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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 L´ evy 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 L´ evy 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) = dP1 dP α dP2 dP 1−α , and hα(P1, P2) =

• Θ

dHα(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) =

• Θ

dHα(P1, P2).

Change of Measure formula and the Hellinger Distance of two L´ evy 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) =

• Θ

dKα(P1, P2),

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 L´ evy 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 L´ evy 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 P 1

i

and P 2

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

Then

h 1

2

• ⊗n

i=1P 1 i , ⊗n i=1P 2 i

• =

n

• i=1

h 1

2

• P 1

i , P 2 i

• .

Change of Measure formula and the Hellinger Distance of two L´ evy 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 → 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 L´ evy Processes – p.7

The Lévy Process L

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

E eiL(1),a = exp

• iy, aλ +
• Rd
• eiλy,a − 1 − iλy1{|y|≤1}
• ν(dy)
• ,

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

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.8

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 → 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)

Change of Measure formula and the Hellinger Distance of two L´ evy 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 → N(t, A) = ♯ {s ∈ (0, t] : ∆L(s) = L(s) − L(s−) ∈ A}

• ne 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 L´ evy 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}

• f 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 L´ evy 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 L´ evy 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 → ν(A) := EP η(·, A) ∈ R

is a measure on (S, S).

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.12

Poisson Random Measure

Let (Z, Z) be a measurable space. If S = Z × R+, S = Z ˆ ×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 ˆ

×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.

Change of Measure formula and the Hellinger Distance of two L´ evy 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 L´ evy Processes – p.14

Poisson random measures

Example 1 Let x0 ∈ R, x0 = 0 and set ν = δx0. Let η be the time homogeneous Poisson random measure on R with intensity ν. Then t → L(t) := t

• R

x η(dx, ds), and P(L(t) = kx0) = exp(−t) tk

k!,

k ∈ I N. Since

• R x γ(dx, dt) = x0 dt, the compensated process is given by

t

• R x ˜

η(dx, ds) = L(t) − x0t.

Change of Measure formula and the Hellinger Distance of two L´ evy 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 → L(t) := t

• R x ˜

η(dx, ds), is an α stable process and E(e−λL(t)) = exp(−λtα).

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.16

Newman’s Result (1976)

By means of the following formula

• h 1

2

• ⊗n

i=1P 1 i , ⊗n i=1P 2 i

• =

n

• i=1

h 1

2

• P 1

i , P 2 i

• ,

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

i

and P 2

i 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 L´ evy Processes – p.17

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) :=

• R z
• νi − ν 1

2

• (dz),

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

Then

H 1

2 (P1, P2) := exp

• −t k 1

2 (ν1, ν2)

• P 1

2 ,

where P 1

2 is the probability measure on I

D(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, but deterministic Lévy processes. See also Liese (1987).

Change of Measure formula and the Hellinger Distance of two L´ evy 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 I

D(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 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 → Y (t) + t

0 Y (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 L´ evy Processes – p.19

Prm - Bismut’s Setting

Let (Ω; F; P) a probability space and µ : B(R) × B(R+) − → I N0 a Poisson random measure over (Ω; F; P) with compensator γ given by B(R) × B(R+) ∋ (A, I) → γ(A, I) := λ(A)λ(I).

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.20

Prm - Bismut’s Setting

Let (Ω; F; P) a probability space and µ : B(R) × B(R+) − → I N0 a Poisson random measure over (Ω; F; P) with compensator γ given by B(R) × B(R+) ∋ (A, I) → 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 → ν(A) :=

• R 1A(c(z)) dz

is a Lévy measure and the process L given by t → L(t) := t

• R

c(z) (µ − γ)(dz, ds), is a (time homogeneous) Lévy process.

Change of Measure formula and the Hellinger Distance of two L´ evy 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 → L(t) := t

• R

|x|

1 α ˜

η(dx, ds), is an α stable process.

Change of Measure formula and the Hellinger Distance of two L´ evy 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 EP t

• Rd |ci(s, z)|2 dz ds < ∞,

and ci(s, z) is differentiable at any z ∈ Rd

∗ := R \ {0}.

Change of Measure formula and the Hellinger Distance of two L´ evy 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 EP t

• Rd |ci(s, z)|2 dz ds < ∞,

and ci(s, z) is differentiable at any z ∈ Rd

∗ := R \ {0}.

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

• Rd ci(s, z) (µ − γ)(dz, ds) +

t bi(s) ds, i = 1, 2, and νi = {νi

t; t ∈ R+} are two unique predictable measure valued processes

given by B(R) × R+ ∋ (A, t) → νi

t(A) :=

• Rd 1A(ci(t, z)) λd(dz),

i = 1, 2.

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.22

The Main Result

Let να

t := Hα(ν1 t , ν2 t ) and ηα be a random measure with compensator γα

defined by γα : B(Rd) × B(R+) ∋ (A, I) →

• I
• Rd νs(A) ds.

ai

t :=

t

• Rd z
• νi

s − να s

• (dz) ds;

Xα

t :=

t

0 z (ηα − γα)(dz, ds) + 1 2

t

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

Qi : B(I D(R+, Rd)) ∋ A → P(Xi + ai ∈ A), i = 1, 2; —————————————– If t

0 (b1(s) − b2(s)) ds = −

t

• Rd (c1(s, z) − c2(s, z)) dz ds,

then dHα(Q1

t, Q2 t) = exp

• −

t

0 kα(ν1 s, ν2 s) ds

• dQα

t

where Qα is the probability measure on I D(R+; Rd) induced by the semimartin- gale Xα = {Xα

t ; t ∈ R+}.

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.23

Consequences

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

z→0 z>0

νi((z, ∞)) z−βi and lim

z→0 z<0

νi((−∞, z)) z−βi i = 1, 2. If β1 = β2 and β1, β2 > 1 then k 1

2 (ν1, ν2) = ∞ and, hence, the induced measures

• n I

D(R+, R) of the corresponding Lévy processes are singular. Let νn

i := νi

• R\(− 1

n , 1 n ), β1 = β2, and Ln

t be the corresponding Lévy processes,

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

i on I

D(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.

Change of Measure formula and the Hellinger Distance of two L´ evy 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 I

D(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 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 → Y (t) + t

0 Y (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 L´ evy 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 → 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 L´ evy Processes – p.26

Change of measure formula

(Bichteler, Gravereaux and Jacod)

• Define a bijective mapping

θ : R∗

a → R∗

possible z → z + sgn(z) |z|− 1

2

aR∗ := R \ {0}.

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.27

Change of measure formula

(Bichteler, Gravereaux and Jacod)

• Define a bijective mapping

θ : R∗ → R∗

possible z → z + sgn(z) |z|− 1

2

• Define a new Poisson random measure µθ given by

B(R) × B(R+) ∋ (A, I) → µθ(A, I) :=

• I
• R

χA(θ(z)) µ(dz, ds). R∗ := R \ {0}.

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.27

Change of measure formula

(Bichteler, Gravereaux and Jacod)

• Define a bijective mapping

θ : R∗ → R∗

possible z → z + sgn(z) |z|− 1

2

• Define a new Poisson random measure µθ given by

B(R) × B(R+) ∋ (A, I) → µθ(A, I) :=

• I
• R

χA(θ(z)) µ(dz, ds).

• Define

a new probability measure

Pθ

• n

(Ω; F)

by saying:

µθ has compensator γ = λ × λ. R∗ := R \ {0}.

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.27

Change of measure formula

B(R) × B(R+) ∋ (A, I) → µθ(A, I) :=

• I
• R

χA(θ(z)) µ(dz, ds).

Let

B(R) × B(R+) ∋ (A, B) → γθ(A, I) =

• I
• A J a(∇θ)(z) λ(dz)λ(ds).

and

B(R) × B(R+) ∋ (A, B) → 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.

aJ denotes the Jacobian determinant.

Change of Measure formula and the Hellinger Distance of two L´ evy 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) 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).

Change of Measure formula and the Hellinger Distance of two L´ evy 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 L´ evy 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 EP t

• Rd |ci(s, z)|2 dz ds < ∞,

and ci(s, z) is differentiable at any z ∈ Rd

∗ := R \ {0}.

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

• Rd ci(s, z) (µ − γ)(dz, ds) +

t

0 bi(s) ds,

i = 1, 2, and νi = {νi

t; t ∈ R+} are two unique predictable measure valued processes

given by B(R) × R+ ∋ (A, t) → νi

t(A) :=

• Rd 1A(ci(t, z)) λd(dz),

i = 1, 2.

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.31

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 = {X0

t , 0 ≤ t < ∞} be the semimartingale given by

t → X0

t :=

t

• Rd c(s, z) (µ − γ) (dz, ds).

Let θi(s, z) :=    c−1(s, ci(s, z)), z ∈ Rd

∗, s ∈ R+,

z = 0, s ∈ R+, i = 1, 2, and ji(s, z) :=    J (∇zθi(s, z)) , z ∈ Rd

∗, s ∈ R+,

z = 0, s ∈ R+ i = 1, 2.

Change of Measure formula and the Hellinger Distance of two L´ evy 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 µi

θ(A, I) :=

• I
• Rd 1A(θi(s, z)) µ(dz, ds)

has compensator γ. Then Pi {X0

t ∈ A}

• = P
• {ξi

t ∈ A}

• ,

A ∈ B(Rd), t ≥ 0, where ξi

t :=

t

• Rd ci(s, z) (µ − γ) (dz, ds) +

t

• Rd (ci(s, z) − c(s, z)) dz ds,

t ≥ 0.

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.33

Proof of the Theorem

Let Pt, P1

t , 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 = Gi(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.

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.34

The end Thank you for your attention !

Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.35