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

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 P 1 and P 2 on (Θ, B) we define H α (P 1,P 2 ) = ( dp1 dp ) α ( dp2 dp ) 1 α, and h α (P 1,P 2 ) = Θ dh α (P 1,P 2 ), 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 ) = Θ dh α (P 1,P 2 ). 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 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 and K α (P 1,P 2 ) = αp 1 + (1 α)p 2 H α (P 1,P 2 ), k α (P 1,P 2 ) = Θ dk α (P 1,P 2 ), 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 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: 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 h1 2 ( n i=1 Pi 1, n i=1pi 2 ) n = i=1 h1 2 ( P 1 i,pi 2 ). 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 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 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: E e i L(1),a = exp { i y,a λ + R d (e iλ y,a 1 iλy1 { y 1} ) where a R d, y E and ν : B(R d ) R + is a Lévy measure. } ν(dy), 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(R d ) R + is called a Lévy measure if ν({0}) = 0 and the function ( ) E a exp (cos( x,a ) 1) ν(dx) C R d a 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 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 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 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 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 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 Eη(,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 Eη(,A) is finite, η(,a) is a Poisson random variable with parameter E η(, A). Remark 3 The mapping is a measure on (S, S). S A ν(a) := E P η(,a) R Change of Measure formula and the Hellinger Distance of two Lévy 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 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 {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 is a local martingale over (Ω; F; P). η(a, (0, t]) γ(a, [0, t]) 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 x 0 R, x 0 0 and set ν = δ x0. Let η be the time homogeneous Poisson random measure on R with intensity ν. Then t t L(t) := 0 R xη(dx,ds), and P(L(t) = kx 0 ) = exp( t) tk k!, k IN. Since R xγ(dx,dt) = x 0 dt, the compensated process is given by t 0 R x η(dx,ds) = L(t) x 0t. 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) = x α 1. Let η be the time homogeneous Poisson random measure on R with intensity ν. Then t L(t) := t 0 R x η(dx,ds), is an α stable process and E(e λl(t) ) = exp( λt α ). 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 ( n i=1 Pi 1, n i=1pi 2 ) n = i=1 h1 2 ( P 1 i,pi 2 ), 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 Lévy 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 Then ν1 2 := H1(ν 1,ν 2 ); 2 a i (t) := R z (ν i ν1 2 ) (dz), i = 1,2; P i : B(ID(R + ; R)) A P(L i + a i A), i = 1,2; where P1 2 ( ) H1(P 1, P 2 ) := exp t k1(ν 1,ν 2 ) P1, 2 2 2 is the probability measure on ID(R + ; R) of the process L1 2. given by the Lévy measure ν1 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 L 1 and L 2 be two semimartingales (here, of pure jump type); P i, i = 1, 2, be the probability measures induced from L 1 and L 2 on ID(R + ; R); Q be a measure such that P 1 Q and P 2 Q. Let z 1 := dp 1 dq, z 1 := dp 2 dq, and for t 0 let z 1 (t) and z 2 (t) be the restriction of z 1 and z 2 on F t. Let Y (t) := (z 1 (t)) 1 2 (z 1 (t)) 1 2, t > 0. Then there exists a predictable increasing process h a 1, called Hellinger process, 2 such that h 1 (0) = 0 and 2 t Y (t) + t 0 Y (s )dh 1(s), 2 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 + ) IN 0 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 Lévy Processes p.20

Prm - Bismut s Setting Let (Ω; F; P) a probability space and µ : B(R) B(R + ) IN 0 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) 2 dz <. Then, the measure ν defined by B(R) A ν(a) := R 1 A(c(z)) dz is a Lévy measure and the process L given by t 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 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 t L(t) := 0 R x 1 α η(dx,ds), is an α stable process. Change of Measure formula and the Hellinger Distance of two Lévy Processes p.21

Main Result - The Setting b i = {b i (s); s R + } are two predictable R d valued processes, c i : Ω R + R d R d, i = 1, 2, be two predictable processes such that t E P 0 R d c i (s, z) 2 dz ds <, and c i (s, z) is differentiable at any z R d := R \ {0}. Change of Measure formula and the Hellinger Distance of two Lévy Processes p.22

Main Result - The Setting b i = {b i (s); s R + } are two predictable R d valued processes, c i : Ω R + R d R d, i = 1, 2, be two predictable processes such that E P t 0 R d c i (s, z) 2 dz ds <, and c i (s, z) is differentiable at any z R d := R \ {0}. X i = {X i (t); t R + }, i = 1, 2, are two R d valued semimartingales given by X i (t) = t 0 R d c i (s, z) (µ γ)(dz, ds) + t 0 b i (s) ds, i = 1, 2, and ν i = {ν i t; t R + } are two unique predictable measure valued processes given by B(R) R + (A, t) ν i t(a) := R d 1 A (c i (t, z)) λ d (dz), i = 1, 2. Change of Measure formula and the Hellinger Distance of two Lévy Processes p.22

The Main Result If then Let νt α := H α (νt 1, νt 2 ) and η α be a random measure with compensator γ α defined by γ α : B(R d ) B(R + ) (A, I) ν I R d s (A) ds. a i t := t z ( ) ν i 0 R d s νs α (dz) ds; Xt α := t 0 z (ηα γ α )(dz, ds) + 1 t 2 0 [b 1(s) + b 2 (s)] ds Q i : B(ID(R +, R d )) A P(X i + a i A), i = 1, 2; t 0 (b 1(s) b 2 (s)) ds = t (c 0 R d 1 (s, z) c 2 (s, z)) dz ds, dh α (Q 1 t, Q 2 t) = exp ( ) t 0 k α(νs, 1 νs) 2 ds where Q α is the probability measure on ID(R + ; R d ) induced by the semimartingale X α = {X α t ; t R + }. dq α t Change of Measure formula and the Hellinger Distance of two Lévy Processes p.23

Consequences Let ν 1 and ν 2 two Lévy measures on B(R), and β 1, β 2 (0, 2] two real number such that ν i ((z, )) ν i ((, z)) z β and i z β i = 1, 2. i lim z 0 z>0 lim z 0 z<0 If β 1 β 2 and β 1, β 2 > 1 then k 1 2 (ν 1, ν 2 ) = and, hence, the induced measures on ID(R +, R) of the corresponding Lévy processes are singular. Let ν n i := ν i R\( 1 n, 1 n ), β 1 β 2, and L n t be the corresponding Lévy processes, i = 1, 2. Then, for any n, the induced probability measures probability measures P n i on ID(R +, R) (shifted by a drift term) are equivalent, but the measures P i, i = 1, 2, given by P i := lim n P n i are singular. Change of Measure formula and the Hellinger Distance of two Lévy Processes p.24

Jacod s and Shiryaev s Approach L 1 and L 2 be two semimartingales (here, of pure jump type); P i, i = 1, 2, be the probability measures induced from L 1 and L 2 on ID(R + ; R); Q be a measure such that P 1 Q and P 2 Q. Let z 1 := dp 1 dq, z 1 := dp 2 dq, and for t 0 let z 1 (t) and z 2 (t) be the restriction of z 1 and z 2 on F t. Let Y (t) := (z 1 (t)) 1 2 (z 1 (t)) 1 2, t > 0. Then there exists a predictable increasing process h a 1, called Hellinger process, 2 such that h 1 (0) = 0 and 2 t Y (t) + t 0 Y (s )dh 1(s), 2 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 Eη(,A) is finite, η(,a) is a Poisson random variable with parameter E η(, A). Remark 6 The mapping is a measure on (S, S). S A ν(a) := E P η(,a) R 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 θ : R a R possible z z + sgn(z) z 1 2 a R := R \ {0}. 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 possible z z + sgn(z) z 1 2 Define a new Poisson random measure µ θ given by B(R) B(R + ) (A,I) µ θ (A,I) := χ A (θ(z)) µ(dz,ds). I R R := R \ {0}. 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 possible z z + sgn(z) z 1 2 Define a new Poisson random measure µ θ given by B(R) B(R + ) (A,I) µ θ (A,I) := χ A (θ(z)) µ(dz,ds). I R Define a new probability measure P θ on (Ω; F) by saying: µ θ 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) µ θ (A,I) := I R χ A (θ(z)) µ(dz,ds). Let B(R) B(R + ) (A,B) γ θ (A,I) = I A J a ( θ)(z) λ(dz)λ(ds). and B(R) B(R + ) (A,B) γ θ 1(A,I) = I A J( (θ) 1 )(z) λ(dz)λ(ds). The following can be shown: µ has compensator γ under P. µ θ has compensator γ under P θ. µ has compensator γ θ under P θ. µ θ has compensator γ θ 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 P θ (t) be the restriction of P and P θ on F t. Then Radon Nikodym derivative is given by dp θ (t) dp(t) = G θ(t); where G θ is the Doleans Dade exponential of ζ θ, where ζ θ is given by dζ θ (t) = A (J( θ(z)) 1) (µ γ)(dz,ds), ζ θ (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 is predictable. θ : Ω R + R d R d Change of Measure formula and the Hellinger Distance of two Lévy Processes p.30

Proof of the Theorem - The Setting b i = {b i (s); s R + } are two predictable R d valued processes, c i : Ω R + R d R d, i = 1, 2, be two predictable processes such that E P t 0 R d c i (s, z) 2 dz ds <, and c i (s, z) is differentiable at any z R d := R \ {0}. X i = {X i (t); t R + }, i = 1, 2, are two R d valued semimartingales given by X i (t) = t c 0 R d i (s, z) (µ γ)(dz, ds) + t 0 b i(s) ds, i = 1, 2, and ν i = {ν i t; t R + } are two unique predictable measure valued processes given by B(R) R + (A, t) ν i t(a) := R d 1 A (c i (t, z)) λ d (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 + R d R d be defined by c(s, z) := 1 2 [c 1(s, z) + c 2 (s, z)] and X 0 = {Xt 0, 0 t < } be the semimartingale given by Let θ i (s, z) := t 0 t Xt 0 := c(s, z) (µ γ) (dz, ds). R d c 1 (s, c i (s, z)), z R d, s R +, 0 z = 0, s R +, i = 1, 2, J ( z θ i (s, z)), z R d, s R +, and j i (s, z) := 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 P i be the probability measure on (Ω, F) under which the Poisson random measure µ θ defined by µ i θ(a, I) := 1 A (θ i (s, z)) µ(dz, ds) I R d has compensator γ. Then P i ( {X 0 t A} ) = P ( {ξ i t A} ), A B(R d ), t 0, where ξ i t := t 0 R d c i (s, z) (µ γ)(dz, ds) + t 0 R d (c i (s, z) c(s, z)) dz ds, t 0. Change of Measure formula and the Hellinger Distance of two Lévy Processes p.33

Proof of the Theorem Let P t, P 1 t, and P 2 t, be the restriction of P, P 1, and P 2, respectively, on F t. Then, we can calculate for any α (0, 1) the process H α (t) := ( ) dp 1 α ( dp 1 dp dp ) 1 α directly, by the knowledge of the Radon Nikodym derivative and the Ito formula. In fact, dp i dp = Gi (t), where G θ = E(ζ i ), where ζ θ is given by dζ i (t) = A (J( θ i(s, z)) 1) (µ γ)(dz, ds), ζ i (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