Change of Measure formula and the Hellinger Distance of two … · The Kakutani–Hellinger Distance Let (Θ,B) be a measurable space. Deﬁnition 1 Let α ∈ (0,1). For two σ–ﬁnite

Change of Measure formula and the HellingerDistance of two Lévy Processes

Erika Hausenblas

University of Salzburg, Austria

Change of Measure formula and the Hellinger Distance of two Levy Processes – p.1

Outline

Hellinger distances

Poisson Random Measures

The Main Result

The Change of Measure formula


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) =

(

dP1

dP

)α (

dP2

dP

)1−α

,

and

hα(P1, P2) =

∫

ΘdHα(P1, P2),

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

hα(P1, P2) =

∫

ΘdHα(P1, P2).



Remark 1 The definition of the Kakutani–Hellinger affinity H isindependent 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) =

∫

ΘdKα(P1, P2),

The latter we call the Kakutani-Hellinger distance of order α betweenP1 and P2.



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).



Some Properties:

hα(P1, P2) = 0 ⇐⇒ P1 and P2 are singular;

Suppose that (Ωi,Fi), 1 ≤ i ≤ n, are measurable spaces and P 1i

and P 2i , probability measures on (Ωi,Fi), 1 ≤ i ≤ n.

Then

h 1

2

(

⊗ni=1P

1i ,⊗n

i=1P2i

)

=

n∏

i=1

h 1

2

(

P 1i , P 2

i

)

.


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;


The Lévy Process L

The Fourier Transform of L is given by the Lévy - Hinchin - Formula:

E ei〈L(1),a〉 = exp

i〈y, a〉λ +

∫

Rd

(

eiλ〈y,a〉 − 1 − iλy1|y|≤1

)

ν(dy)

,

where a ∈ Rd, y ∈ E and ν : B(Rd) → R+ is a Lévy measure.


The Lévy Process L

Definition 3 (see Linde (1986), Section 5.4) A σ–finite symmetric a

Borel-measure ν : B(Rd) → R+ is called a Lévy measure if ν(0) = 0

and the function

E′ ∋ a 7→ exp

(∫

Rd

(cos(〈x, a〉) − 1) ν(dx)

)

∈ C

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.

aν(A) = ν(−A) for all A ∈ B(Rd)


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 variablesN(t, A1), . . . , N(t, An) are pairwise independent;



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 Levy Processes – p.11


A random measure η on (S,S) is called Poisson random measure ifffor each A ∈ S such that E η(·, A) is finite, η(·, A) is a Poisson randomvariable with parameter E η(·, A).




Remark 3 The mapping

S ∋ A 7→ ν(A) := EP η(·, A) ∈ R

is a measure on (S,S).



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+

• 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), A ∈ Z, I ∈ B(R+),

ν is called the intensity of η.



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 filtrationinduced 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 timehomogeneous Poisson random measure given by

γ(A, [0, t]) = t ν(A), A ∈ B(E).


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) :=

∫ t

0

∫

R

x η(dx, ds),

and

P(L(t) = kx0) = exp(−t) tk

k! , k ∈ IN.

Since∫

Rx γ(dx, dt) = x0 dt, the compensated process is given by

∫ t

0

∫

Rx η(dx, ds) = L(t) − x0t.



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) :=∫ t

0

∫

Rx η(dx, ds),

is an α stable process and

E(e−λL(t)) = exp(−λtα).


Newman’s Result (1976)

By means of the following formula

• h 1

2

(

⊗ni=1P

1i ,⊗n

i=1P2i

)

=n

∏

i=1

h 1

2

(

P 1i , P 2

i

)

,

where above (Ωi,Fi), 1 ≤ i ≤ n, are different probability spaces and P 1i

and P 2i two probability measures, and,

• since the counting measure of a Lévy process isindependently scattered,

Newman was able to show the following:


Newman’s Result (1976)

Let L1 and L2 be two Lévy processes with Lévy measures ν1 and ν2.Let

ν 1

2

:= H 1

2

(ν1, ν2);

ai(t) :=∫

Rz

(

νi − ν 1

2

)

(dz), i = 1, 2;

Pi : B(ID(R+; R)) ∋ A 7→ P (Li + ai ∈ A) , i = 1, 2;

ThenH 1

2

(P1,P2) := exp(

−t k 1

2

(ν1, ν2))

P 1

2

,

where P 1

2

is the probability measure on ID(R+; R) of the process L 1

2

given by the Lévy measure ν 1

2

.

Inoue (1996) extended the result to non time homogeneous, butdeterministic Lévy processes. See also Liese (1987).


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.

Letz1 := 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

Y (t) := (z1(t))1

2 (z1(t))1

2 , t > 0.

Then there exists a predictable increasing process h 1

2

a, called Hellinger process,such that h 1

2

(0) = 0 and

t 7→ Y (t) +∫ t

0Y (s−)dh 1

2

(s),

is a Q-martingale. (see also Jacod (1989), Grigelionis (1994))

aIn terms of Newman, h should be k.Change of Measure formula and the Hellinger Distance of two Levy 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).


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|c(z)|2dz < ∞.

Then, the measure ν defined by

B(R) ∋ A 7→ ν(A) :=∫

R1A(c(z)) dz

is a Lévy measure and the process L given by

t 7→ L(t) :=

∫ t

0

∫

R

c(z) (µ − γ)(dz, ds),

is a (time homogeneous) Lévy process. Change of Measure formula and the Hellinger Distance of two Levy Processes – p.20


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) :=

∫ t

0

∫

R

|x|1

α η(dx, ds),

is an α stable process.


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

EP

∫ t

0

∫

Rd

|ci(s, z)|2 dz ds < ∞,

and ci(s, z) is differentiable at any z ∈ Rd∗ := R \ 0.


Main Result - The Setting



EP

∫ t

0

∫

Rd

|ci(s, z)|2 dz ds < ∞,


—————————————–

Xi = Xi(t); t ∈ R+, i = 1, 2, are two Rd–valued semimartingales given by

Xi(t) =

∫ t

0

∫

Rd

ci(s, z) (µ − γ)(dz, ds) +

∫ t

0

bi(s) ds, i = 1, 2,

and νi = νit ; t ∈ R+ are two unique predictable measure valued processes

given by

B(R) × R+ ∋ (A, t) 7→ νi

t(A) :=∫

Rd 1A(ci(t, z)) λd(dz), i = 1, 2.


The Main Result

Let ναt := Hα(ν1

t , ν2t ) and ηα be a random measure with compensator γα

defined by

γα : B(Rd) × B(R+) ∋ (A, I) 7→∫

I

∫

Rd νs(A) ds.

ait :=

∫ t

0

∫

Rd z(

νis − να

s

)

(dz) ds;

Xαt :=

∫ t

0z (ηα − γα)(dz, ds) + 1

2

∫ t

0[b1(s) + b2(s)] ds

Qi : B(ID(R+, Rd)) ∋ A 7→ P(Xi + ai ∈ A), i = 1, 2;

—————————————–

If∫ t

0(b1(s) − b2(s)) ds = −

∫ t

0

∫

Rd (c1(s, z) − c2(s, z)) dz ds,

thendHα(Q1

t ,Q2t ) = exp

(

−∫ t

0kα(ν1

s , ν2s ) ds

)

dQαt

where Qα is the probability measure on ID(R+; Rd) induced by the semimartin-

gale Xα = Xαt ; t ∈ R

+.Change of Measure formula and the Hellinger Distance of two Levy Processes – p.23

Consequences

Let ν1 and ν2 two Lévy measures on B(R), and β1, β2 ∈ (0, 2] two real numbersuch that

limz→0

z>0

νi((z,∞))

z−βiand lim

z→0

z<0

νi((−∞, z))

z−βii = 1, 2.

If β1 6= β2 and β1, β2 > 1 then k 1

2

(ν1, ν2) = ∞ and, hence, the induced measureson ID(R+, R) of the corresponding Lévy processes are singular.

Let νni := νi

∣

∣

R\(− 1

n, 1

n), β1 6= β2, and Ln

t be the corresponding Lévy processes,

i = 1, 2. Then, for any n, the induced probability measures probability measuresP n

i on ID(R+, R) (shifted by a drift term) are equivalent, but the measures Pi,i = 1, 2, given by Pi := limn→∞ P n

i are singular.


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.

Letz1 := dP1

dQ , z1 := dP2

dQ ,

and for t ≥ 0 let z1(t) and z2(t) be the restriction of z1 and z2 on Ft.

LetY (t) := (z1(t))

1

2 (z1(t))1

2 , t > 0.

Then there exists a predictable increasing process h 1

2

a, called Hellinger process,such that h 1

2

(0) = 0 and

t 7→ Y (t) +∫ t

0Y (s−)dh 1

2

(s),

is a Q-martingale. (see also Jacod (1989), Grigelionis (1994))

aIn terms of Newman, h should be k.Change of Measure formula and the Hellinger Distance of two Levy Processes – p.25

Change of Measure Formula


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

(Bichteler, Gravereaux and Jacod)

• Define a bijective mapping

θ : R∗a → R∗ possible z 7→ z + sgn(z) |z|−

1

2

aR∗ := R \ 0. Change of Measure formula and the Hellinger Distance of two Levy Processes – p.27




θ : R∗ → R∗ possible z 7→ z + sgn(z) |z|−1

2

• Define a new Poisson random measure µθ given by

B(R) × B(R+) ∋ (A, I) 7→ µθ(A, I) :=

∫

I

∫

R

χA(θ(z)) µ(dz, ds).

R∗ := R \ 0. Change of Measure formula and the Hellinger Distance of two Levy Processes – p.27




θ : R∗ → R∗ possible z 7→ z + sgn(z) |z|−1

2

• Define a new Poisson random measure µθ given by

B(R) × B(R+) ∋ (A, I) 7→ µθ(A, I) :=

∫

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 Levy Processes – p.27


B(R) × B(R+) ∋ (A, I) 7→ µθ(A, I) :=

∫

I

∫

R


LetB(R) × B(R+) ∋ (A,B) 7→ γθ(A, I) =

∫

I

∫

AJa(∇θ)(z) λ(dz)λ(ds).

andB(R) × B(R+) ∋ (A,B) 7→ γθ−1(A, I) =

∫

I

∫

AJ(∇(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.

aJ denotes the Jacobian determinant. Change of Measure formula and the Hellinger Distance of two Levy Processes – p.28


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)

dP(t)= Gθ(t);

where Gθ is the Doleans Dade exponential of ζθ, where ζθ is given by

dζθ(t) =∫

A(J(∇θ(z)) − 1) (µ − γ)(dz, ds),

ζθ(0) = 0.

In the following the Doleans Dade exponential of a process X is denoted

by E(X).



Remark 7 The same idea works also, if

θ : Ω × R+ × R

d → Rd

is predictable.


Proof of the Theorem - The Setting



EP

∫ t

0

∫

Rd

|ci(s, z)|2 dz ds < ∞,


—————————————–

Xi = Xi(t); t ∈ R+, i = 1, 2, are two Rd–valued semimartingales given by

Xi(t) =∫ t

0

∫

Rd ci(s, z) (µ − γ)(dz, ds) +∫ t

0bi(s) ds, i = 1, 2,

and νi = νit ; t ∈ R+ are two unique predictable measure valued processes

given by

B(R) × R+ ∋ (A, t) 7→ νi

t(A) :=∫

Rd 1A(ci(t, z)) λd(dz), i = 1, 2.


Proof of the Theorem

Let c : Ω × R+ × Rd → Rd be defined by

c(s, z) :=1

2[c1(s, z) + c2(s, z)]

and X0 = X0t , 0 ≤ t < ∞ be the semimartingale given by

t 7→ X0t :=

∫ t

0

∫

Rd

c(s, z) (µ − γ) (dz, ds).

Let θi(s, z) :=

c−1(s, ci(s, z)), z ∈ Rd∗, s ∈ R+,

0 z = 0, s ∈ R+, i = 1, 2,

and ji(s, z) :=

J (∇zθi(s, z)) , z ∈ Rd∗, s ∈ R

+,

0 z = 0, s ∈ R+ i = 1, 2.



For i = 1, 2 let Pi be the probability measure on (Ω,F) under which the Poissonrandom measure µθ defined by

µiθ(A, I) :=

∫

I

∫

Rd

1A(θi(s, z)) µ(dz, ds)

has compensator γ. Then

Pi(

X0t ∈ A

)

= P(

ξit ∈ A

)

, A ∈ B(Rd), t ≥ 0,

where

ξit :=

∫ t

0

∫

Rd

ci(s, z) (µ − γ) (dz, ds) +

∫ t

0

∫

Rd

(ci(s, z) − c(s, z)) dz ds, t ≥ 0.



Let Pt, P1t , and P2

t , be the restriction of P, P1, and P2, respectively, on Ft.

Then, we can calculate for any α ∈ (0, 1) the process

Hα(t) :=

(

dP1

dP

)α (

dP1

dP

)1−α

directly, by the knowledge of the Radon Nikodym derivative and the Ito formula.In fact,

dPi

dP= G

i(t),

where Gθ = E(ζi), where ζθ is given by

dζi(t) =∫

A(J(∇θi(s, z)) − 1) (µ − γ)(dz, ds),

ζi(0) = 0.


The end

Thank you for your attention !


Documents

Change of Measure formula and the Hellinger Distance of two … · The Kakutani–Hellinger Distance Let (Θ,B) be a measurable space. Deﬁnition 1 Let α ∈ (0,1). For two σ–ﬁnite