Change of Measure formula and the Hellinger Distance of two Lvy - PowerPoint PPT Presentation

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 P 1 and P 2 on (Θ , B ) we define � α � dP 2 � 1 − α � dP 1 H α ( P 1 , P 2 ) = , dP dP and � h α ( P 1 , P 2 ) = dH α ( P 1 , P 2 ) , Θ where P is a σ –finite measure such that P 1 , P 2 ≪ P . We call H α ( P 1 , P 2 ) the Hellinger-Kakutani inner product of order α of P 1 and P 2 . The total mass of H α ( P 1 , P 2 ) is written as � h α ( P 1 , P 2 ) = dH α ( P 1 , P 2 ) . Θ 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 P 1 , P 2 ≪ P holds. Definition 2 Let α ∈ (0 , 1) . For two σ –finite measures P 1 and P 2 on (Θ , B ) we define K α ( P 1 , P 2 ) = αP 1 + (1 − α ) P 2 − H α ( P 1 , P 2 ) , and � k α ( P 1 , P 2 ) = dK α ( P 1 , P 2 ) , Θ The latter we call the Kakutani-Hellinger distance of order α between P 1 and P 2 . 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: h α ( P 1 , P 2 ) = 0 ⇐ ⇒ P 1 and P 2 are singular; Suppose that (Ω i , F i ) , 1 ≤ i ≤ n , are measurable spaces and P 1 i and P 2 i , probability measures on (Ω i , F i ) , 1 ≤ i ≤ n . Then n i =1 P 1 i =1 P 2 � P 1 i , P 2 ⊗ n i , ⊗ n � � � � = h 1 h 1 . i i 2 2 i =1 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 R d –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 �→ E φ ( 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 iλ � y,a � − 1 − iλy 1 {| y |≤ 1 } E e i � L (1) ,a � = exp � � i � y, a � λ + ν ( dy ) , R d where a ∈ R d , y ∈ E and ν : B ( R d ) → R + is a Lévy measure. Change of Measure formula and the Hellinger Distance of two L´ evy Processes – p.8

The Lévy Process L a Definition 3 (see Linde (1986), Section 5.4) A σ –finite symmetric Borel-measure ν : B ( R d ) → R + is called a Lévy measure if ν ( { 0 } ) = 0 and the function �� � E ′ ∋ a �→ exp R d (cos( � x, a � ) − 1) ν ( dx ) ∈ C is a characteristic function of a certain Radon measure on R d . 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 ( R d ) 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 } 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 A 1 , . . . , A n , the random variables N ( t, A 1 ) , . . . , N ( t, A n ) 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 } 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 A 1 , . . . , A n ∈ S , the random variables η ( · , A 1 ) , . . . , η ( · , A n ) 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 ) := E P 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 + A ∈ Z , I ∈ B ( R + ) , such that E η ( � , A, I ) = ν ( A ) λ ( I ) , 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 {F t , 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 x 0 ∈ R , x 0 � = 0 and set ν = δ x 0 . Let η be the time homogeneous Poisson random measure on R with intensity ν . Then � t � t �→ L ( t ) := x η ( dx, ds ) , 0 R and P ( L ( t ) = kx 0 ) = exp( − t ) t k k ∈ I N . k ! , � Since R x γ ( dx, dt ) = x 0 dt , the compensated process is given by � t � η ( dx, ds ) = L ( t ) − x 0 t. R x ˜ 0 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 ) = x α − 1 . Let η be the time homogeneous Poisson random measure on R with intensity ν . Then � t � t �→ L ( t ) := R x ˜ η ( dx, ds ) , 0 is an α stable process and E ( e − λL ( t ) ) = exp( − λt α ) . 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 n � i =1 P 1 i =1 P 2 P 1 i , P 2 ⊗ n i , ⊗ n � � � � • = h 1 h 1 , i i 2 2 i =1 where above (Ω i , F i ) , 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 L 1 and L 2 be two Lévy processes with Lévy measures ν 1 and ν 2 . Let 2 := H 1 2 ( ν 1 , ν 2 ) ; ν 1 � � � ν i − ν 1 i = 1 , 2 ; a i ( t ) := R z ( dz ) , 2 P i : B (I D( R + ; R )) ∋ A �→ P ( L i + a i ∈ A ) , i = 1 , 2 ; Then � � 2 ( P 1 , P 2 ) := exp − t k 1 2 ( ν 1 , ν 2 ) P 1 H 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