Logarithmic derivatives of densities for jump processes Atsushi - PowerPoint PPT Presentation

Logarithmic derivatives of densities for jump processes Atsushi TAKEUCHI Osaka City University (JAPAN) June 30, 2009 City University of Hong Kong Workshop on ”Stochastic Analysis and Finance” (June 29 - July 3, 2009) A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 1 / 25

Preliminaries dν : the L´ evy measure on R 0 := R \{ 0 } A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 2 / 25

Preliminaries dν : the L´ evy measure on R 0 := R \{ 0 } ∫ ∫ | θ | p dν < ∞ for any p ≥ 1 , | θ | dν + | θ |≤ 1 | θ | > 1 ∫ ( | θ/ρ | 2 ∧ 1 ) ρ α there exists α > 0 such that lim inf dν > 0 , ρ ↘ 0 R 0 there exists a C 1 -density g ( θ ) such that lim | g ( θ ) | = 0 . | θ |→∞ A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 2 / 25

Preliminaries dν : the L´ evy measure on R 0 := R \{ 0 } ∫ ∫ | θ | p dν < ∞ for any p ≥ 1 , | θ | dν + | θ |≤ 1 | θ | > 1 ∫ ( | θ/ρ | 2 ∧ 1 ) ρ α there exists α > 0 such that lim inf dν > 0 , ρ ↘ 0 R 0 there exists a C 1 -density g ( θ ) such that lim | g ( θ ) | = 0 . | θ |→∞ a 0 ( ε, y ) , a ( ε, y ) ∈ C 1 , ∞ 1+ ,b ( R × R ) b ( ε, y, θ ) ∈ C 1 , ∞ , ∞ ( R × R × R 0 ) 1+ ,b � 1 + b ′ ( ε, y, θ ) � > 0 , � � y ∈ R inf inf | θ |↘ 0 b ( ε, y, θ ) = 0 lim θ ∈ R 0 y ∈ R a ( ε, y ) 2 > 0 , θ ∈ R 0 ∂ θ b ( ε, y, θ ) 2 > 0 inf y ∈ R inf inf A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 2 / 25

Example 1.1 Let a, b, c > 0 , and 0 ≤ β < 1 . Define dθ { } e bθ I ( θ< 0) + e − cθ I ( θ> 0) dν = a | θ | 1+ β which is a special case of CGMY process. A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 3 / 25

Example 1.1 Let a, b, c > 0 , and 0 ≤ β < 1 . Define dθ { } e bθ I ( θ< 0) + e − cθ I ( θ> 0) dν = a | θ | 1+ β which is a special case of CGMY process. In particular, gamma process: b = + ∞ , β = 0 variance gamma process: β = 0 tempered stable process: b = + ∞ , 0 < β < 1 inverse Gaussian process: b = + ∞ , β = 1 / 2 . A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 3 / 25

For each ( ε, x ) ∈ R 2 , consider the stochastic differential equation: ✓ ✏ ∫ dx t = a 0 ( ε, x t ) dt + a ( ε, x t ) ◦ dW t + b ( ε, x t − , θ ) dJ, x 0 = x R 0 ✒ ✑ { W t ; t ∈ [0 , T ] } : 1 -dimensional Brownian motion dJ : the Poisson random measure on [0 , T ] × R 0 dt dν : the intensity d ˜ J = dJ − dt dν, dJ = I ( | θ |≤ 1) d ˜ J + I ( | θ | > 1) dJ A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 4 / 25

For each ( ε, x ) ∈ R 2 , consider the stochastic differential equation: ✓ ✏ ∫ dx t = a 0 ( ε, x t ) dt + a ( ε, x t ) ◦ dW t + b ( ε, x t − , θ ) dJ, x 0 = x R 0 ✒ ✑ { W t ; t ∈ [0 , T ] } : 1 -dimensional Brownian motion dJ : the Poisson random measure on [0 , T ] × R 0 dt dν : the intensity d ˜ J = dJ − dt dν, dJ = I ( | θ |≤ 1) d ˜ J + I ( | θ | > 1) dJ The associated infinitesimal generator is 0 f ( y ) + A ε A ε ∫ L ε f ( y ) = A ε { B ε θ f ( y ) − I ( | θ |≤ 1) B ε } f ( y ) + θ f ( y ) dν 2 R 0 B ε ( ) θ f ( y ) := f ( y + b ( ε, y, θ )) − f ( y ) A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 4 / 25

Proposition 1.2 → x t ∈ R has a C 1 -modification such that The mapping R ∋ x �− Z t := ∂ x x t satisfies the linear SDE: ∫ dZ t = a ′ 0 ( ε, x t ) Z t dt + a ′ ( ε, x t ) Z t ◦ dW t + b ′ ( ε, x t − , θ ) Z t − dJ. R 0 Z t is invertible a.s. A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 5 / 25

Proposition 1.2 → x t ∈ R has a C 1 -modification such that The mapping R ∋ x �− Z t := ∂ x x t satisfies the linear SDE: ∫ dZ t = a ′ 0 ( ε, x t ) Z t dt + a ′ ( ε, x t ) Z t ◦ dW t + b ′ ( ε, x t − , θ ) Z t − dJ. R 0 Z t is invertible a.s. → x t ∈ R has a C 1 -modification such that The mapping R ∋ ε �− H t := ∂ ε x t satisfies the SDE: ∫ dH t = a ′ 0 ( ε, x t ) H t dt + a ′ ( ε, x t ) H t ◦ dW t + b ′ ( ε, x t − , θ ) H t − dJ R 0 ∫ + ∂ ε a 0 ( ε, x t ) dt + ∂ ε a ( ε, x t ) ◦ dW t + ∂ ε b ( ε, x t − , θ ) dJ. R 0 A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 5 / 25

Existence of smooth densities Let ˜ ∂ θ b/ (1 + b ′ ) [ ] b ( ε, y, θ ) := ( ε, y, θ ) θ. A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 6 / 25

Existence of smooth densities Let ˜ ∂ θ b/ (1 + b ′ ) [ ] b ( ε, y, θ ) := ( ε, y, θ ) θ. Under the conditions y ∈ R a ( ε, y ) 2 > 0 and inf θ ∈ R 0 ∂ θ b ( ε, y, θ ) 2 > 0 , inf y ∈ R inf there exists α > 0 such that ∫ | θ/ρ | 2 ∧ 1 ( ) ρ α lim inf dν > 0 , ρ ↘ 0 R 0 there exists γ > 0 such that { ∫ } � 2 ∧ 1 | a ( ε, y ) /ρ | 2 + (� ) � ˜ ≥ c ρ − γ � inf b ( ε, y, θ ) /ρ dν y ∈ R R 0 for 0 < ρ < 1 . A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 6 / 25

Existence of smooth densities Let ˜ ∂ θ b/ (1 + b ′ ) [ ] b ( ε, y, θ ) := ( ε, y, θ ) θ. Under the conditions y ∈ R a ( ε, y ) 2 > 0 and inf θ ∈ R 0 ∂ θ b ( ε, y, θ ) 2 > 0 , inf y ∈ R inf there exists α > 0 such that ∫ | θ/ρ | 2 ∧ 1 ( ) ρ α lim inf dν > 0 , ρ ↘ 0 R 0 there exists γ > 0 such that { ∫ } � 2 ∧ 1 | a ( ε, y ) /ρ | 2 + (� ) � ˜ ≥ c ρ − γ � inf b ( ε, y, θ ) /ρ dν y ∈ R R 0 for 0 < ρ < 1 . Then, for each ( x, ε ) ∈ R 2 , there exists a smooth density p x,ε T ( y ) for x T . (cf. [T. 2002]) A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 6 / 25

＜ ＞ Main problem C LG ( R ) = { f ∈ C ( R ) ; | f ( y ) | ≤ c (1 + | y | ) } { n } ∑ F = α k f k I A k ; α k ∈ R , f k ∈ C LG ( R ) , A k ⊂ R : interval k =1 A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 7 / 25

Main problem C LG ( R ) = { f ∈ C ( R ) ; | f ( y ) | ≤ c (1 + | y | ) } { n } ∑ F = α k f k I A k ; α k ∈ R , f k ∈ C LG ( R ) , A k ⊂ R : interval k =1 ＜ Goal ＞ For ϕ ∈ F , compute the differentials of E [ ϕ ( x T )] in x ∈ R and ε ∈ R : ϕ ( x T ) Γ x [ ] ∂ x ( E [ ϕ ( x T )]) = E T ϕ ( x T ) Γ ε [ ] ∂ ε ( E [ ϕ ( x T )]) = E T [ ] ∂ 2 ϕ ( x T ) ˜ Γ x x ( E [ ϕ ( x T )]) = E T (logarithmic derivatives of p x,ε T ( y ) , computations of the Greeks) A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 7 / 25

Sensitivity with respect to the initial point Theorem 1 (Sensitivity in x ∈ R , [T. ’08] ) For ϕ ∈ F , it holds that T = L x T − V x + K x T T ϕ ( x T ) Γ x , Γ x [ ] ∂ x ( E [ ϕ ( x T )]) = E . T A 2 A T T ∫ T [ 1 + b ′ ∫ ] | θ | 2 dJ, v ( ε, t, θ ) = ( ε, x t , θ ) Z t | θ | 2 , A T = T + ∂ θ b 0 R 0 ∫ t ∫ t { } Z s ∂ θ g ( θ ) v ( ε, s, θ ) ∫ d ˜ L x a ( ε, x s ) dW s , V x t = t = J, g ( θ ) 0 0 R 0 ∫ t ∫ K x t = 2 θv ( ε, s, θ ) dJ 0 R 0 A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 8 / 25

Remark 4.1 ∫ T ∫ ∫ e − λ | θ | 2 − 1 ( ) | θ | 2 dJ . Let N λ Recall A T = T + T = T dν . 0 R 0 R 0 Under the condition on dν : ∫ | θ/ρ | 2 ∧ 1 ρ α ( ) lim inf dν > 0 , ρ ↘ 0 R 0 A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 9 / 25

Remark 4.1 ∫ T ∫ ∫ e − λ | θ | 2 − 1 ( ) | θ | 2 dJ . Let N λ Recall A T = T + T = T dν . 0 R 0 R 0 Under the condition on dν : ∫ | θ/ρ | 2 ∧ 1 ρ α ( ) lim inf dν > 0 , ρ ↘ 0 R 0 it holds that, for any p > 1 , ∫ ∞ 1 [ ] [ ( )] A − p e N λ λ p − 1 E − λA T − N λ T dλ E = exp T T Γ( p ) 0 ∫ ∞ [ ∫ ] λ p − 1 exp ( λ | θ | 2 ) ∧ 1 { } ≤ c − λT − c T dν dλ 0 R 0 ∫ ∞ λ p − 1 exp [ ] − λT − c λ α/ 2 T ≤ c dλ 0 < ∞ . A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 9 / 25

Key Lemmas Let ϕ ∈ C 2 K ( R ) for a while. Recall stochastic differential equation: ∫ dx t = a 0 ( ε, x t ) dt + a ( ε, x t ) ◦ dW t + b ( ε, x t − , θ ) dJ R 0 x 0 = x infinitesimal generator: 0 + A ε A ε ∫ L ε = A ε B ε θ − I ( | θ |≤ 1) B ε { } + dν θ 2 R 0 B ε ( ) θ f ( y ) := f ( y + b ( ε, y, θ )) − f ( y ) A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 10 / 25

)� [ ( ] Let u ( t, x ) := E ϕ x T − t � x 0 = x ( t ∈ [0 , T ) , x ∈ R ) . A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 11 / 25

)� [ ( ] Let u ( t, x ) := E ϕ x T − t � x 0 = x ( t ∈ [0 , T ) , x ∈ R ) . Then, we see u ∈ C 1 , 2 ([0 , T ) × R ) , ∂ t u + L ε u = 0 , lim t ↗ T u ( t, x ) = ϕ ( x ) . b A. Takeuchi (Osaka City Univ. ) Logarithmic derivatives for jump processes June 30, 2009 11 / 25