数理解析研究所講究録
第 1672 巻 2010 年 94-110

94

Logarithmic derivatives of densities for jumpprocesses
Atsushi TAKEUCHI

Abstract
The purpose of this paper is to study the sensitivity analysis for jump-type
stochastic differential equations under the condition on the L\’evy measure, and the
H\"ormander type condition on the coefficients. Our approach is based on the martingale property via the Kolmogorov backward equation for the integro-differential
operator associated with the equation.

1

Introduction

Malliavin introduced the stochastic calculus of variations in order to exhibit the probabilistic proof of the hypoelliptic problem for differential operators (cf. [18]). The integrationby-parts formula plays a key role in the Malliavin calculus, and the formula over a probability space can be also established via the Girsanov transform on Brownian motions(cf. [3]). In [12], the Malliavin calculus on the Wiener space was applied to the Greeks
computations for an asset price dynamics.
Recently, various types
the Malliavin calculus for jump processes have been introduced by many authors over the Poisson space, or the Wiener-Poisson space. See[1, 2, 4, 5, 9, 10, 17, 19] for details. The measure change technique for jump processes firstfound by [4] enables us to $ot$ $tain$ that the uniformly elliptic condition, or the H\"ormander
type condition on the coefficients of stochastic differential equations yields the existenceof smooth densities for the solution. Here the H\"ormander type condition is the condition, the Lie brackets of them, and theon the linear subspace generated by the
integrals of the jump term effects. See [16, 20] for details. FUrthermore, there are a lotof works in which the Malli.ivin calculus for jump processes are applied to the sensitiv. [1, 7, 8, 10] . This can be also regarded as
ity analysis in mathematical financethe logarithmic derivatives of the density with respect to various parameters in certain
sense. Although the process discussed in those works has jumps, most of them are focusedon the effect only from the diffusion terms. In [21], t,he sensitivities for jump processes
determined by stochast,ic differential equations are studied under the uniformly elliptic
condition on the diffusion and the $jum$ ] terms. Then. it is a natural question whether asimilar problem can be studied in the hypoelliptic situation.
$|\supset n$)$(oeffi\mathfrak{c}\cdot ients$$(\langle f$

$)$

$)$

95

In this paper, we shall study the sensitivity analysis for the solution to the stochastic
differential equation with jumps via the martingale approach based upon the Kolmogorov
backward equation for the associated infinitesimal generator, in the hypoelliptic situation, that is, the case where the coefficients of the equation satisfy the H\"ormander typecondition. The result obtained in the present paper includes the effects from not only the
diffusion terms, but also the jump terms. Moreover, the equation can be of a pure-jump
type, and an infinite activity type.
The paper is organized as follows: Section 2 is devoted to the introduction of ourframework, and the criterion on the existence of smooth densities as stated in [16, 20].
In Section 3, the sensitivity analysis with respect to the initial point of the equation isinvestigated in the hypoelliptic situation, which will be proved in Section 5. Some key

lemmas in order to prove the main result are given in Section 4, and the example isexhibited in Section 6.
, and the-zero matrix by
In the whole sequel, we shal denote the.
denotes the family of k-times differentiable functions withidentity byindicateandbounded derivatives of any orders more than 1. The symbols$0_{a,\beta}\in \mathbb{R}^{\beta}\otimes \mathbb{R}^{\alpha}$$\alpha\cross[t$$I_{\gamma}\in \mathbb{R}^{\gamma}\otimes \mathbb{R}^{\gamma}$,$C_{b}^{k}$$\nabla_{x},$$\nabla,$the gradient operators. Define the mapping$\pi(A)=(\begin{array}{l}\pi_{1}(A)\vdots\pi_{p}(A)\end{array})$

for$A=(A_{j,k})_{1\leq J\leq d,1\leq k\leq p}\in \mathbb{R}^{p}\otimes \mathbb{R}^{d}$

.

$\pi$

:

$\nabla_{X}$$\partial_{\theta}$by$\mathbb{R}^{p}\otimes \mathbb{R}^{d}arrow \mathbb{R}^{d\ell}$$\pi_{k}(A)=(\begin{array}{l}A_{1,k}\vdots A_{d_{t}k}\end{array})(1\leq k\leq\ell)$.

Preliminaries

2

Fix $T>0$ . Let
such that$(\Omega, \mathcal{F}.\mathbb{P})$be a probability space, and$d\nu$

the L\’evy measure over$\mathbb{R}\backslash \{0\}$Assumption 1.

$(a)$

for any$p\geq 1$ ,,$\int_{|\theta|-<\mathfrak{l}}|\theta|(l’/+\int_{|\theta|>1}|\theta|^{p}d\iota/<+\infty$$(b)$

there exists a constant $\delta>0$ such that
$1 i_{111}i_{11}f\rho^{\delta}\rho\backslash 0\int_{1R_{0}}(|\theta/\rho|^{2}\wedge 1)d\iota/>0$,$\mathbb{R}_{0};=$

96

$(c)$

there ensts athat$C^{1}$-density$g(\theta)$with respect to the Lebesgue measure over$\lim$$\mathbb{R}_{0}$such$|g(\theta)|=0$ .

$|\theta|arrow+\infty$

Example 1. The L\’evy measures of tempered stable processes, inverse Gaussian processes
and CGMY processes (cf. [6]) satisfy Assumption 1.

$\square$Remark 1. In order to study the existence of (smooth) densities, the following conditionis assumed in 14, 19 .
$[$

$(d)$

$]$

there $ex’\iota sts0<\alpha<2$ such that
$1 ini\inf_{\backslash \rho 0}p^{-\circ J_{|\theta|\leq\rho}}|\theta|^{2}d\nu>0$.

We can check that the condition (d) implies (b) in Assumption 1.

$\square$be a l-dimensional Brownian motion with $W_{0}=0$ , and $dJ$ a PoissonLetwith the intensity $d\hat{J}=dtd\iota/$ . Denote bytherandom measure over
and $dJ$ . For simplicity of notations, write
augmented filtration generated by$d\tilde{J}=dJ-d\hat{J}$. Let $a_{0}(x),$
, and
and
such that$\{W_{t}\}_{t\in|0,T)}$$[0, T]\cross \mathbb{R}_{0}$$\{\mathcal{F}_{t}\}_{t\in[0_{1}T]}$$\{W_{t}\}_{\ell\in|0T|}$$d\overline{J}=I_{(|\theta|\leq 1)}d\tilde{J}+I_{(|\theta|>1)}dJ$$a_{1}(x)\in C_{b}^{1,\infty}(\mathbb{R}^{d},$$\mathbb{R}^{d})$$b_{\theta}(x)\in C_{b}^{1,\infty}(\mathbb{R}^{d}\cross \mathbb{R}_{0};\mathbb{R}^{d})$$\inf_{x\in \mathbb{R}^{d}}\inf_{\theta\in \mathbb{R}_{0}}|\det\nabla\overline{b}_{\theta}(x)|>0$,,$\lim_{|\theta|\backslash 0}b_{\theta}(x)=0$. consider the. For-valued process
wheredetermined by the stochastic differential equat.ion (SDE):$\overline{b}_{\theta}(x)=.c+b_{\theta}(x)$$r\in \mathbb{R}^{d}$$\mathbb{R}^{d}$$x_{0}=x$$dx_{t}= \alpha_{0}(J:_{t})(lt+a_{1}(.\mathfrak{l}_{f})\circ dl\uparrow t+\int_{\mathbb{R}_{(\}}}b_{\theta}(J_{\ell-})d\overline{J},$

Under the conditions on theassociated infinitesimal generator$\mathfrak{c}\cdot oeffi_{t}\cdot iet1|\backslash$$\mathcal{L}$$\{x_{t}(\equiv x_{t}^{x})\}_{t\in|0,T]}$.

. there exists a unique solution to (2.1), and theis$\mathcal{L}f=\mathcal{A}_{0}f+\frac{1}{2}\mathcal{A}_{1}\mathcal{A}_{1}f+./R_{1\}}\{(f\circ\overline{f)}\theta)-f-I_{(|\theta_{1}\leq\iota)}\mathcal{B}_{\theta}f\}d\nu$

where
For $y,$be the$\mathcal{A}_{\triangleleft}=\alpha_{i}(.\iota\cdot)\cdot\nabla$

and$\mathcal{B}_{\theta}=b_{\theta}(1)\cdot\nabla_{C}\iota rt^{s}\iota^{r}rightarrow\langle$

tor fields over$\mathbb{R}^{d}$

,

.

and
with det $(/\neq 0$ . clef $z\neq 0$ . let-valued processes determi ] bv he linear SDEs: $y_{0}=y,$ $z_{0}=z$ and$z\in \mathbb{R}^{d}\otimes \mathbb{R}^{d}$$\mathbb{R}^{d}\otimes \mathbb{R}^{d}$(2.1)$\{.|/t(\equiv y_{t}^{x,y})\}_{t\in|0,T]}$

$i$

$e(1$

$\{z_{t}(\equiv z_{t}^{x,z})\}_{t\in|0,T]}$$t$,$dy_{t}= \nabla a_{0}(.r_{t}).y_{\ell^{(}}lt+\nabla(/l(.1_{f})_{1/f}\circ(l1f_{f}+\int_{\mathbb{R}_{()}}\nabla b_{\theta}(x_{\ell-})y_{t-}d\overline{J}$

(2.2)

97

$dz_{\ell}=-z_{t} \{\nabla a_{0}(x_{t})-\int_{|\theta|\leq 1}(\nabla\overline{b}_{\theta}(x_{\ell}))^{-1}(\nabla b_{\theta}(x_{t}))^{2}d\nu\}dt$(2.3)$-z_{t} \nabla_{t}\iota_{1}(x;_{\ell})\circ d\mathcal{W}_{t}^{7}-\int_{\mathbb{R}_{0}}z_{t-}(\nabla\overline{b}_{\theta}(.\iota_{t-}))^{-1}\nabla b_{\theta}(x_{t-})d\overline{J}$

has a
, let
. For
and
-valued process defined by the following SDE: $v_{0}=v$ andThen, for each $t\in[0, T]$ , the mappingthat$\mathbb{R}^{d}\ni x x_{t}^{x}\in \mathbb{R}^{d}$$y_{t}^{x,z^{-1}}z_{t}^{x,z}=I_{d}$$\nabla_{x^{J}}.\iota_{\ell}^{x}=y_{t}^{x,I_{d}}$$\iota’\in \mathbb{R}^{d}\otimes \mathbb{R}^{d}$$C^{1}$.

-modification suchbe the$\{v_{t}(\equiv v_{t}^{x,z_{7}v})\}_{t\in[0,T]}$$\mathbb{R}^{d}\otimes \mathbb{R}^{d}$$dv_{t}= \nabla a_{0}(x_{t})\uparrow)tdt+\nabla a_{1}(x_{t})_{1i_{t}}\circ dW_{t}+\int_{\mathbb{R}_{0}}\nabla b_{\theta}(x_{\ell-})v_{t-}d\overline{J}$(2.4)$+a_{1}(x_{t})a_{1}(x_{t})^{*}z_{\ell}^{*}dt+ \int_{\mathbb{R}_{0}}\nabla\overline{b}_{\theta}(x_{t-})\tilde{b}_{\theta}(x_{t-})\tilde{b}_{\theta}(x_{t-})^{*}z_{t-}^{*}dJ$

where

,

. Then, it can be easily checked that$\tilde{b}_{\theta}(x)=(\nabla\overline{b}_{\theta}(x))^{-1}\partial_{\theta}b_{\theta}(.r)\theta$$v_{t}^{x,z,v}=y_{t}^{x,z^{-1}} \{z\dagger)+\int_{0}^{t}’$.

$+ \int_{0}^{t}\int_{\mathbb{R}_{0}}z_{s-}^{x,z}\tilde{b}_{\theta}(x:_{s-}^{x})\tilde{b}_{\theta}(’\iota_{s-}^{x})^{*}(z_{s-}^{x,z})^{*}dJ\}$

We shall introduce the criterion on the existence of smooth densities for solutions to. and define
SDEs (cf. [16, 20]). Write$\Theta=\{0,1\}\cup \mathbb{R}_{0}$$\tilde{\alpha}_{0}(J^{\cdot})=a_{0}(J:)+\frac{1}{2}\mathcal{A}_{1^{(}}x_{1}(.\downarrow:)$,$(l\mu=d\delta_{\{0\}}+d\delta_{\{1\}}+d\nu$,$[\varphi.\psi](*\iota\cdot)=\nabla_{\mathfrak{h}^{I}}(.’\cdot)(r^{q}(.\iota:)-\nabla\varphi(x)\psi(x)$for$\varphi,$$\psi\in C^{1}(\mathbb{R}^{d};\mathbb{R}^{d})$. Denote the families of$\mathcal{V}_{0}=\{a_{1},\tilde{b}_{\theta};\theta_{\sim}^{\sim}-\mathbb{R}_{0}\}$,$\mathbb{R}^{d}$-valued functions on$\mathbb{R}^{d}$by$\mathcal{V}_{k}=\{K)\theta\varphi;\varphi\in \mathcal{V}_{k-1}, \theta\in\Theta\}(k\geq 1)$,

and define$\wp_{\theta}\varphi=I_{(\theta=0)}[\tilde{a}_{0}, \varphi]+I_{(\theta=1)}[(r_{1\cdot\hat{\gamma}}]+I_{(\theta\in \mathbb{R}_{0})}\{(\nabla\overline{b}_{\theta})^{-1}(\varphi\circ\overline{b}_{\theta})-\varphi\}$.

Fact 1 (cf. [16, 20]). Suppose that the measuresatisfies Assumption 1. If the coefficients of (2.1) satisfy the Horm ander type conditions, that is, there exist a positive$d\nu$

98

constant

and a non-negatiz integer vt such that$\cdot$$\iota$

$e$

(2.5)

$\lim_{\rho\backslash }\inf_{0}p^{\iota}\sum_{k=0}^{n}\angle\nabla\int_{0^{k}}\varphi\in\overline{v}_{k}\{(\iota\cdot\wp_{\theta_{k}}\cdots\wp_{\theta_{1}}\varphi(.\iota:)/\rho)^{2}\wedge 1\}d\mu^{\otimes k}>0$

for any

and $v\in S^{d-1}$ , then the probability law ofhas a density
respect to the Lebesgue measure oversuch that the functionsmooth.
$x\in \mathbb{R}^{d}$$\mathbb{R}^{d}$

3

with
is

$\mathbb{R}^{d}\ni’\tilde{\chi}\prime r(x, ’\tilde{x})$Main result

In this section, we shall present the sensitivity formula with respect towith $\det z\neq|]$ , write$z,$$l^{y}r(\prime x, ’\tilde{x})$$J:_{T}^{x}$$x\in \mathbb{R}^{d}$. For$v\in \mathbb{R}^{d}\otimes \mathbb{R}^{d}$$X=(\begin{array}{l}x\pi(z^{*})\pi(v)\end{array}),$$\overline{X}=(\begin{array}{l}l\cdot\pi(I_{d})\pi(0_{d,d})\end{array})$$X_{t}(\equiv X_{\ell}^{X})=(\begin{array}{l}x_{t}^{x}\pi((z_{t}^{x,z})^{*})\pi(v_{\iota}^{x,z,v})\end{array})$.

Define$\tilde{A}_{0}(X)=(\begin{array}{l}0_{d.1}\pi(a_{1}(r^{\backslash }).z^{*}+J_{|\theta|\leq 1}(\nabla\overline{b}_{\theta}(x))\tilde{b}_{\theta}(x)\tilde{b}_{\theta}(x)^{*}d\nu z^{*})\pi.(|\theta|\leq 1\theta\end{array})$$A_{i}(X)=(,|l)$$(i=0.1)$ ,$B_{\theta}(X)=(\begin{array}{l}b_{\theta}(\iota\cdot)\pi(\backslash \cdot,\backslash -\pi((z(\nabla\overline{b}_{\theta}\iota\cdot))^{-1}\nabla b_{\theta}(.l^{\backslash }))^{*})\end{array})$

Let $N=d+2d^{2}$ . Then, theof the form: $X_{0}=X$ and$\mathbb{R}^{N}$,-valued process.

$\{_{\lrcorner}\lambda_{t}^{r}(\equiv X_{\ell}^{X})\}_{t\in|0,T]}$satisfies the equation$dX_{t}=(A4_{0}(X_{t})+\tilde{A}_{0}(\lrcorner\lambda_{f}’))(lf+4_{1}(X_{t})\circ dW_{t}+/\mathbb{R}_{0}B_{\theta}(X_{t-})d\overline{J}$.

99

Moreover, for each $t\in[0, T]$ , the mapping $\mathbb{R}^{N}\ni X X_{t}^{X}\in \mathbb{R}^{N}$ has a$C^{1}$-modification,and its Jacobi matrix$Y_{t}:=\nabla_{X}X_{t}^{X}=(’\backslash \nabla_{x}\pi((z_{t}^{x.z},)^{*})\nabla_{x}\uparrow\pi(v_{t}^{x,zv})/^{I\overline{\wedge}^{-l}}t$$\partial_{\pi(v)}\pi(v_{t}^{x_{r}z_{2}v})0_{d,d^{2}})$$0_{d^{2},d^{2}}$$\dot{\iota}J_{\pi(z^{*})}\pi((z_{\ell}^{x,z})^{*})$$\partial_{\pi(z^{\nu})}\pi(v_{t}^{x,z_{1}v})0_{d,d^{2}}$is invertible, because of the condition on
matrix of by , which can be computed as follows:$b: \inf_{x,\theta}|\det\nabla\overline{b}_{\theta}(x)$I $>0$ . Denote the inverse$Z_{t}$$Y_{t}$$Z_{t}=(\begin{array}{lll}z_{t}^{x,z} 0_{d,d^{2}} 0_{d,d^{2}}\pi-\nabla_{x}((z_{t}^{x,z})^{*}) \{\partial_{\pi(z)}\pi((z_{t}^{x,z})^{*})\}^{-1} 0_{d^{2},d^{2}}\pi-\nabla_{x}(v^{\frac{\tau}{\ell},z,v}) -\partial_{\pi(z)}\pi(v_{t}^{x,z_{y}v}) \{\partial_{\pi(v)}\pi(v_{t}^{x,z_{l}v})\}^{-1}\end{array})$.

$+(\begin{array}{lll}0_{d,d} 0_{d,d^{2}} 0_{d,d^{2}}0_{d^{2},d} 0_{d^{2},d^{2}} 0_{d^{2},d^{2}}\partial_{\pi(z)}\pi(\iota)^{x,z,v}t)\nabla_{x}\pi((z_{t}^{x,z})^{*}) 0_{d_{l}^{2}d^{2}} 0_{d^{2},d^{2}}\end{array})$Let$\{V_{t}\}_{t\in[0,T]}$be the$\mathbb{R}^{N}\otimes \mathbb{R}^{N}$-valued process determined by the SDE: $V_{0}=0_{N,N}$ and$dV_{\ell}=(\nabla A_{1)}(\lrcorner\lambda_{t}’)+\nabla\tilde{A}_{0}(\lrcorner\lambda_{t}’))V_{t}dt+\nabla A_{1}(X_{t})V_{\ell}\circ dW_{t}$$+_{L}[\mathbb{R}_{0}\nabla B_{\theta}(X_{t-})1_{t-}^{j}d\overline{J}+A_{1}(X_{\ell})A_{1}(X_{t})^{*}Z_{\ell}^{*}dt$$+\backslash (\mathbb{R}_{0}\nabla\overline{B}_{\theta}(X_{t-})\tilde{B}_{\theta}(d\lambda_{t-}^{r})\tilde{B}_{\theta}(X_{t-})^{*}Z_{\ell-}^{*}dJ$

where $\overline{B}_{\theta}(X)=X+B_{\theta}(X)$ , andenables us to see that$\tilde{B}_{\theta}(X)=(\nabla\overline{B}_{\theta}(X))^{-1}\partial_{\theta}B_{\theta}(X)\theta$,. Then, the It\^o formula$V_{t}=Y_{t} \{\int_{0}^{t}Z_{s}A_{1}(X_{s})A_{1}(X_{s})^{*}Z_{s}^{*}ds$$+ \int_{r)}^{\ell}1_{\mathbb{R}_{0}}^{Z_{s-}\tilde{B}_{\theta}(X_{s-})\tilde{B}_{\theta}(X_{s-})^{*}}$

Define$V_{t}^{11},$$V_{t}^{21}$and$V_{t}^{31}$zg $-dJ\}$ .

by$V_{t}|_{X=\overline{X}}(\begin{array}{ll}I_{(t} 0_{2d^{2}} d\end{array})=(\begin{array}{l}V_{t}^{11}V_{t}^{21}1_{t}^{31}\end{array})$

.

(3.1)

100

Denote byclasses$\mathcal{U}$$C_{LG}(\mathbb{R}^{d})$the family of bounded doniains and their complements inof R-valued functions by
and$\mathbb{R}^{d}$. Define the$\{\zeta(\mathbb{R}^{d})$$C_{LG}(\mathbb{R}^{d})=\{f\in C’(\mathbb{R}^{d})$; $|f(.\{:)|\leq$ const. $(1+|x|)\}$ ,$\mathfrak{F}(\mathbb{R}^{d})=\{f=\sum_{k=1}^{n}\alpha_{k}f_{k}I_{A_{k}};77\in N,$$\alpha_{k}\in \mathbb{R},$$f_{k}\in C_{LG}(\mathbb{R}^{d}),$$A_{k}\in \mathcal{U}\}$.

Theorem 1. Suppose that the Hormander type condition (2.5) stated in Fact 1 is satisfied.
, it holds that
Then, for$\varphi\in \mathfrak{F}(\mathbb{R}^{d})$,$\nabla_{-}(E[\varphi(.\iota:_{T})])=E[\varphi(J_{T})\Gamma_{T}|_{X=\overline{X}}]$

where$\Gamma_{T}=(\Gamma_{T}^{1}, \ldots, \Gamma_{T}^{d}),$$m_{\overline{J}}=(m_{T}^{1}, \ldots .

m_{T}^{d})$and$m_{T}=(\int_{0}^{T}z_{t}c\iota_{1}(.\iota_{t})dI\dagger_{t}^{r})^{*}-\int_{0}^{T}\int_{\mathbb{R}_{0}}\frac{\partial_{\theta}[g(\theta)\theta\tilde{b}_{\theta}(x_{t-})^{*}z_{t-}^{*}]}{g(\theta)}d\tilde{J}$$\Gamma_{T}^{k}=[m\tau\tau^{1}’\tau,\{;V_{TT}^{21,-1}.y_{T}]_{(\beta-1)d+,\beta}\zeta\}[y_{T}]_{a,k}$$+ \sum_{\alpha,\beta=1}^{d}[\uparrow\prime_{T}^{-1}]_{l}f_{0}[1_{T}^{\gamma 31}]_{1\cdot’-1)d+0,\beta}[?1_{T}^{-1}y_{T}]_{\beta,k}$.

Remark 2. (1) Although a similar result to Theorem 1 can be also obtained via theGirsanov transform approach (cf. [4, 16, 20]), or the Malliavin calculus on the
Wiener space (cf. [18]), most of them are $|)aitl$ attention to only the diffusion term.
in t.erms of not only the diffusion term, but also
Our formula in Theorem 1 is$st_{\dot{\epsilon}}\iota|\rho(]$the jump term.
(2) Similarly to Theorem 1, the sensitivities in the other parameters which govem theprocess can be studied. This will be discussed elsewhere.

4

Key lemmas

In this section, we shall prepare some key lemmas, which will play an important role in
the proof of Theorem 1. Write$X_{t}=X_{t}^{X}$$]_{t}’=\}_{1}^{\prime\lambda J_{N}}$.

$Z_{f}=Z_{t}^{\lambda,J_{N}}$,$\mathfrak{l}_{t}^{i}=V_{t}^{X,J_{N},0_{N.N}}$101with compact support. For $t\in[0, T]$ and$E[\Phi(X_{T-t})|X_{0}=\tilde{X}]$ . Define the operator
byLet$\Phi\in C^{2}(\mathbb{R}^{N})$$\tilde{X}\in \mathbb{R}^{N}$, write $U(t,\tilde{X})=$$\mathfrak{B}_{\theta}$$\mathfrak{B}_{\theta}\Psi(X):=\Psi(\overline{B}_{\theta}(X))-\Psi(X)$.

Then, the Kolmogorov backviard equation (cf. [13]) implies that

Lemma 4.1 (cf. [21], Lemma 4.1). For
equality holds.

$\Phi\in(:2(\mathbb{R}^{N})$with compact support, the following$\Phi(X_{T})=E[\Phi(X_{T})]+\int_{0}^{T}\nabla U(s, X_{s})A_{1}(X_{s})dW_{s}$(4.1)$+ \int_{0}^{T}\int_{\mathbb{R}_{0}}\mathfrak{B}_{\theta}U(s, X_{s-})d\tilde{J}$.

the key equalities as stated below, which can be regarded
Lemma 4.1 helps us toas the integration by parts formula.
$ob\tau_{J}ain$Lemma 4.2. For$\Phi\in C^{2}(\mathbb{R}^{N})$with compact support, it holds that

$E[\nabla_{X}(\Phi(\lambda_{T}^{r}))\int_{0}^{T}Z_{s}A_{1}(X_{s})A_{1}(X_{s})^{*}Z_{s}^{*}ds]$(4.2)$= E[\Phi(\lambda_{T}’)(\int_{0}^{T}Z_{s}A_{1}(X_{s})dW_{s})^{*}]$

Proof. Multiplying both$\llcorner cides$.

of the equality (4.1) in Lemma 4.1 by $( \int_{0}^{T}Z_{s}A_{1}(X_{s})dW_{s})^{*}$

we see that
$E[\Phi(X_{T})(_{L}1_{0^{T}}^{Z_{s}A_{1}}(X_{s})dM_{s}^{7}/)^{*}]$$= E[\int_{0}^{T}\nabla lJ(s, X_{s})A_{1}(X_{s})d\dagger/V_{s}(\int_{0}^{T}Z_{s}A_{1}(X_{s})dW_{s})^{*}]$$= \int_{0}^{T}E[\nabla_{X}(U(s, X_{s}))Z_{s}A_{1}(X_{s})A_{1}(X_{s})^{*} zg]ds$$= E[d\int_{0}^{T}Z_{s}A_{1}(X_{s})A_{1}(X_{s})^{*}Z_{s}^{*}ds]$.
$\square$102

Lemma 4.3. For$\Phi\in C^{2}(\mathbb{R}^{N})$with compact support, it holds that

$E[\lrcorner t’\int_{0}^{T}\int_{\mathbb{R}_{0}}Z_{s-}\tilde{B}_{\theta}(X_{s-})\tilde{B}_{\theta}(X_{s-})^{*}Z_{s-}^{*}dJ]$(4.3)$=- E[,d\int_{0}^{T}\int_{R_{0}}\frac{\dot{c})_{\theta}[g(\theta)\theta\tilde{B}_{\theta}(X_{s-})^{*}Z_{s-}^{*}]}{g(\theta)}d\tilde{J}]$.

Proof. Write,$M_{T}= \int_{0}^{T}\int_{R_{0}}Z_{s-}\tilde{B}_{\theta}(X_{s-})\tilde{B}_{\theta}(X_{s-})^{*}Z_{s-}^{*}dJ$.

$\hat{M}_{T}=\int_{0}^{T}\int_{\mathbb{R}_{0}}Z_{s}\tilde{B}_{\theta}(X_{s})\overline{B}_{\theta}(X_{s})^{*}Z_{s}^{*}d\hat{J}$

Since$E[\Phi(X_{T})\hat{M}_{T}]=E[\int_{0}^{T}\int_{R_{0}}U(s, X_{s})Z_{s}\tilde{B}_{\theta}(X_{s})\tilde{B}_{\theta}(X_{s})^{*}Z_{s}^{*}dj]$,

multiplying both sides of the equality (4.1) in Lemma 4.1 by $M_{T}-\hat{M}_{T}$ enables us to see
that$E[\Phi(X_{T})M_{T}]=E[\Phi(X_{T})(\Lambda f_{T}-\Lambda^{\wedge}f_{T}+\hat{M}_{T})]$$= E[\int_{0}^{T}\int_{\mathbb{R}_{(}}\mathfrak{B}_{\theta}U(s.X_{s})Z_{s}\tilde{B}_{\theta}(X_{s})\tilde{B}_{\theta}(X_{s})^{*}Z_{s}^{*}d\hat{J}]$$+ E[\int_{0}^{T}\int_{\mathbb{R}_{0}}U(\backslash \backslash \backslash .X_{s})Z_{s}\tilde{B}_{\theta}(X_{s})\tilde{B}_{\theta}(X_{s})^{*}Z_{s}^{*}d\hat{J}]$$= E[\int_{0}^{T}\int_{\mathbb{R}_{0}}U(.\backslash \cdot.\overline{B}_{\theta}(X_{s}))Z_{s}\tilde{B}_{\theta}(X_{s})\tilde{B}_{\theta}(X_{s})^{*}Z_{s}^{*}d\hat{J}]$

Taking the derivative in$X\in \mathbb{R}^{N}$

yields that the right hand side is equal to$\nabla_{X}(E[\int_{0}^{T}\int_{\mathbb{R}_{0}}U(s, \overline{B}_{\theta}(X_{s}))Z_{s}\tilde{B}_{\theta}(X_{s})\tilde{B}_{\theta}(X_{s})^{*}Z_{s}^{*}d\hat{J}])$$= E[\int_{0}^{T}\int_{\mathbb{R}_{0}}\nabla U(s, \overline{B}_{\theta}(X_{s}))\nabla\overline{B}_{\theta}(X_{s})\tilde{B}_{\theta}(X_{s})\tilde{B}_{\theta}(X_{s})^{*}Z_{s}^{*}d\hat{J}]$$+ E[\int_{0}^{T}\int_{\mathbb{R}_{0}}U(s, \overline{B}_{\theta}(X_{s}))\nabla_{X}(Z_{s}\tilde{B}_{\theta}(\lrcorner\lambda_{s}’)\tilde{B}_{\theta}(X_{s})^{*}Z_{s}^{*})d\hat{J}]$$=- E[\int_{0}^{T}\int_{\mathbb{R}_{0}}\mathfrak{B}_{\theta}U_{\backslash }’s.X_{s})(J_{\theta}[L/(\theta)\theta\tilde{B}_{\theta}(X_{s})^{*}Z_{s}^{*}]d\theta ds]$

.

103

$+ E[\int_{0}^{T}\int_{\mathbb{R}_{0}}\{\mathfrak{B}_{\theta}U(s, X_{s})+U(.\backslash .X_{s})\}\nabla_{X}(Z_{s}\tilde{B}_{\theta}(X_{s})\overline{B}_{\theta}(X_{s})^{*}Z_{s}^{*})d\hat{J}]$$=- E[\Phi(X_{T})\int_{0}^{T}\int_{\mathbb{R}_{0}}\frac{\partial_{\theta}[g(\theta)\theta\tilde{B}_{\theta}(X_{s-})^{*}Z_{s-}^{*}]}{g(\theta)}d\tilde{J}]+E[\Phi(X_{T})\nabla_{X}M_{T}]$.

Here we have used the integration-by-parts formula via Assumption 1 (iii) in the second
equality, and (4.1) of Lemma 4.1 in the third equality. On the other hand, the left handside is equal to$\nabla_{X}E[\Phi(X_{T})M_{T}]=E[\nabla_{X}\Phi(X_{T})M_{T}]+E[\Phi(X_{T})\nabla_{X}M_{T}]$.

Therefore, we can get the assertion.

$\square$Combining Lemma 4.2 and Lemma 4.3, we have
Corollary 4.1. For$\Phi\in C^{2}(\mathbb{R}^{N})$with compact support, it holds that

$E[\nabla_{X}\Phi(X_{T})Z_{T}1_{T}’]=E[\Phi(X_{T})\mathfrak{M}_{T}]$

.

(4.4)

where
$\mathfrak{M}_{T}:=(\int_{0}\tau_{Z_{t}A_{1}(X_{t})/}dM_{f}^{\dot{\prime}})^{*}-\int_{0}^{T}\int_{\mathbb{R}_{0}}\frac{\partial_{\theta}[g(\theta)\theta\tilde{B}_{\theta}(X_{t-})^{*}Z_{\ell-}^{*}]}{g(\theta)}d\tilde{J}$5.

Proof of Theorem 1

admits a smooth
In this section, we shall prove Theorem 1. Since the probability law offrom Fact 1, it is sufficient to study
density with respect to the Lebesgue measure overwith compact support via the standard density argument as
the case of$J_{:\tau}^{x}$$\mathbb{R}^{d}$$\varphi\in C^{2}(\mathbb{R}^{d})$.
stated in [7, 15, 21], instead ofwith compact support. Multiplying both sides of (4.4) in CorollaryLet, we have
4.1 by$\varphi\in \mathfrak{F}(\mathbb{R}^{d})$$\Phi\in C^{2}(\mathbb{R}^{N})$$(I_{d}, 0_{d,d^{2}},0_{d,d^{2}})^{*}\in \mathbb{R}^{d}\otimes \mathbb{R}^{N}$$E[\nabla\Phi(X_{T})(V_{T}^{11*}, V_{T}^{21*}.

]’\tau^{31*})^{*}]=E[\Phi(X_{T})\mathfrak{M}_{T}(I_{d}, 0_{d,2d^{2}})^{*}|_{X=\overline{X}}]$$=E[\Phi(X_{T})m_{T^{\downarrow J_{T}^{-1}}}|_{X=\overline{X}}]$

Take $\Phi(X)=\varphi(.l:)[v^{-1}z^{-1}]_{j,k}(1\leq j$. $k\leq(l)$ . Remark that, for$\nabla_{x_{0}}\Phi(X),\prime 7^{\prime^{-1}}$.

$1\leq\alpha,$ $\beta\leq d$,$\partial_{[z]_{0,\beta}}\Phi(X)=-^{\neg}(.r\cdot)[z^{-1}]_{a,k}[u^{-1}z^{-1}]_{j,\beta}$

,

,

104

.

$i?v]_{\alpha,\beta}\Phi(X)=-t_{\hat{\prime}}(.\iota\cdot)[t^{1^{-1}}]_{j.\sigma}\delta_{j}^{\beta}[1^{f^{-}}1z^{-1}]_{j,k}$Then, we have$\sum_{j=1}^{d}E[[\nabla\Phi(X_{T})(V_{T}^{11*}, V_{T}^{21*}, V_{T}^{31*})^{*}]_{j}]$$= \sum_{j=1}^{d}\{E[\sum_{a=1}^{d}\nabla_{a}\varphi(\alpha_{T})[\iota_{T}^{-1}y_{T}]_{j,k}[V_{T}^{11}]_{oj}]$$-E[[y_{T}]_{0,k}[f^{-}\tau^{1}$$- E[\varphi(.r_{T})\sum_{o,\beta=1}^{d}[\iota_{T}^{-1}]_{j_{tY}},\delta_{j}^{l}’[\iota_{T}^{-1}y_{T}]_{j,k}[V_{T}^{31}]_{(\beta-1)d+\mathfrak{a},j]}\}X=\overline{X}$$=\{.\cdot$

ll

$-1$

$-E[[y_{T}]_{0.A}[/T$$-E[1)^{-l},-1T\}X=\overline{X}$ .
Since$V_{T}^{11}=v_{T}$, we can get$E[[\nabla\varphi(.\iota:_{T})y_{T}]_{k}]$$=E[\varphi(.r_{T})[m\uparrow I^{-}]_{k}]|_{X=\overline{X}}$$+E[TT|_{X=\overline{X}}$$+E[[1_{T}^{-1}]_{l’.O}[1_{T}^{-1}X=\overline{X}$$=E[\varphi(J:_{T})\Gamma_{T}^{k}|_{X=\overline{X}}]$

which completes the proof.

,$\square$105

6

ExampleLet $m=1$ andAssumption 1. For

satisfies
Suppose that the measure
given by-valued L\’evy process$(\gamma, \sigma_{1}, \sigma_{2})\in \mathbb{R}\cross(0, +\infty)\cross(0, +\infty)$$\zeta\in \mathbb{R}$, consider the$\mathbb{R}$.

$d\nu$$\{\zeta_{t}(\equiv\zeta_{t}^{\zeta})\}_{t\in[0,T]}$

(6.1)

.

$\zeta_{t}=\zeta+\gamma t+\sigma_{1}W_{t}+\sigma_{2}\int_{0}^{\ell}\int_{\mathbb{R}_{0}}\theta d\overline{J}$Let$f\in C^{\infty}(\mathbb{R})$with $f’\neq 0$ . For$\eta\in \mathbb{R}$, define the process$?h= \eta+\int_{0}^{t}f((s)ds$by$\{’\gamma\int^{\ell}\}_{\ell\in[0_{2}T]}$

(6.2)

.

Now, we are in position that$x=(\zeta, \eta)^{*},$

Since $f’\neq 0,$$\sigma_{1}>0$$a_{1}(x)=(\sigma_{1},0)^{*}$$a_{0}(x)=(\gamma, f(\zeta))^{*}$$b_{\theta}(x)=(\sigma_{2}\theta, 0)^{*}$

and$\wp_{0^{(}}\iota_{1}(J:)=[(l_{1}.\tilde{(J}_{0}](.\iota\cdot)=(0, \sigma_{1}f’(\zeta))^{*}$,

with respect to the,admits a smooth density
the probability law of
withfrom Fact 1. Our interest is to study the sensitivity forLebesgue measure on.
respect toexplicitly.
In order to get our desired result, we have to computeandDefine$x_{T}^{x}=$$(\zeta_{T}^{\zeta}$$p_{T}(x,\tilde{x})$$\eta_{T}^{(})$$\mathbb{R}^{2}$$r \int\tau$$(\in \mathbb{R}$$V_{T}^{11},$$\Phi[\Psi]_{t}=\int_{0}^{t}\Psi_{s}d\Phi_{s}$.

$G_{t}= \sigma_{1}^{2}t+\int_{0}^{t}\int_{\mathbb{R}_{0}}(\sigma_{2}\theta)^{2}d,l$$K_{t}= \int_{0}^{f}\int_{1R_{0}}\sigma_{2}\theta^{2}d.J$

Then we see that, for$z,$$v\in \mathbb{R}^{2}\otimes \mathbb{R}^{2}$.

$F_{t}= \int_{0}^{t}f’(\zeta_{s})ds$.

$V_{T}^{31}$$V_{T}^{21}$,,$H= \sigma_{1}^{2}+\int_{|\theta|\leq 1}(\sigma_{2}\theta)^{2}d\nu$$L_{t}= \int_{0}^{t}\int_{\mathbb{R}_{0}}2(\sigma_{2}\theta)^{3}dJ$.

with $\det z\neq 0$ ,$z_{t}^{x,z*}=(\begin{array}{ll}1 -F_{f}0 1\end{array})z^{*}$$\lfloor 1_{t}^{x,zv}=(\begin{array}{ll}l 0F_{t} 1\end{array})(’+(\begin{array}{ll}C_{z\ell} -G[F]_{t}F[G]_{\ell} -F[G[F]]_{t}\end{array})z^{*}$

.

106

Write
$\overline{X}=(\begin{array}{l}d\pi(I_{2})\pi(0_{2,2})\end{array})$$X=(\begin{array}{l}x\pi(z^{*})\pi(v)\end{array}),$Then, the process$\{X_{t}\}_{t\in|0_{7}T]}$$\lambda_{\ell}’(\equiv X_{\ell}^{X})=(\begin{array}{l}x_{t}^{x}\pi(z_{\ell}^{x,z*})\pi(v_{\ell}^{x,zv}))\end{array})$.

satisfies the following SDE: $X_{0}=X$ and$dX_{t}=(A_{0}(X_{\ell})+ \tilde{A}_{0}(X_{t}))dt+A_{1}(X_{t})\circ dW_{t}+\int_{R_{O}}B_{\theta}(X_{t-})d\overline{J}$ ,where $A_{1}(X)=(\sigma_{1},0_{1,9})^{*},\tilde{A}_{0}(X)=(0_{1,6}, z^{11} H. 0, z^{21}H, 0)^{*}$ and$A_{0}(X)=(\gamma, f(\zeta), -z^{12}f’(\zeta).

0, -z^{22}f’(\zeta), 0,0, v^{11}f’(\zeta), 0, v^{12}f’(\zeta))^{*}$$B_{\theta}(X)=(\sigma_{2}\theta, 0_{1,5}, z^{11}(\sigma_{2}\theta)^{2}.0, z^{21}(\sigma_{2}\theta)^{2},0)^{*}$

Then, we have$Y_{t}|_{X=\overline{X}}=\nabla_{X}X_{t}^{X}|_{X=\overline{X}}=(\begin{array}{lll}Y_{t}^{11} 0_{2,4} 0_{2,4}Y_{\ell}^{21} Y_{l}^{22} 0_{4,4}Y_{\ell}^{31} Y_{\ell}^{32} Y_{t}^{33}\end{array})$

,

where

$Y_{\ell}^{11}=(\begin{array}{ll}1 0F_{t} l\end{array})$$Y_{t}^{21}=(\begin{array}{ll}0_{2,1} 0_{2,1}-(J_{\backslash }\cdot F_{t} 00 0\end{array})$$Y_{\ell}^{22}=(\begin{array}{ll}(Y_{\ell}^{11*})^{-1} 0_{2.2}0_{2,2} (\}_{t}^{11*})^{-1}\end{array})$.

$\}_{t}^{\prime 31}=(\begin{array}{ll}0 0(\partial_{i}F)[G]_{t} 0-G[\partial_{\zeta}F]_{\ell} 0-\subset 9_{i}(F[G[F]]_{t}) 0\end{array})$,$Y_{t}^{32}=(\begin{array}{llll}G_{t} -G[F]_{\ell} 0 0F[G]_{t} -F[|_{I}^{\cap}[F]]_{t} 0 00 0 C_{z^{1}}\prime -G[F]_{t}0 0 F[C]_{t} -F[G[F]]_{t}\end{array})\cdot Y_{t}^{33}=(\begin{array}{ll}Y_{t}^{11} 0_{2,2}0_{2,2} Y_{\ell}^{11}\end{array})$

,

107

Moreover, we see that

$Z_{\ell}=Y_{t}^{-1}|_{X=\overline{X}}=(\begin{array}{lll}Z_{t}^{11} 0_{2,4} 0_{2,4}Z_{t}^{21} Z_{t}^{22} 0_{4,4}Z_{t}^{31} Z_{t}^{32} Z_{t}^{33}\end{array})$

where$Z_{t}^{11}=(\begin{array}{ll}1 0-F_{t} 1\end{array}),$$Z_{t}^{21}=(\begin{array}{ll}0_{2_{\prime}1} 0_{2.1}\partial_{\dot{(}}F_{t} 00 0\end{array})$$Z_{\ell}^{22}=(\begin{array}{ll}(Z_{\ell}^{11*})^{-1} 0_{2,2}0_{2,2} (\Delta_{\ell}^{11*}\prime)^{-1}\end{array})$$Z_{t}^{31}=(\begin{array}{ll}0 0-(\partial_{\zeta}F)[G]_{t} 0-(\partial_{\zeta}F)[G]_{t} 02(\partial_{\zeta}F)[G[F]]_{\ell} 0\end{array})$$Z_{t}^{32}=(\begin{array}{llll}-G_{t} C_{J}[F]_{t} 0 0-F[G]_{t} F[G[F]]_{t} 0 00 0 -C_{J}t G[F]_{t}0 0 -F[G]_{t} F[G[F]]_{t}\end{array}),$,$Z_{\ell}^{33}=(\begin{array}{ll}Z_{t}^{11} 0_{2,2}0_{2,2} Z_{t}^{11}\end{array})$.

Hence, we can get$V_{T}^{11}=Y_{T}^{11} \int_{0}^{T}Z_{t-}^{11}(\begin{array}{ll}1 00 0\end{array})(Z_{t-}^{11})^{*}(lG_{t}= (\begin{array}{ll}G_{T} -G[F]_{T}F[G]_{T} -F[G[F]]_{T}\end{array})$,$V_{T}^{21}=Y_{T}^{21} \int_{0}^{T}Z_{t-}^{11}(\begin{array}{ll}1 00 0\end{array})(Z_{t-}^{11})^{*}dG_{t}+ \}’\tau^{22}\int_{0}^{T}Z_{t-}^{21}(\begin{array}{ll}1 00 0\end{array})(Z_{t-}^{11})^{*}dG_{t}$$=(\begin{array}{l}0_{2,1}0_{2,1}-(\partial_{(}F)[G]_{T}(\partial_{\zeta}F)[C_{I}[F]]_{T}00\end{array})$.

$V_{T}^{31}=Y_{T}^{31} \int_{0}^{T}Z_{t-}^{11}(\begin{array}{ll}1 00 0\end{array})(Z_{t-}^{11})^{*}dG_{f}+Y_{T}^{32} \int_{0}^{T}Z_{t-}^{21}(\begin{array}{ll}l 00 0\end{array})(Z_{t-}^{11})^{*}dG_{t}$$+Y_{T}^{33} \int_{0}^{T}Z_{t-}^{31}(\begin{array}{ll}1 00 0\end{array})( (\begin{array}{ll}1 00_{3,1} 0_{3,1}\end{array})(Z_{t-}^{11})^{*}dL_{t}$108$=(\begin{array}{ll}0 0((\partial_{\zeta}F)[G])[G]_{T} -((cJ_{\tilde{\zeta}}F^{\backslash })[G])[C_{J}’[F]]_{T}0 0-((\partial_{\zeta}F)[G])[G[F]]_{T} ((()_{i}F)[G[F]])[G[F]]_{T}\end{array})$$+(\begin{array}{ll}L_{T} -L[F]_{T}F[L]_{T} -F[L[F]]_{T}-G[(\partial_{\dot{\zeta}}F)[G]]_{T} C_{J}[(\partial_{i}F)[G[F]]]_{T}-F[G[(\partial_{(}\cdot F)[(J\urcorner]]]_{T} F[(G[F])[(\partial_{\zeta}F)[G]]]_{T}\end{array})$

From Theorem 1, we can calculate the weight$\Gamma_{T}$$\Gamma_{T}=(\sigma_{1}W_{T}+\sigma_{2}\int_{0}^{T}\int_{\mathbb{R}_{0}}\frac{\partial_{\theta}[g(\theta)\theta^{2}]}{J((\theta)}d.\tilde{J},$$+([v_{T}^{-1}V_{T}^{31}]_{1,1},$.

concretely as follows.
$0)?1_{T}^{-1}+([V_{TT}^{2\iota_{t\prime}-1}y_{T}]_{3,2},0)y_{T}$$\sum_{j=1}^{2}[’\prime^{-1}\tau]_{2_{j}},[1_{T}^{\gamma 31}]_{j+2,2)\prime^{-1}}’\iota_{T}y_{T}$.

$\square$

References[1] Bally, V., Bavouzet, M. and Messaoud, M., (2007). Integration by parts formula for
locally smooth laws and applications to sensitivity computations, Ann. Appl. Probab.,17, 33-66.
[2] Bichteler, K., Gravereaux, J. B. and Jacod, J., (1987). Malliavin calculus for processes
with jumps, Gordon and Breach Science Publishers, New York.
[3] Bismut, J. M., (1981). Martingales, tbe Malliavin calculus and hypoellipticity undergeneral H\"ormander’s conditions, Z. Wahrsch. Verw. Gebiete, 56, 469-505.

[4] Bismut, J. M., (1983). Calcul des variations stochastique et processus de sauts, Z.
Wahrsch. Verw. Gebiete, 63, 147-235.
[5] Carlen, E. and Pardoux, E.. (1990). Differential ( $al(ulus$ and integration by partson Poisson space. In: Albeverio, S., Blaiichard, Ph., Testard, D. (eds.) Stochastics,
algebra and analysis in classical and quantum dynamics (Marseille, 1988) (Math.
Appl., vol. 59). Dordrecht: Kluwer Aca, . Publ., 63-73.
[6] Carr, P., Geman, H., Madan, D. an(1 Yor, M., (2003).
processes, Math. Finance, 13, 345-382.

$St_{O(}\cdot ha_{\sim^{i}}^{t}tic$

Volatility for L\’evy

109

$[$

7 Cass, T. R. and Friz, P. K., (2007). The Bismut-Elworthy-Li formula for jumpdiffusions and applications to Monte arlo methods in finance, available atarXiv $:math/060431lv3$ .
$]$$C^{1}$[8] Davis, M. H. A. and Johansson, M. P., (2006). Malliavin Monte Carlo Greeks forjump diffusions, Stoch. Processes Appl., 116, 101-129.

[9] Di Nunno, G., Oksendal, B. and Proske, F., (2008). Malliavin Calculus for L\’evy
Processes with Applications to Finance, Springer-Verlag, Berlin.
[10] El-Khatib, Y. and Privault, N., (2004). Computations of Greeks in a market withjumps via the Malliavin calculus, Finance Stoch., 8, 161-179.

[11] Elwothy, K. D. and Li, X. -M., (1994). Fornmlae for the derivatives of heat semigroups, J. Funct. Anal., 125, 252-286.
[12] Fourni\’e, E., Lasry, J., Lebuchoux, J., Lions, P. and Touzi, N., (1999). Applicationsof Malliavin calculus to Monte Carlo methods in finance, Finance Stoch., 3, 391-412.
[13] Gihman, I. I. and Skorokhod, A. V., (1972). Stochastic Differential Equations,
Springer-Verlag, New York.
[14] Ishikawa, Y. and Kunita H., (2006). Malliavin calculus on the Wiener-Poisson space
and its applications to canonical SDE with jumps, Stoch. Processes Appl., 116, 1743-

1769.
[15] Kawai, R. and Takeuchi, A., (2009). Greeks formulae for an asset price dynamics
model with gamma processes, in revision for publication.

[16] Komatsu, T. and Takeuchi, A., (2001). On the smoothness of pdf of solutions to SDEwith jumps, Int. J. Differ. $E/]^{‘}\cdot A_{l1)})l_{!}.2,141-197$ .
, F. and Vives. J., (2002). On L\’evy processes, Malliavin[17] Leon, J.A., Sol\’e, J.L.,
calculus and market models with jumps, Finance Stoch., 6, 197-225.
$Utzp\uparrow,$[18] Nualart, D., (2006). The Malliavin calculus and related topics, 2nd edition, SpringerVerlag, Berlin.
[19] Picard, J., (1996). On the existence of smooth densities for jump processes, Probab.
Theory Relat. Fields, 105, 481-511.

[20] Takeuchi, A. (2002). The Malliavin calculus for SDE with jumps and the partiallyhypoelliptic problem, Osaka J. Math. 39, 523-559.

110[21] Takeuchi, A. (2009). The Bismut-Elworthy-Li type formulae for stochastic differential
equasions with jumps, in revision for publication.

Department of Mathematics,
Osaka City University,
Sugimoto 3-3-138, Sumiyoshi-ku, Osaka 558-8585, JAPANE-mail: takeuchiQsci. $osaka-cu$ .ac. jp