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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 R0 := R\{0}

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 2 / 25

Preliminaries

dν: the L´ evy measure on R0 := R\{0} ∫

|θ|≤1

|θ|dν + ∫

|θ|>1

|θ|pdν < ∞ for any p ≥ 1, there exists α > 0 such that lim inf

ρ↘0

ρα ∫

R0

( |θ/ρ|2 ∧ 1 ) dν > 0, there exists a C1-density g(θ) such that lim

|θ|→∞

|g(θ)| = 0.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 2 / 25

Preliminaries

dν: the L´ evy measure on R0 := R\{0} ∫

|θ|≤1

|θ|dν + ∫

|θ|>1

|θ|pdν < ∞ for any p ≥ 1, there exists α > 0 such that lim inf

ρ↘0

ρα ∫

R0

( |θ/ρ|2 ∧ 1 ) dν > 0, there exists a C1-density g(θ) such that lim

|θ|→∞

|g(θ)| = 0. a0(ε, y), a(ε, y) ∈ C1,∞

1+,b(R × R)

b(ε, y, θ) ∈ C1,∞,∞

1+,b

(R × R × R0) inf

y∈R inf θ∈R0

• 1 + b′(ε, y, θ)
• > 0,

lim

|θ|↘0 b(ε, y, θ) = 0

inf

y∈R a(ε, y)2 > 0,

inf

y∈R inf θ∈R0 ∂θb(ε, y, θ)2 > 0

• 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 dν = a { ebθI(θ<0) + e−cθI(θ>0) } dθ |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 dν = a { ebθI(θ<0) + e−cθI(θ>0) } dθ |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) ∈ R2, consider the stochastic differential equation:

✓ ✏

dxt = a0(ε, xt)dt+a(ε, xt)◦dWt+ ∫

R0

b(ε, xt−, θ)dJ, x0 = x

✒ ✑

{Wt ; t ∈ [0, T ]}: 1-dimensional Brownian motion dJ: the Poisson random measure on [0, T ] × R0 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) ∈ R2, consider the stochastic differential equation:

✓ ✏

dxt = a0(ε, xt)dt+a(ε, xt)◦dWt+ ∫

R0

b(ε, xt−, θ)dJ, x0 = x

✒ ✑

{Wt ; t ∈ [0, T ]}: 1-dimensional Brownian motion dJ: the Poisson random measure on [0, T ] × R0 dt dν: the intensity d ˜ J = dJ − dt dν, dJ = I(|θ|≤1)d ˜ J + I(|θ|>1)dJ The associated infinitesimal generator is

Lεf(y) = Aε

0f(y) + AεAε

2 f(y) + ∫

R0

{ Bε

θf(y) − I(|θ|≤1)Bε θf(y)

} dν ( Bε

θf(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

The mapping R ∋ x − → xt ∈ R has a C1-modification such that Zt := ∂xxt satisfies the linear SDE:

dZt = a′

0(ε, xt)Ztdt + a′(ε, xt)Zt ◦ dWt +

∫

R0

b′(ε, xt−, θ)Zt−dJ.

Zt is invertible a.s.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 5 / 25

Proposition 1.2

The mapping R ∋ x − → xt ∈ R has a C1-modification such that Zt := ∂xxt satisfies the linear SDE:

dZt = a′

0(ε, xt)Ztdt + a′(ε, xt)Zt ◦ dWt +

∫

R0

b′(ε, xt−, θ)Zt−dJ.

Zt is invertible a.s. The mapping R ∋ ε − → xt ∈ R has a C1-modification such that Ht := ∂εxt satisfies the SDE:

dHt = a′

0(ε, xt)Htdt + a′(ε, xt)Ht ◦ dWt +

∫

R0

b′(ε, xt−, θ)Ht−dJ + ∂εa0(ε, xt)dt + ∂εa(ε, xt) ◦ dWt + ∫

R0

∂εb(ε, xt−, θ)dJ.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 5 / 25

Existence of smooth densities

Let ˜ b(ε, y, θ) := [ ∂θb/(1 + b′) ] (ε, y, θ) θ.

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Logarithmic derivatives for jump processes June 30, 2009 6 / 25

Existence of smooth densities

Let ˜ b(ε, y, θ) := [ ∂θb/(1 + b′) ] (ε, y, θ) θ. Under the conditions inf

y∈R a(ε, y)2 > 0 and inf y∈R inf θ∈R0 ∂θb(ε, y, θ)2 > 0,

there exists α > 0 such that lim inf

ρ↘0

ρα ∫

R0

( |θ/ρ|2 ∧ 1 ) dν > 0, there exists γ > 0 such that inf

y∈R

{ |a(ε, y)/ρ|2 + ∫

R0

( ˜ b(ε, y, θ)/ρ

• 2 ∧ 1

) dν } ≥ c ρ−γ for 0 < ρ < 1.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 6 / 25

Existence of smooth densities

Let ˜ b(ε, y, θ) := [ ∂θb/(1 + b′) ] (ε, y, θ) θ. Under the conditions inf

y∈R a(ε, y)2 > 0 and inf y∈R inf θ∈R0 ∂θb(ε, y, θ)2 > 0,

there exists α > 0 such that lim inf

ρ↘0

ρα ∫

R0

( |θ/ρ|2 ∧ 1 ) dν > 0, there exists γ > 0 such that inf

y∈R

{ |a(ε, y)/ρ|2 + ∫

R0

( ˜ b(ε, y, θ)/ρ

• 2 ∧ 1

) dν } ≥ c ρ−γ for 0 < ρ < 1. Then, for each (x, ε) ∈ R2, there exists a smooth density px,ε

T (y) for xT . (cf. [T. 2002])

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 6 / 25

Main problem

CLG(R) = {f ∈ C(R) ; |f(y)| ≤ c (1 + |y|)} F = { n ∑

k=1

αk fk IAk ; αk ∈ R, fk ∈ CLG(R), Ak ⊂ R: interval } ＜ ＞

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 7 / 25

Main problem

CLG(R) = {f ∈ C(R) ; |f(y)| ≤ c (1 + |y|)} F = { n ∑

k=1

αk fk IAk ; αk ∈ R, fk ∈ CLG(R), Ak ⊂ R: interval } ＜ Goal ＞ For ϕ ∈ F, compute the differentials of E [ϕ(xT )] in x ∈ R and ε ∈ R: ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) Γx

T

] ∂ε (E[ϕ(xT )]) = E [ ϕ(xT ) Γε

T

] ∂2

x (E[ϕ(xT )]) = E

[ ϕ(xT ) ˜ Γx

T

] (logarithmic derivatives of px,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 ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) Γx

T

] , Γx

T = Lx T − V x T

AT + Kx

T

A2

T

.

AT = T + ∫ T ∫

R0

|θ|2 dJ, v(ε, t, θ) = [1 + b′ ∂θb ] (ε, xt, θ) Zt |θ|2, Lx

t =

∫ t Zs a(ε, xs) dWs, V x

t =

∫ t ∫

R0

∂θ { g(θ)v(ε, s, θ) } g(θ) d ˜ J, Kx

t =

∫ t ∫

R0

2θv(ε, s, θ)dJ

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 8 / 25

Remark 4.1

Recall AT = T + ∫ T ∫

R0

|θ|2dJ. Let N λ

T = T

∫

R0

( e−λ|θ|2 − 1 ) dν. Under the condition on dν: lim inf

ρ↘0

ρα ∫

R0

( |θ/ρ|2 ∧ 1 ) dν > 0,

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 9 / 25

Remark 4.1

Recall AT = T + ∫ T ∫

R0

|θ|2dJ. Let N λ

T = T

∫

R0

( e−λ|θ|2 − 1 ) dν. Under the condition on dν: lim inf

ρ↘0

ρα ∫

R0

( |θ/ρ|2 ∧ 1 ) dν > 0, it holds that, for any p > 1, E [ A−p

T

] = 1 Γ(p) ∫ ∞ λp−1E [ exp ( −λAT − N λ

T

)] eNλ

T dλ

≤ c ∫ ∞ λp−1 exp [ −λT − c T ∫

R0

{ (λ|θ|2) ∧ 1 } dν ] dλ ≤ c ∫ ∞ λp−1 exp [ −λT − c λα/2T ] dλ < ∞.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 9 / 25

Key Lemmas

Let ϕ ∈ C2

K(R) for a while. Recall

stochastic differential equation: dxt = a0(ε, xt)dt + a(ε, xt) ◦ dWt + ∫

R0

b(ε, xt−, θ)dJ x0 = x infinitesimal generator: Lε = Aε

0 + AεAε

2 + ∫

R0

{ Bε

θ − I(|θ|≤1)Bε θ

} dν ( Bε

θf(y) := f(y + b(ε, y, θ)) − f(y)

)

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Logarithmic derivatives for jump processes June 30, 2009 10 / 25

Let u(t, x) := E [ ϕ ( xT −t ) x0 = x ] (t ∈ [0, T ), x ∈ R).

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Logarithmic derivatives for jump processes June 30, 2009 11 / 25

Let u(t, x) := E [ ϕ ( xT −t ) x0 = x ] (t ∈ [0, T ), x ∈ R). Then, we see u ∈ C1,2

b

([0, T ) × R), ∂tu + Lεu = 0, lim

t↗T u(t, x) = ϕ(x).

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 11 / 25

Let u(t, x) := E [ ϕ ( xT −t ) x0 = x ] (t ∈ [0, T ), x ∈ R). Then, we see u ∈ C1,2

b

([0, T ) × R), ∂tu + Lεu = 0, lim

t↗T u(t, x) = ϕ(x).

Applying the Itˆ

• formula to u(t, xt) for 0 ≤ t < T , we have

u(t, xt) = u(0, x) + ∫ t u′(s, xs)a(ε, xs)dWs + ∫ t ∫

R0

Bε

θu(s, xs−)d ˜

J.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 11 / 25

Let u(t, x) := E [ ϕ ( xT −t ) x0 = x ] (t ∈ [0, T ), x ∈ R). Then, we see u ∈ C1,2

b

([0, T ) × R), ∂tu + Lεu = 0, lim

t↗T u(t, x) = ϕ(x).

Applying the Itˆ

• formula to u(t, xt) for 0 ≤ t < T , we have

u(t, xt) = u(0, x) + ∫ t u′(s, xs)a(ε, xs)dWs + ∫ t ∫

R0

Bε

θu(s, xs−)d ˜

J.

Taking the limit as t ↗ T enables us to get the following lemma.

Lemma 5.1

ϕ(xT ) = E [ϕ(xT )] + ∫ T u′(s, xs)a(ε, xs)dWs + ∫ T ∫

R0

Bε

θu(s, xs−)d ˜

J

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 11 / 25

Lemma 5.2

T ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) Lx

T

]

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Logarithmic derivatives for jump processes June 30, 2009 12 / 25

Lemma 5.2

T ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) Lx

T

] Proof of Lemma 5.2 We have already seen in Lemma 5.1 that

ϕ(xT ) = E[ϕ(xT )] + ∫ T u′(s, xs)a(ε, xs)dWs + ∫ T ∫

R0

Bε

θu(s, xs−)d ˜

J

Multiplying the above equality by Lx

T =

∫ T Zt a(ε, xt)dWt, we have (RHS) = E [∫ T u′(t, xt)a(ε, xt)dWt ∫ T Zt a(ε, xt) dWt ] = ∫ T E [ u′(t, xt)Zt ] dt = ∫ T ∂x (E[u(t, xt)]) dt = (LHS)

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Logarithmic derivatives for jump processes June 30, 2009 12 / 25

Lemma 5.3

∂x ( E [ ϕ(xT ) ∫ T ∫

R0

|θ|2dJ ]) = −E [ ϕ(xT ) V x

T

]

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Logarithmic derivatives for jump processes June 30, 2009 13 / 25

Lemma 5.3

∂x ( E [ ϕ(xT ) ∫ T ∫

R0

|θ|2dJ ]) = −E [ ϕ(xT ) V x

T

] Proof of Lemma 5.3 We have already seen in Lemma 5.1 that

ϕ(xT ) = E[ϕ(xT )] + ∫ T u′(s, xs)a(ε, xs)dWs + ∫ T ∫

R0

Bε

θu(s, xs−)d ˜

J

Multiplying the above equality by ∫ T ∫

R0

|θ|2d ˜ J, we have

E [ ϕ(xT ) ∫ T ∫

R0

|θ|2d ˜ J ] = E [∫ T ∫

R0

Bε

θu(t, xt)|θ|2dt dν

] .

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 13 / 25

Recall V x

T =

∫ T ∫

R0

∂θ{g(θ) v(ε, t, θ)} g(θ) d ˜

• J. Since

lim

|θ|→∞ g(θ) = 0, we

see that

E [ ϕ(xT ) V x

T

] = E [∫ T ∫

R0

Bε

θu(t, xt) ∂θ {g(θ)v(ε, t, θ)}

g(θ) dt dν ] = −E [∫ T ∫

R0

u′(t, xt + b(ε, xt, θ)) (1 + b′(ε, xt, θ)) Zt|θ|2dt dν ] ( ∵ lim

|θ|→∞ g(θ) = 0, v(ε, t, θ) =

[1 + b′ ∂θb ] (ε, xt, θ)Zt|θ|2 ) = −∂x ( E [∫ T ∫

R0

u(t, xt + b(ε, xt, θ))|θ|2dt dν ]) = −∂x ( E [ ϕ(xT ) ∫ T ∫

R0

|θ|2dJ ]) .

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 14 / 25

Proof of Theorem 1

It is sufficient to study ϕ ∈ C2

K(R), instead of ϕ ∈ F, via the standard

density argument.

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Logarithmic derivatives for jump processes June 30, 2009 15 / 25

Proof of Theorem 1

It is sufficient to study ϕ ∈ C2

K(R), instead of ϕ ∈ F, via the standard

density argument. Our goal is ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) {Lx

T − V x T

AT + Kx

T

A2

T

}] .

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 15 / 25

Proof of Theorem 1

It is sufficient to study ϕ ∈ C2

K(R), instead of ϕ ∈ F, via the standard

density argument. Our goal is ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) {Lx

T − V x T

AT + Kx

T

A2

T

}] . We have already obtained ∂x (E[ϕ(xT ) AT ]) = E [ ϕ(xT ) { Lx

T − V x T

}] .

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 15 / 25

Proof of Theorem 1

It is sufficient to study ϕ ∈ C2

K(R), instead of ϕ ∈ F, via the standard

density argument. Our goal is ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) {Lx

T − V x T

AT + Kx

T

A2

T

}] . We have already obtained ∂x (E[ϕ(xT ) AT ]) = E [ ϕ(xT ) { Lx

T − V x T

}] . Thus, we have to consider how to get rid of AT = T + ∫ T ∫

R0

|θ|2dJ.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 15 / 25

Remark that E[ϕ(xT )] = E [ϕ(xT )AT AT ] = ∫ ∞ M λ

T Eλ[ϕ(xT )AT ]dλ,

where dPλ dP

• FT

= exp { − ∫ T ∫

R0

λ|θ|2dJ − ∫ T ∫

R0

( e−λ|θ|2 − 1 ) dt dν } M λ

T = exp

{ −λT − ∫ T ∫

R0

( 1 − e−λ|θ|2) dt dν } : deterministic

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 16 / 25

Remark that E[ϕ(xT )] = E [ϕ(xT )AT AT ] = ∫ ∞ M λ

T Eλ[ϕ(xT )AT ]dλ,

where dPλ dP

• FT

= exp { − ∫ T ∫

R0

λ|θ|2dJ − ∫ T ∫

R0

( e−λ|θ|2 − 1 ) dt dν } M λ

T = exp

{ −λT − ∫ T ∫

R0

( 1 − e−λ|θ|2) dt dν } : deterministic From the Girsanov theorem, we see that, under Pλ, {Wt ; t ∈ [0, T ]}: the Brownian motion, dJ: the Poisson random measure with the intensity e−λ|θ|2dt dν, d ˜ Jλ = dJ − e−λ|θ|2dt dν: the martingale measure.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 16 / 25

We have already seen that ∂x ( Eλ[ϕ(xT )AT ] ) = Eλ[ ϕ(xT ) { Lx

T − V x,λ T

}] .  V x,λ

t

:= ∫ t ∫

R0

∂θ { e−λ|θ|2g(θ)v(ε, s, θ) } e−λ|θ|2g(θ) d ˜ Jλ  

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 17 / 25

We have already seen that ∂x ( Eλ[ϕ(xT )AT ] ) = Eλ[ ϕ(xT ) { Lx

T − V x,λ T

}] .  V x,λ

t

:= ∫ t ∫

R0

∂θ { e−λ|θ|2g(θ)v(ε, s, θ) } e−λ|θ|2g(θ) d ˜ Jλ   Hence we have

∫ ∞ M λ

T Eλ[

ϕ(xT )Lx

T

] dλ = E [(∫ ∞ e−λAT dλ ) ϕ(xT )Lx

T

] = E [ ϕ(xT ) Lx

T

AT ] .

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 17 / 25

We have already seen that ∂x ( Eλ[ϕ(xT )AT ] ) = Eλ[ ϕ(xT ) { Lx

T − V x,λ T

}] .  V x,λ

t

:= ∫ t ∫

R0

∂θ { e−λ|θ|2g(θ)v(ε, s, θ) } e−λ|θ|2g(θ) d ˜ Jλ   Hence we have

∫ ∞ M λ

T Eλ[

ϕ(xT )Lx

T

] dλ = E [(∫ ∞ e−λAT dλ ) ϕ(xT )Lx

T

] = E [ ϕ(xT ) Lx

T

AT ] .

Since V x,λ

T

= V x

T − λKx T , we see that

∫ ∞ M λ

T Eλ[

ϕ(xT ) V x,λ

T

] dλ = E [∫ ∞ e−λAT ϕ(xT ) { V x

T − λKx T

} dλ ] = E [ ϕ(xT ) {V x

T

AT − Kx

T

A2

T

}] .

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 17 / 25

✓ ✏

∂x (E[ϕ(xT )]) = E [ ϕ(xT ) {Lx

T − V x T

AT + Kx

T

A2

T

}]

✒ ✑

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 18 / 25

✓ ✏

∂x (E[ϕ(xT )]) = E [ ϕ(xT ) {Lx

T − V x T

AT + Kx

T

A2

T

}]

✒ ✑

In a similar manner, we can derive other sensitivity formulae: ∂2

x (E[ϕ(xT )]) = E

[ ϕ(xT ) ˜ Γx

T

] : Gamma ∂ε (E[ϕ(xT )]) = E [ ϕ(xT ) Γε

T

] : Vega etc.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 18 / 25

✓ ✏

∂x (E[ϕ(xT )]) = E [ ϕ(xT ) {Lx

T − V x T

AT + Kx

T

A2

T

}]

✒ ✑

In a similar manner, we can derive other sensitivity formulae: ∂2

x (E[ϕ(xT )]) = E

[ ϕ(xT ) ˜ Γx

T

] : Gamma ∂ε (E[ϕ(xT )]) = E [ ϕ(xT ) Γε

T

] : Vega etc. Under the hypoelliptic situation, instead of the uniform ellipticity, the martingale method can be applied to the sensitivity on E [ ϕ ( 1 T ∫ T xt dt )] .

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 18 / 25

Remark

We have already obtained in Theorem 1 that

✓ ✏

Under the uniformly elliptic condition: (a) inf

y∈R a(ε, y)2 > 0,

(b) inf

y∈R inf θ∈R0 ∂θb(ε, y, θ)2 > 0,

it holds that Γx

T = Lx T − V x T

AT + Kx

T

A2

T

, AT = T + ∫ T ∫

R0

|θ|2dJ.

✒ ✑

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 19 / 25

Remark

We have already obtained in Theorem 1 that

✓ ✏

Under the uniformly elliptic condition: (a) inf

y∈R a(ε, y)2 > 0,

(b) inf

y∈R inf θ∈R0 ∂θb(ε, y, θ)2 > 0,

it holds that Γx

T = Lx T − V x T

AT + Kx

T

A2

T

, AT = T + ∫ T ∫

R0

|θ|2dJ.

✒ ✑

[Question]: Can we study the same problem in the case where either (a)

• r (b) is satisfied?
• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 19 / 25

[Lemma 5.2] T ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) Lx

T

]

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 20 / 25

[Lemma 5.2] T ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) Lx

T

]

Corollary 7.1

Under the condition inf

y∈R a(ε, y)2 > 0, then Γx T = Lx T

T holds.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 20 / 25

[Lemma 5.2] T ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) Lx

T

]

Corollary 7.1

Under the condition inf

y∈R a(ε, y)2 > 0, then Γx T = Lx T

T holds.

Remark 7.2

The condition on the measure dν is not necessary.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 20 / 25

[Lemma 5.2] T ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) Lx

T

]

Corollary 7.1

Under the condition inf

y∈R a(ε, y)2 > 0, then Γx T = Lx T

T holds.

Remark 7.2

The condition on the measure dν is not necessary.

Remark 7.3

In case of b(ε, y, θ) ≡ 0, the formula Γx

T = Lx T

T ( = 1 T ∫ T Zt a(ε, xt) dWt ) is well known as the Bismut formula.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 20 / 25

[Lemma 5.3] ∂x ( E [ ϕ(xT ) ∫ T ∫

R0

|θ|2dJ ]) = −E [ ϕ(xT ) V x

T

]

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 21 / 25

[Lemma 5.3] ∂x ( E [ ϕ(xT ) ∫ T ∫

R0

|θ|2dJ ]) = −E [ ϕ(xT ) V x

T

]

Corollary 7.4

Under the condition inf

y∈R inf θ∈R0 ∂θb(ε, y, θ)2 > 0, then it holds that

Γx

T = −

V x

T

∫ T ∫

R0

|θ|2dJ + Kx

T

(∫ T ∫

R0

|θ|2dJ )2 .

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 21 / 25

[Lemma 5.3] ∂x ( E [ ϕ(xT ) ∫ T ∫

R0

|θ|2dJ ]) = −E [ ϕ(xT ) V x

T

]

Corollary 7.4

Under the condition inf

y∈R inf θ∈R0 ∂θb(ε, y, θ)2 > 0, then it holds that

Γx

T = −

V x

T

∫ T ∫

R0

|θ|2dJ + Kx

T

(∫ T ∫

R0

|θ|2dJ )2 .

Remark 7.5

The condition on the measure dν is essential.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 21 / 25

Example 1: L´ evy processes

xt = x + γt + σWt + ∫ t ∫

R0

δ θ dJ (γ ∈ R, σ ≥ 0, δ ≥ 0)

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 22 / 25

Example 1: L´ evy processes

xt = x + γt + σWt + ∫ t ∫

R0

δ θ dJ (γ ∈ R, σ ≥ 0, δ ≥ 0) If σ > 0 and δ > 0, we have

Γx

T =

WT σ − ∫ T ∫

R0

{ g(θ)|θ|2}′ δg(θ) d ˜ J T + ∫ T ∫

R0

|θ|2 dJ + ∫ T ∫

R0

2θ3 δ dJ ( T + ∫ T ∫

R0

|θ|2 dJ )2 ,

If σ > 0, we have Γx

T = WT

σ T ,

If δ > 0, we have

Γx

T =

− ∫ T ∫

R0

{ g(θ)|θ|2}′ δg(θ) d ˜ J ∫ T ∫

R0

|θ|2 dJ + ∫ T ∫

R0

2θ3 δ dJ (∫ T ∫

R0

|θ|2 dJ )2 .

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 22 / 25

Example 2: Geometric L´ evy processes

(γ, σ, δ) ∈ R × [0, +∞) × [0, +∞), x ∈ (0, +∞) Xt = γt + σWt + ∫ t ∫

R0

δθ dJ : L´ evy process xt = x exp [Xt] : geometric L´ evy process

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 23 / 25

Example 2: Geometric L´ evy processes

(γ, σ, δ) ∈ R × [0, +∞) × [0, +∞), x ∈ (0, +∞) Xt = γt + σWt + ∫ t ∫

R0

δθ dJ : L´ evy process xt = x exp [Xt] : geometric L´ evy process For the Itˆ

• formula, we have

dxt = { γ + ∫

|θ|≤1

(eδθ − 1 − δθ)dν } xtdt + σxt ◦ dWt + ∫

R0

(eδθ − 1)xt−dJ.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 23 / 25

Example 2: Geometric L´ evy processes

(γ, σ, δ) ∈ R × [0, +∞) × [0, +∞), x ∈ (0, +∞) Xt = γt + σWt + ∫ t ∫

R0

δθ dJ : L´ evy process xt = x exp [Xt] : geometric L´ evy process For the Itˆ

• formula, we have

dxt = { γ + ∫

|θ|≤1

(eδθ − 1 − δθ)dν } xtdt + σxt ◦ dWt + ∫

R0

(eδθ − 1)xt−dJ. Write h(y) := ey. Since ∂x (ϕ(xt)) = ϕ′(xt) xt x = 1 x∂X ((ϕ ◦ h)(X + Xt))

• X=log x,

we can compute the weight Γx

T by using the results in Example 1.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 23 / 25

Conclusion

Stochastic differential equations with jumps Under the uniform ellipticity on a and b (, a or b), we have ∂x (E[ϕ(xT )]) = E [ ϕ(xT ) Γx

T

] , etc. There 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 Lε. The models can be of pure-jump type, and of infinite-activity type.

• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 24 / 25

References

[1] Cass, T. R., and Friz, P. K. (2007): arXiv:math/0604311v3. [2] Davis, M. H. A., and Johansson, M. P. (2006): Stoch. Processes Appl., 116 101–129. [3] El-Khatib, Y., and Privault, N. (2004): Finance Stoch., 8 161–179. [4] Kawai, R. and T. (2008): under revision. [5] Kawai, R. and T. (2008): submitted for publication. [6]

• T. (2002): Osaka J. Math., 39 523 – 559.

[7]

• T. (2008): submitted for publication.

[8]

• T. (2009): in preparation.
• A. Takeuchi (Osaka City Univ. )

Logarithmic derivatives for jump processes June 30, 2009 25 / 25