and η be the time homogeneous Poisson
random measure on R with intensity λ.
Then
t 7→ L(t) :=
Z tZ
0
1
α
|x| η̃(dx, ds),
R
is an α stable process.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.21
Main Result - The Setting
bi = {bi (s); s ∈ R+ } are two predictable Rd –valued processes,
ci : Ω × R+ × Rd → Rd , i = 1, 2, be two predictable processes such that
Z tZ
EP
|ci (s, z)|2 dz ds < ∞,
0
Rd
and ci (s, z) is differentiable at any z ∈ Rd∗ := R \ {0}.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.22
Main Result - The Setting
bi = {bi (s); s ∈ R+ } are two predictable Rd –valued processes,
ci : Ω × R+ × Rd → Rd , i = 1, 2, be two predictable processes such that
Z tZ
EP
|ci (s, z)|2 dz ds < ∞,
0
Rd
and ci (s, z) is differentiable at any z ∈ Rd∗ := R \ {0}.
—————————————–
X i = {X i (t); t ∈ R+ }, i = 1, 2, are two Rd –valued semimartingales given by
X i (t) =
Z tZ
0
Rd
ci (s, z) (µ − γ)(dz, ds) +
Z
t
bi (s) ds,
i = 1, 2,
0
and ν i = {νti ; t ∈ R+ } are two unique predictable measure valued processes
given by
R
+
i
B(R) × R ∋ (A, t) 7→ νt (A) := Rd 1A (ci (t, z)) λd (dz), i = 1, 2.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.22
The Main Result
Let νtα := Hα (νt1 , νt2 ) and η α be a random measure with compensator γ α
defined by
R R
α
d
+
γ : B(R ) × B(R ) ∋ (A, I) 7→ I Rd νs (A) ds.
ait
Xtα
:=
RtR
:=
0
Rt
0
Rd
νsi
z
α
−
νsα
α
(dz) ds;
z (η − γ )(dz, ds) +
1
2
Rt
0
[b1 (s) + b2 (s)] ds
Qi : B(ID(R+ , Rd )) ∋ A 7→ P(X i + ai ∈ A), i = 1, 2;
If
then
—————————————–
RtR
Rt
(b1 (s) − b2 (s)) ds = − 0 Rd (c1 (s, z) − c2 (s, z)) dz ds,
0
R
t
dHα (Q1t , Q2t ) = exp − 0 kα (νs1 , νs2 ) ds dQα
t
where Qα is the probability measure on ID(R+ ; Rd ) induced by the semimartingale X α = {Xtα ; t ∈ R+ }.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.23
Consequences
Let ν1 and ν2 two Lévy measures on B(R), and β1 , β2 ∈ (0, 2] two real number
such that
νi ((z, ∞))
νi ((−∞, z))
lim
and lim
i = 1, 2.
−βi
−βi
z→0
z→0
z
z
z>0
z<0
If β1 6= β2 and β1 , β2 > 1 then k 21 (ν1 , ν2 ) = ∞ and, hence, the induced measures
on ID(R+ , R) of the corresponding Lévy processes are singular.
Let
νin
:= νi R\(− 1 , 1 ) , β1 6= β2 , and Lnt be the corresponding Lévy processes,
n n
i = 1, 2. Then, for any n, the induced probability measures probability measures
Pin on ID(R+ , R) (shifted by a drift term) are equivalent, but the measures Pi ,
i = 1, 2, given by Pi := limn→∞ Pin are singular.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.24
Jacod’s and Shiryaev’s Approach
• L1 and L2 be two semimartingales (here, of pure jump type);
• Pi , i = 1, 2, be the probability measures induced from L1 and L2 on ID(R+ ; R);
• Q be a measure such that P1 ≪ Q and P2 ≪ Q.
Let
z1 :=
dP1
dQ ,
z1 :=
dP2
dQ ,
and for t ≥ 0 let z1 (t) and z2 (t) be the restriction of z1 and z2 on Ft .
Let
1
2
1
2
Y (t) := (z1 (t)) (z1 (t)) ,
t > 0.
Then there exists a predictable increasing process h 12 a , called Hellinger process,
such that h 21 (0) = 0 and
Rt
t 7→ Y (t) + 0 Y (s−)dh 21 (s),
is a Q-martingale. (see also Jacod (1989), Grigelionis (1994))
a
In terms of Newman, h should be k .
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.25
Change of Measure Formula
A random measure η on (S, S) is called Poisson random measure iff
for each A ∈ S such that E η(·, A) is finite, η(·, A) is a Poisson random
variable with parameter E η(·, A).
Remark 6 The mapping
S ∋ A 7→ ν(A) := EP η(·, A) ∈ R
is a measure on (S, S).
specifying the intensity ν
⇐⇒
specifying P on (Ω, F ).
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.26
Change of measure formula
(Bichteler, Gravereaux and Jacod)
• Define a bijective mapping
a
θ : R∗ → R∗
a
R∗ := R \ {0}.
− 12
possible z 7→ z + sgn(z) |z|
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.27
Change of measure formula
(Bichteler, Gravereaux and Jacod)
• Define a bijective mapping
θ : R∗ → R∗
− 21
possible z 7→ z + sgn(z) |z|
• Define a new Poisson random measure µθ given by
Z Z
B(R) × B(R+ ) ∋ (A, I) 7→ µθ (A, I) :=
χA (θ(z)) µ(dz, ds).
I
R∗ := R \ {0}.
R
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.27
Change of measure formula
(Bichteler, Gravereaux and Jacod)
• Define a bijective mapping
θ : R∗ → R∗
− 21
possible z 7→ z + sgn(z) |z|
• Define a new Poisson random measure µθ given by
Z Z
B(R) × B(R+ ) ∋ (A, I) 7→ µθ (A, I) :=
χA (θ(z)) µ(dz, ds).
I
R
• Define a new probability measure Pθ on (Ω; F ) by saying:
µθ has compensator γ = λ × λ.
R∗ := R \ {0}.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.27
Change of measure formula
B(R) × B(R+ ) ∋ (A, I) 7→ µθ (A, I) :=
Z Z
I
χA (θ(z)) µ(dz, ds).
R
Let
+
B(R) × B(R ) ∋ (A, B) 7→ γθ (A, I) =
R R
I
a
J
(∇θ)(z) λ(dz)λ(ds).
A
and
R R
+
B(R) × B(R ) ∋ (A, B) 7→ γθ−1 (A, I) = I A J(∇(θ)−1 )(z) λ(dz)λ(ds).
The following can be shown:
µ has compensator γ under P.
µθ has compensator γ under Pθ .
µ has compensator γθ under Pθ .
µθ has compensator γθ−1 under P.
a
J denotes the Jacobian determinant.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.28
Change of measure formula
For t ≥ 0 let P(t) and Pθ (t) be the restriction of P and Pθ on Ft . Then
Radon Nikodym derivative is given by
dPθ (t)
= Gθ (t);
dP(t)
where Gθ is the Doleans Dade exponential of ζθ , where ζθ is given by

 dζ (t) = R (J(∇θ(z)) − 1) (µ − γ)(dz, ds),
θ
A
 ζθ (0) = 0.
In the following the Doleans Dade exponential of a process X is denoted
by E(X).
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.29
Change of measure formula
Remark 7 The same idea works also, if
θ : Ω × R+ × Rd → Rd
is predictable.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.30
Proof of the Theorem - The Setting
bi = {bi (s); s ∈ R+ } are two predictable Rd –valued processes,
ci : Ω × R+ × Rd → Rd , i = 1, 2, be two predictable processes such that
Z tZ
EP
|ci (s, z)|2 dz ds < ∞,
0
Rd
and ci (s, z) is differentiable at any z ∈ Rd∗ := R \ {0}.
—————————————–
X i = {X i (t); t ∈ R+ }, i = 1, 2, are two Rd –valued semimartingales given by
Rt
RtR
i
X (t) = 0 Rd ci (s, z) (µ − γ)(dz, ds) + 0 bi (s) ds, i = 1, 2,
and ν i = {νti ; t ∈ R+ } are two unique predictable measure valued processes
given by
R
+
i
B(R) × R ∋ (A, t) 7→ νt (A) := Rd 1A (ci (t, z)) λd (dz), i = 1, 2.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.31
Proof of the Theorem
Let c : Ω × R+ × Rd → Rd be defined by
1
c(s, z) := [c1 (s, z) + c2 (s, z)]
2
and X 0 = {Xt0 , 0 ≤ t < ∞} be the semimartingale given by
t 7→ Xt0 :=
Z tZ
0
Let
and
c(s, z) (µ − γ) (dz, ds).
Rd

c−1 (s, ci (s, z)), z ∈ Rd , s ∈ R+ ,
∗
θi (s, z) :=
0
z = 0, s ∈ R+ , i = 1, 2,
ji (s, z) :=

J (∇z θi (s, z)) , z ∈ Rd∗ , s ∈ R+ ,
0
z = 0, s ∈ R+ i = 1, 2.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.32
Proof of the Theorem
For i = 1, 2 let Pi be the probability measure on (Ω, F) under which the Poisson
random measure µθ defined by
Z Z
1A (θi (s, z)) µ(dz, ds)
µiθ (A, I) :=
I
Rd
has compensator γ. Then
i
P
where
ξti :=
Z tZ
0
Rd
{Xt0
i
∈ A} = P {ξt ∈ A} ,
ci (s, z) (µ − γ) (dz, ds) +
Z tZ
0
A ∈ B(Rd ), t ≥ 0,
(ci (s, z) − c(s, z)) dz ds,
t ≥ 0.
Rd
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.33
Proof of the Theorem
Let Pt , P1t , and P2t , be the restriction of P, P1 , and P2 , respectively, on Ft .
Then, we can calculate for any α ∈ (0, 1) the process
Hα (t) :=
1
dP
dP
α 1
dP
dP
1−α
directly, by the knowledge of the Radon Nikodym derivative and the Ito formula.
In fact,
dPi
= Gi (t),
dP
where Gθ = E(ζi ), where ζθ is given by

 dζ (t) = R (J(∇θ (s, z)) − 1) (µ − γ)(dz, ds),
i
i
A
 ζi (0) = 0.
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.34
The end
Thank you for your attention !
Change of Measure formula and the Hellinger Distance of two Lévy Processes – p.35