Logarithmic derivatives of densities for jump processes

Transcription

1 Logarithmic derivatives of densities for jump processes Atsushi AKEUCHI Osaka City University (JAPAN) June 3, 29 City University of Hong Kong Workshop on Stochastic Analysis and Finance (June 29 - July 3, 29) A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 1 / 25

2 Preliminaries dν: the Lévy measure on := R\{} A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 2 / 25

3 Preliminaries dν: the Lévy measure on := R\{} θ dν + θ p dν < for any p 1, θ 1 θ >1 there exists α > such that lim inf ρ ρα there exists a C 1 -density g(θ) such that ( ) θ/ρ 2 1 dν >, lim g(θ) =. θ A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 2 / 25

4 Preliminaries dν: the Lévy measure on := R\{} θ dν + θ p dν < for any p 1, θ 1 θ >1 there exists α > such that lim inf ρ ρα there exists a C 1 -density g(θ) such that a (ε, y), a(ε, y) C 1, 1+,b (R R) b(ε, y, θ) C 1,, 1+,b (R R ) inf inf 1 + b (ε, y, θ) >, y R θ inf a(ε, y R y)2 >, ( ) θ/ρ 2 1 dν >, lim g(θ) =. θ lim b(ε, y, θ) = θ inf inf θ b(ε, y, θ) 2 > y R θ A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 2 / 25

5 Example 1.1 Let a, b, c >, and β < 1. Define } dθ dν = a {e bθ I (θ<) + e cθ I (θ>) θ 1+β which is a special case of CGMY process. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 3 / 25

6 Example 1.1 Let a, b, c >, and β < 1. Define } dθ dν = a {e bθ I (θ<) + e cθ I (θ>) θ 1+β which is a special case of CGMY process. In particular, gamma process: b = +, β = variance gamma process: β = tempered stable process: b = +, < β < 1 inverse Gaussian process: b = +, β = 1/2. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 3 / 25

7 For each (ε, x) R 2, consider the stochastic differential equation: dx t = a (ε, x t )dt+a(ε, x t ) dw t + b(ε, x t, θ)dj, x = x {W t ; t [, ]}: 1-dimensional Brownian motion dj: the Poisson random measure on [, ] dt dν: the intensity d J = dj dt dν, dj = I ( θ 1) d J + I ( θ >1) dj A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 4 / 25

8 For each (ε, x) R 2, consider the stochastic differential equation: dx t = a (ε, x t )dt+a(ε, x t ) dw t + b(ε, x t, θ)dj, x = x {W t ; t [, ]}: 1-dimensional Brownian motion dj: the Poisson random measure on [, ] dt dν: the intensity he associated infinitesimal generator is L ε f(y) = A ε f(y) + Aε A ε f(y) + 2 d J = dj dt dν, dj = I ( θ 1) d J + I ( θ >1) dj { B ε θ f(y) I ( θ 1)B ε θ f(y)} dν ( B ε θf(y) := f(y + b(ε, y, θ)) f(y)) A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 4 / 25

9 Proposition 1.2 he mapping R x x t R has a C 1 -modification such that Z t := x x t satisfies the linear SDE: dz t = a (ε, x t)z t dt + a (ε, x t )Z t dw t + b (ε, x t, θ)z t dj. Z t is invertible a.s. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 5 / 25

10 Proposition 1.2 he mapping R x x t R has a C 1 -modification such that Z t := x x t satisfies the linear SDE: dz t = a (ε, x t)z t dt + a (ε, x t )Z t dw t + b (ε, x t, θ)z t dj. Z t is invertible a.s. he mapping R ε x t R has a C 1 -modification such that H t := ε x t satisfies the SDE: dh t = a (ε, x t)h t dt + a (ε, x t )H t dw t + b (ε, x t, θ)h t dj R + ε a (ε, x t )dt + ε a(ε, x t ) dw t + ε b(ε, x t, θ)dj. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 5 / 25

11 Existence of smooth densities Let b(ε, y, θ) := [ θ b/(1 + b ) ] (ε, y, θ) θ. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 6 / 25

12 Existence of smooth densities Let b(ε, y, θ) := [ θ b/(1 + b ) ] (ε, y, θ) θ. Under the conditions inf a(ε, y R y)2 > and inf inf θ b(ε, y, θ) 2 >, y R θ there exists α > such that lim inf ρ ρα there exists γ > such that { a(ε, y)/ρ 2 + inf y R for < ρ < 1. ( ) θ/ρ 2 1 dν >, ( ) } b(ε, y, θ)/ρ 2 1 dν c ρ γ A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 6 / 25

13 Existence of smooth densities Let b(ε, y, θ) := [ θ b/(1 + b ) ] (ε, y, θ) θ. Under the conditions inf a(ε, y R y)2 > and inf inf θ b(ε, y, θ) 2 >, y R θ there exists α > such that lim inf ρ ρα there exists γ > such that { a(ε, y)/ρ 2 + inf y R ( ) θ/ρ 2 1 dν >, ( ) } b(ε, y, θ)/ρ 2 1 dν c ρ γ for < ρ < 1. hen, for each (x, ε) R 2, there exists a smooth density p x,ε (y) for x. (cf. [. 22]) A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 6 / 25

14 Main problem C LG (R) = {f C(R) ; f(y) c (1 + y )} { n } F = α k f k I Ak ; α k R, f k C LG (R), A k R: interval k=1 A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 7 / 25

15 Main problem C LG (R) = {f C(R) ; f(y) c (1 + y )} { n } F = α k f k I Ak ; α k R, f k C LG (R), A k R: interval k=1 Goal x R and ε R: For φ F, compute the differentials of E [φ(x )] in x (E[φ(x )]) = E [ φ(x ) Γ x ] ε (E[φ(x )]) = E [ φ(x ) Γ ε ] [ ] x 2 (E[φ(x )]) = E φ(x ) Γ x (logarithmic derivatives of p x,ε (y), computations of the Greeks) A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 7 / 25

16 Sensitivity with respect to the initial point heorem 1 (Sensitivity in x R, [. 8] ) For φ F, it holds that x (E[φ(x )]) = E [ φ(x ) Γ x ], Γ x = Lx V x + Kx A A 2. [ 1 + b A = + θ 2 ] dj, v(ε, t, θ) = (ε, x t, θ) Z t θ 2, θ b t L x t = Z t { } s a(ε, x s ) dw s, V x θ g(θ)v(ε, s, θ) t = d J, g(θ) t K x t = 2θv(ε, s, θ)dj A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 8 / 25

17 Remark 4.1 ( ) Recall A = + θ 2 dj. Let N λ = e λ θ 2 1 dν. Under the condition on dν: ( lim inf ρ ρα θ/ρ 2 1 ) dν >, A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 9 / 25

18 Remark 4.1 ( ) Recall A = + θ 2 dj. Let N λ = e λ θ 2 1 dν. Under the condition on dν: ( lim inf ρ ρα θ/ρ 2 1 ) dν >, it holds that, for any p > 1, [ E A p ] = 1 c c Γ(p) <. [ ( )] λ p 1 E exp λa N λ e N λ dλ [ λ p 1 exp λ c [ λ p 1 exp λ c λ α/2 { (λ θ 2 ) 1 } dν ] dλ ] dλ A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 9 / 25

19 Key Lemmas Let φ CK 2 (R) for a while. Recall stochastic differential equation: dx t = a (ε, x t )dt + a(ε, x t ) dw t + b(ε, x t, θ)dj x = x infinitesimal generator: L ε = A ε + Aε A ε { + B ε 2 θ I ( θ 1) B ε } θ dν ( B ε θ f(y) := f(y + b(ε, y, θ)) f(y) ) A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 1 / 25

20 Let u(t, x) := E [ φ ( x t ) x = x ] (t [, ), x R). A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

21 Let u(t, x) := E [ φ ( x t ) x = x ] (t [, ), x R). hen, we see u C 1,2 b ([, ) R), t u + L ε u =, lim u(t, x) = φ(x). t A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

22 Let u(t, x) := E [ φ ( x t ) x = x ] (t [, ), x R). hen, we see u C 1,2 b ([, ) R), t u + L ε u =, lim u(t, x) = φ(x). t Applying the Itô formula to u(t, x t ) for t <, we have u(t, x t ) = u(, x) + t u (s, x s )a(ε, x s )dw s + t B ε θ u(s, x s )d J. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

23 Let u(t, x) := E [ φ ( x t ) x = x ] (t [, ), x R). hen, we see u C 1,2 b ([, ) R), t u + L ε u =, lim u(t, x) = φ(x). t Applying the Itô formula to u(t, x t ) for t <, we have u(t, x t ) = u(, x) + t u (s, x s )a(ε, x s )dw s + t B ε θ u(s, x s )d J. aking the limit as t enables us to get the following lemma. Lemma 5.1 φ(x ) = E [φ(x )] + + u (s, x s )a(ε, x s )dw s B ε θ u(s, x s )d J A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

24 Lemma 5.2 x (E[φ(x )]) = E [ φ(x ) L x ] A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

25 Lemma 5.2 x (E[φ(x )]) = E [ φ(x ) L x ] Proof of Lemma 5.2 We have already seen in Lemma 5.1 that φ(x ) = E[φ(x )] + u (s, x s )a(ε, x s )dw s + B ε θ u(s, x s )d J Multiplying the above equality by L x = Z t a(ε, x t ) dw t, we have [ ] (RHS) = E u Z t (t, x t )a(ε, x t )dw t a(ε, x t ) dw t = = E [ u (t, x t )Z t ] dt x (E[u(t, x t )]) dt = (LHS) A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

26 Lemma 5.3 [ x (E φ(x ) ]) θ 2 dj = E [ φ(x ) V x ] A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

27 Lemma 5.3 [ x (E φ(x ) Proof of Lemma 5.3 ]) θ 2 dj We have already seen in Lemma 5.1 that φ(x ) = E[φ(x )] + Multiplying the above equality by u (s, x s )a(ε, x s )dw s + = E [ φ(x ) V x ] B ε θ u(s, x s )d J [ ] [ ] E φ(x ) θ 2 d J = E B ε θ u(t, x t) θ 2 dt dν. θ 2 d J, we have A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

28 Recall V x see that = θ {g(θ) v(ε, t, θ)} g(θ) d J. Since lim g(θ) =, we θ E [ φ(x ) V x ] [ = E B ε θ u(t, x t) ] θ {g(θ)v(ε, t, θ)} dt dν g(θ) [ ] = E u (t, x t + b(ε, x t, θ)) (1 + b (ε, x t, θ)) Z t θ 2 dt dν ( [ 1 + b ) lim g(θ) =, v(ε, t, θ) = ](ε, x t, θ)z t θ 2 θ θ b [ ]) = x (E u(t, x t + b(ε, x t, θ)) θ 2 dt dν [ ]) = x (E φ(x ) θ 2 dj. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

29 Proof of heorem 1 It is sufficient to study φ CK 2 (R), instead of φ F, via the standard density argument. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

30 Proof of heorem 1 It is sufficient to study φ CK 2 (R), instead of φ F, via the standard density argument. Our goal is [ x (E[φ(x )]) = E φ(x ) { L x V x A }] + Kx A 2. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

31 Proof of heorem 1 It is sufficient to study φ CK 2 (R), instead of φ F, via the standard density argument. Our goal is [ x (E[φ(x )]) = E φ(x ) { L x V x A }] + Kx A 2. We have already obtained x (E[φ(x ) A ]) = E [ φ(x ) { L x V x }]. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

32 Proof of heorem 1 It is sufficient to study φ CK 2 (R), instead of φ F, via the standard density argument. Our goal is [ x (E[φ(x )]) = E φ(x ) We have already obtained { L x V x A }] + Kx A 2. x (E[φ(x ) A ]) = E [ φ(x ) { L x V x }]. hus, we have to consider how to get rid of A = + θ 2 dj. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

33 Remark that [ ] φ(x )A E[φ(x )] = E = where dp λ { dp = exp F M λ = exp { λ A λ θ 2 dj M λ Eλ [φ(x )A ]dλ, ( 1 e λ θ 2) dt dν ( ) } e λ θ 2 1 dt dν } : deterministic A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

34 Remark that [ ] φ(x )A E[φ(x )] = E = where dp λ { dp = exp F M λ = exp { λ A λ θ 2 dj M λ Eλ [φ(x )A ]dλ, ( 1 e λ θ 2) dt dν From the Girsanov theorem, we see that, under P λ, {W t ; t [, ]}: the Brownian motion, ( ) } e λ θ 2 1 dt dν } : deterministic dj: the Poisson random measure with the intensity e λ θ 2 dt dν, d J λ = dj e λ θ 2 dt dν: the martingale measure. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

35 We have already seen that ) x (E λ [φ(x )A ] t V x,λ t := θ = E λ[ { φ(x ) }] L x V x,λ. } { e λ θ 2 g(θ)v(ε, s, θ) e λ θ 2 g(θ) d J λ A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

36 We have already seen that ) x (E λ [φ(x )A ] t V x,λ t := Hence we have M λ Eλ[ φ(x )L x θ = E λ[ { φ(x ) }] L x V x,λ. } { e λ θ 2 g(θ)v(ε, s, θ) e λ θ 2 g(θ) ] dλ = E [( [ = E φ(x ) Lx A ) e λa dλ ]. d J λ φ(x )L x ] A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

37 We have already seen that ) x (E λ [φ(x )A ] t V x,λ t := Hence we have Since V x,λ M λ Eλ[ φ(x )L x = V x M λ Eλ[ φ(x ) V x,λ θ = E λ[ { φ(x ) }] L x V x,λ. } { e λ θ 2 g(θ)v(ε, s, θ) e λ θ 2 g(θ) ] dλ = E [( [ = E φ(x ) Lx A λkx, we see that ] [ dλ = E [ = E φ(x ) e λa ) e λa dλ { V x A ]. φ(x ) { V x Kx A 2 d J λ φ(x )L x ] ] } λkx dλ }]. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

38 [ { L x x (E[φ(x )]) = E φ(x ) V x }] + Kx A A 2 A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

39 [ { L x x (E[φ(x )]) = E φ(x ) V x }] + Kx A A 2 In a similar manner, we can derive other sensitivity formulae: [ ] x 2 (E[φ(x )]) = E φ(x ) Γ x : Gamma ε (E[φ(x )]) = E [ φ(x ) Γ ε ] : Vega etc. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

40 [ { L x x (E[φ(x )]) = E φ(x ) V x }] + Kx A A 2 In a similar manner, we can derive other sensitivity formulae: [ ] x 2 (E[φ(x )]) = E φ(x ) Γ x : Gamma ε (E[φ(x )]) = E [ φ(x ) Γ ε ] : Vega etc. Under the hypoelliptic situation, instead of the uniform ellipticity, the martingale method can be applied to the sensitivity on [ ( 1 )] E φ x t dt. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

41 Remark We have already obtained in heorem 1 that Under the uniformly elliptic condition: (a) inf y R a(ε, y)2 >, (b) inf y R inf θ θ b(ε, y, θ) 2 >, it holds that Γ x = Lx V x A + Kx A 2, A = + θ 2 dj. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

42 Remark We have already obtained in heorem 1 that Under the uniformly elliptic condition: (a) inf y R a(ε, y)2 >, (b) inf y R inf θ θ b(ε, y, θ) 2 >, it holds that Γ x = Lx V x A + Kx A 2, A = + θ 2 dj. [Question]: Can we study the same problem in the case where either (a) or (b) is satisfied? A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

43 [Lemma 5.2] x (E[φ(x )]) = E [ φ(x ) L x ] A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 2 / 25

44 [Lemma 5.2] x (E[φ(x )]) = E [ φ(x ) L x ] Corollary 7.1 Under the condition inf y R a(ε, y)2 >, then Γ x = Lx holds. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 2 / 25

45 [Lemma 5.2] x (E[φ(x )]) = E [ φ(x ) L x ] Corollary 7.1 Under the condition inf y R a(ε, y)2 >, then Γ x = Lx holds. Remark 7.2 he condition on the measure dν is not necessary. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 2 / 25

46 [Lemma 5.2] x (E[φ(x )]) = E [ φ(x ) L x ] Corollary 7.1 Under the condition inf y R a(ε, y)2 >, then Γ x = Lx holds. Remark 7.2 he condition on the measure dν is not necessary. Remark 7.3 In case of b(ε, y, θ), the formula Γ x = Lx ( = 1 is well known as the Bismut formula. ) Z t a(ε, x t ) dw t A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, 29 2 / 25

47 [ [Lemma 5.3] x (E φ(x ) ]) θ 2 dj = E [ φ(x ) V x ] A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

48 [ [Lemma 5.3] x (E φ(x ) Corollary 7.4 ]) θ 2 dj = E [ φ(x ) V x ] Under the condition inf y R inf θ θ b(ε, y, θ) 2 >, then it holds that Γ x = V x θ 2 dj K x + ( ) 2. θ 2 dj A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

49 [ [Lemma 5.3] x (E φ(x ) Corollary 7.4 ]) θ 2 dj = E [ φ(x ) V x ] Under the condition inf y R inf θ θ b(ε, y, θ) 2 >, then it holds that Γ x = V x θ 2 dj K x + ( ) 2. θ 2 dj Remark 7.5 he condition on the measure dν is essential. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

50 Example 1: Lévy processes t x t = x + γt + σw t + δ θ dj (γ R, σ, δ ) A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

51 Example 1: Lévy processes t x t = x + γt + σw t + δ θ dj (γ R, σ, δ ) If σ > and δ >, we have W { } g(θ) θ 2 Γ x = σ d J δg(θ) + θ 2 dj + ( + R 2θ 3 δ dj θ 2 dj ) 2, If σ >, we have Γ x = W σ, If δ >, we have { } g(θ) θ 2 d J Γ x = δg(θ) θ 2 dj + ( R 2θ 3 δ dj ) 2. θ 2 dj A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

52 Example 2: Geometric Lévy processes (γ, σ, δ) R [, + ) [, + ), x (, + ) t X t = γt + σw t + δθ dj : Lévy process x t = x exp [X t ] : geometric Lévy process A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

53 Example 2: Geometric Lévy processes (γ, σ, δ) R [, + ) [, + ), x (, + ) t X t = γt + σw t + δθ dj : Lévy process x t = x exp [X t ] : geometric Lévy process For the Itô formula, we have { } dx t = γ + (e δθ 1 δθ)dν x t dt + σx t dw t θ 1 + (e δθ 1)x t dj. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

54 Example 2: Geometric Lévy processes (γ, σ, δ) R [, + ) [, + ), x (, + ) t X t = γt + σw t + δθ dj : Lévy process x t = x exp [X t ] : geometric Lévy process For the Itô formula, we have { } dx t = γ + (e δθ 1 δθ)dν x t dt + σx t dw t θ 1 Write h(y) := e y. Since + (e δθ 1)x t dj. x (φ(x t )) = φ (x t ) x t x = 1 x X ((φ h)(x + X t )) X=log x, we can compute the weight Γ x by using the results in Example 1. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

55 Conclusion Stochastic differential equations with jumps Under the uniform ellipticity on a and b (, a or b), we have x (E[φ(x )]) = E [ φ(x ) Γ x ], etc. here are some approaches to attack the sensitivity analysis. the Girsanov transform the Malliavin calculus on the Wiener-Poisson space the martingale method We make use of the Kolmogorov backward equation for L ε. he models can be of pure-jump type, and of infinite-activity type. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25

56 References [1] Cass,. R., and Friz, P. K. (27): arxiv:math/64311v3. [2] Davis, M. H. A., and Johansson, M. P. (26): Stoch. Processes Appl., [3] El-Khatib, Y., and Privault, N. (24): Finance Stoch., [4] Kawai, R. and. (28): under revision. [5] Kawai, R. and. (28): submitted for publication. [6]. (22): Osaka J. Math., [7]. (28): submitted for publication. [8]. (29): in preparation. A. akeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 3, / 25