THE MALLIAVIN CALCULUS FOR SDE WITH JUMPS AND THE PARTIALLY HYPOELLIPTIC PROBLEM

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1 Takeuchi, A. Osaka J. Math. 39, THE MALLIAVIN CALCULUS FOR SDE WITH JUMPS AND THE PARTIALLY HYPOELLIPTIC PROBLEM ATSUSHI TAKEUCHI Received October 11, 1. Introduction It has been studied by many authors to give natural conditions under which the law of the solution to a stochastic differential equation admits a smooth density. There are many approaches to this problem in the theory of partial differential equations, and it is well known that the smoothness of the fundamental solution for a parabolic differential operator holds under the Hörmander condition which is a condition on the dimension of the Lie algebra generated by the vector fields associated with the coefficients of the differential operator cf. 8. In 17 and 18, Malliavin introduced differential calculus on the Wiener space, and applied it to a probabilistic proof of the Hörmander theorem. It is well known that the Kusuoka-Stroock-Norris lemma plays an important role cf. 3, 9, 14, 19 and. On the other hand, Bismut discussed the case of jump processes in 4. He used the Girsanov transformation, and proved formulas of integration by parts. A generalization to stochastic differential equations with jumps has been done since Bismut cf., 7, 1, 15 and 16. Picard introduced new calculus on the Poisson space in 1. He considered, what is called, the duality formula without using the Girsanov formula, instead of formulas of integration by parts, and applied his calculus to the existence of smooth densities. In 1, Komatsu and the present author gave a new approach to the existence of smooth densities. There the Girsanov transformation is used, and the formula of integration by parts is considered not on the Poisson space but on the cád-lág space. It is essential to discuss the integrability of the inverse of the Malliavin covariance matrix via the Sobolev inequalities. For this problem, certain fundamental inequlalities about semimartingales are considered, and the exponential decay of the Laplace transform for the law of the Malliavin covariance is proved. This inequality can be proved by an elementary stochastic calculus. In particular, in the case of diffusion processes, the fundamental inequality can be showed so easily that a simple proof of the Hörmander theorem is obtained 11. Moreover using the fundamental inequality, a generalization of the Hörmander condition is obtained for SDE with jumps 1. In 13, Kunita discussed the case of canonical stochastic differential equations with jumps by using Picard s method and certain fundamental inequalities on semimartingales obtained in 1.

2 54 A. TAKEUCHI In this paper, we shall consider the smoothness of the density for the image of the solution to a stochastic differential equation with jumps through a smooth mapping on a Euclidean space. The smoothness of the density for image processes is often called the partial hypoellipticity. In the case of diffusion processes, Bismut-Michel 5, Stroock 4, and Taniguchi 6 have already studied it by using the Malliavin calculus. Their results are a natural generalization of well-known Hörmander s theorem. Taniguchi considered the partially hypoelliptic problem on Riemannian manifolds. Our main purpose is an attempt to generalize their results for diffusion processes to jump processes on Euclidean spaces. We shall discuss the partially hypoelliptic problem by using the Malliavin calculus on the cád-lág space, for which we have to investigate the exponential decay of the Laplace transform for the law of the Malliavin covariance stated above. Hence the crucial point is to use certain fundamental inequalities about some semimartingales considered in 11 and 1. When we apply the fundamental inequalities, we are faced with a difficulty how to treat random variables at a terminal time of processes included in the Malliavin covariance. In order to overcome the difficulty, we use time-reversed stochastic differential equations. After all, we need to consider an estimate similar to the one considered in 11 and 1. It might be expected that the partial hypoellipticity for stochastic differential equations with jumps holds under a condition similar to the one obtained by Bismut-Michel, Stroock, and Taniguchi. However, we have to give some additional condition to the coefficient of the jump term in stochastic differential equations. The condition may seem to be technical, but it remains open to remove the condition. Adding the condition, we obtain main results on the partially hypoelliptic problem for stochastic differential equations with jumps. The non-degenerate condition in our main theorem seems to be complicated, but it is essentially a generalization of the one for diffusion processes. The organization of this paper is as follows: in Section, we shall give some preparation and state main results. In order to understand the result for general processes with jumps, we divide into two cases. In Section 3, we shall study timereversed stochastic differential equations. In Section 4, we shall give a proof of the result for diffusion processes. This yields a new proof of Taniguchi s result 6. In Section 5, we shall finish the proof for general processes with jumps.. Preliminaries and main results Let be the cád-lág space R R. For, let be the projection from to R such that ω = ω for ω. Put σ-fields W = ε> σ ; ε and W = W as usual. Set = and ν θ = θ α θ for < α <. We shall consider a probability measure on the measurable space W such that θ = ; = θ

3 THE MALLIAVIN CALCULUS FOR SDE 55 is a Poisson random measure on R R with the intensity ν θ, and that process defined by = θ θ 1 θ θ θ θ >1 is a -dimensional W -Brownian motion starting at R, where θ = θ ν θ We remark that the measure θ generates W -martingales. Throughout this paper, s denote certain positive absolute constants, which may change every lines. Let = = be a family of smooth vector fields on R such that derivatives of all orders of are bounded, where the symbol denotes the derivative with respect to R, that is, φ = φ 1 for any R -valued 1 -mapping φ defined on R. Let θ be an R -valued 1 -mapping on R R such that θ and θ θ are smooth in R. Moreover, we assume that it satisfies the following conditions: = θ θν θ 1 θ 1 ν θ 1 θ < 1 θ ν θ θ 1 ν θ R θ ν θ R R is continuous θ θ θ θ θ θ θ θ is bounded and continuous, R θ R is a homeomorphism where integrals by the measure ν θ are defined in the sense of the principal value: ψ θ ν θ = lim ψ θ ν θ ε θ >ε for any R -valued mapping ψ θ defined on R R. Let θ be the R -valued mapping on R R satisfying = θ = θ

4 56 A. TAKEUCHI Let = R \ and = Define an operator θ θ by θ θ φ θ 1 if θ = = θ if θ 1... θ 1 φ θ φ if θ for any vector field = φ, where θ = θ θ. For R, we shall consider the following stochastic differential equation 1 = θ θ where denotes the stochastic integral in the Stratonovich sense. Here we give a remark. The precise meaning of the last term of 1 is the following sum: θ 1 θ θ θ 1 θ ν θ θ >1 θ θ From now on, we often make use of such simple representations without any comments. Using the stochastic integral in the Ito sense, we can rewrite 1 as follows: = θ θ where = 1 From the assumption on the coefficients of 1, there exists a pathwise unique solution =. For, define = θ where θ is the shift operator on such that θ =. By a discussion as in 6, we see that the mapping defines a stochastic flow of diffeomorphisms on R cf. 5.

5 THE MALLIAVIN CALCULUS FOR SDE 57 Let π = π be an R -valued smooth mapping on R such that all derivatives of any orders are bounded, where. We shall call π the image process of by the mapping π. Our main purpose is to study the partially hypoelliptic problem for jump processes. The main theorem on the problem may seem strange at first sight, and its proof also seems to be complicated. However, considering the theorem only for diffusion processes, it will be understood that the theorem is essentially a generalization of well-known Hörmander s theorem, and the proof is simple and natural. So we shall give a proof of the result restricted to the case for diffusion processes before doing it of the general result. This would make our method of the proof clear and our main theorem easy to understand. First we shall mention the result about diffusion processes, which yields a new proof of Taniguchi s results 6 on Euclidean spaces. Let A be families of vector fields on R defined by A = 1... A = ; A 1 = Theorem 1. Assume that there exist non-negative integer such that inf 1 = φ A π φ > for any R. Then the law of random variable π admits a smooth density with respect to the Lebesgue measure on R. The proof of Theorem 1 will be given in Section 4. REMARK 1. Condition in Theorem 1 is called the Hörmander condition for the partially hypoelliptic problem. It is equivalent to the following condition: 3 dim π L A = for any R, where π denotes the differential of mapping π at and LA denotes the linear space generated by a family A of vector fields at. =

6 58 A. TAKEUCHI EXAMPLE 1. Let = 3, =, = 1 and π 1 3 = 1. We shall consider the following stochastic differential equation: 1 = = 3 3 = 3 Since 3 = 1, it is clear that the law of doesn t admit a smooth density. In this case = = = = 1 1 Since the dimension of the linear space spanned by the vectors 1 = π = π 1 1 is equal to, the law of random variable π admits a smooth density from Theorem 1 cf. Remark 1. It may be expected that adding a certain regularity condition on θ, the partial hypoellipticity for SDE with jumps holds under a condition corresponding to Hörmander s one. But this conjecture remains open. Here we shall consider the case: the coefficient θ of the jump term of 1 has the following expression 1 1 θ 1 θ = 1 θ where = 1 R R, 1 θ is an R -valued mapping defined on R R, and θ is an R -valued mapping defined on R R. We say that an R -valued mapping θ defined on R R belongs to the class ν if the mapping µγ θ = µ θ θ γ θ is bounded and continuous for any µ Z and any γ Z, and satisfies that µγ θ ν θ µγ θ ν θ <

7 THE MALLIAVIN CALCULUS FOR SDE 59 where µ = 1 µ1 µ for µ = µ 1... µ Z. From now on, we always assume the following condition. ASSUMPTION. Each component of θ belongs to the class ν. We shall introduce some notations. Choose ρ such that < ρ < α 1/4. Set τ = α/α, 1 4ρ/4 if θ 1... σθ = α ρ/α if θ and τθ 1... θ = σθ 1 σθ τ for θ 1... θ. Define a probability measure θ on as follows: θ = = θ ν θ θ where ν θ = θ 1 ν θ. Let θ λ be an operator acting on vector fields on R such that θ λ θ λφ θ if θ 1... = λ 1 σθ θ/λ 1 σθ if θ For θ \, and θ 1... θ, let λ θ θ 1 θ = φ λ θ θ 1 θ be the vector fields on R defined by λ θ = θ if θ 1... λ θ if θ λ θ θ 1 θ = θ λ τθ 1... θ 1 θ1 λ τ λ θ where θ = θ θθ and λ θ = λ τ θ/λ τ. Theorem. Assume that there exist non-negative integer and three positive constants κ, ε and σ with σ 5ε < ε such that 4 inf inf λ ε 1 = θ = θ θ 1 θ λ σ π φ λ θ θ 1 θ 1 λ κ

8 53 A. TAKEUCHI for sufficiently large λ, where ε = τ 1 4ρ 4 α ρ α Then the law of random variable π Lebesgue measure on R. admits a smooth density with respect to the Let be the vector fields on R defined by = θ Set = = 1... for any vector field, and let A be a family of vector fields on R defined by A = A = ; = A 1 1 Corollary 1. Assume that there exists a non-negative integer such that 5 inf inf 1 = φ A π φ > Then the assumption of Theorem is satisfied, and the law of random variable π admits a smooth density with respect to the Lebesgue measure on R. In Section 5, we will prove Theorem and Corollary 1. REMARK. We remark that condition 5 is a generalization of condition in Theorem 1 and the conditions obtained by Léandre in 15 and 16. EXAMPLE. Let = 4, = 3, = 1 and π = 1 3. Let δ > be a sufficiently small constant, and set δ 1 = 1 δ. We shall consider the

9 THE MALLIAVIN CALCULUS FOR SDE 531 following stochastic differential equation: 1 = 1θ θ = = 3 θ θ 4 = 1 1 θ 1 1θ 3 θ θ where 1θ and 3 θ are certain bounded functions on R such that 1θ = 1 θ δ θ and 3 θ = 1 θ for θ 1. Since 4 = 1 1 3, it is clear that the law of doesn t admit a smooth density. In this case 1 = 1 θ = 1 θ δ θ θ θ δ θ 1 θ 4 1 θ δ θ = θ = 1 δ 1 θ δ θ θ δ 1 θ δ θ 1 θ 1 δ 1 θ δ = θ = 1 1 θ = 1 δ 1 θ δ θ θ 1 θ 1 η 1 θ = 4 1 δ 1 θ δ η δ θη 1 1 for θ, η 1. This example doesn t satisfy condition 5 in Corollary 1, but satisfies the assumption in Theorem. In fact, = φ λ θ 1 θ θ φ λ θ1 1 θ θ φ λ θ1η 1 η θ θ η λ τδ δ 1 θ δ θ 4 ν θ θ 1 θ 1 θ 1 λ τδ δ 1 θ δ θ 4 ν θ η 1 λ τδ 1 λ σδ δ 1 θ δ η δ 3 1 θ 4 η 4 ν η ν θ θ 1 δ 1 θ δ 1 θ 4 ν θ 3 =

10 53 A. TAKEUCHI λ σδ 3 θ 1 η 1 3 = 3 3 = λ σδ δ 1 θ δ η δ 1 θ 4 η 4 ν η ν θ 3 = where σ = τ τµ /α. If < δ < σ/ σ, condition 4 in Theorem is satisfied. Hence the law of random variable π admits a smooth density. REMARK 3. Let A 1 be a family of vector fields on R defined by A = A A 1 = λθ A = λ τθ 1... θ 1 ; = 1... = A 1 θ λ τθ 1... θ 1 ; θ = 1 A λ where = ; θ. Then vector field θ θ 1 θ is an element in 1 = A. Theorem 1 and Corollary 1 corresponds to the case of A. Theorem says that it is possible to extend Theorem 1 and Corollary 1 to the case of 1 = A. A A 1 Theorem 1 Corollary 1 A 1 A 1 1 A 1 A A 1 A A 3 Theorem A 3 A 1 3 A 3 A 3 3 A 4 3 We shall prove Theorem 1, Theorem and Corollary 1 by using the Malliavin calculus. The Jacobi matrix = / 1 of diffeomorphism

11 satisfies the following linear SDE: THE MALLIAVIN CALCULUS FOR SDE = θ θ where = δ 1. Let be the solution to the linear SDE 7 = θ 1 θ From the Ito formula, we see that = = Since,, and are bounded, it is a routine work to prove that 8 < for all > 1 and cf. 5. For a matrix, we will denote its transpose by. Define an -matrices valued process = as follows: = 1 θ θ θ where = 1 1 and θ = θ 1 θ θθ. Let = be a non-negative definite, -matrices valued process defined by = π π We call the Malliavin covariance matrix for the functional π on. Finally we shall consider a variation of the process =. First we shall introduce some notations. Set = 1 = 1...

12 534 A. TAKEUCHI θ = 1 θ ξ θ = expξ θθ for ξ R such that ξ < 1. Define a perturbed process ξ = ξ by ξ = ξ ξ θ θ Then and the law ξ = ξ 1 of the process ξ are mutually absolutely continuous 3. We shall denote the conditional expectation of the Radon-Nikodym derivative ξ / W by ξ. Set ξ = ξ ξ = ξ ξ = ξ ξ = ξ ξ = ξ Lemma.1. For >, if det 1 >1, then the law of π admits a smooth density with respect to the Lebesgue measure on R. Proof. We shall give an outline of the proof. From the absolute continuity between and ξ, the equality π π 9 1 = π ξ π ξ ξ ξ 1 ξ holds for any test function on R, 1 and 1. By using the same argument as stated in 1, we see that random variables ξ π ξ π ξ ξ ξ 1 ξ are integrable by uniformly in ξ. Taking the differential of both sides of 9 at ξ =, we see that π 1 = π π ξ ξ ξ 1 ξ ξ= ξ Define operators D = D 1... D acting for smooth functionals on by D = ξ π ξ ξ ξ 1 ξ ξ ξ=

13 Then equality 1 is expressed as follows: THE MALLIAVIN CALCULUS FOR SDE 535 π = π D 1 Set = D 1 and = /. By an argument as in 1, we see that random variables are integrable by ξ π ξ π ξ ξ ξ 1 uniformly in ξ. Hence we have π π = = π D 1 π = D 1 = π D D 1 We remark D D 1 >1 by the same method as stated in 1. Repeating such arguments, we obtain µ π = ξ ξ π D µ 1 D µ 1 >1 for any multi-index µ Z. The assertion of the lemma follows from the Sobolev lemma. 3. Time-reversed processes In this section, we shall give some remarks on time-reversed stochastic differential equations. Fix. Let =. Then is also a Lévy process, and the law of coincides with that of. In particular, defined by = is also a -dimensional Brownian motion starting at R, and θ defined by ; = ; BR is also a Poisson random measure with the intensity ν θ θ = θ ν θ. cf. 1,. Put

14 536 A. TAKEUCHI For R, let be the solution to the following SDE: 11 = θ θ From the assumption on the coefficients of 11, there exists a pathwise unique solution. For any, let = θ Then the mapping defines a stochastic flow of diffeomorphisms on R as in 6. The Jacobi matrix = / 1 of the mapping of diffeomorphism satisfies the linear SDE: 1 = Let be a solution to the linear SDE: 13 = θ θ θ 1 θ Since θ 1 = θ θ, equations 1 and 13 are parallel with equations 6 and 7. By using the Ito formula, we can easily check that = = Moreover, the following proposition holds. Proposition ζ µ τ µ τ τ ζ for any > 1,, µ Z and ζ > 1. Proof. The continuity and the differentiability of with respect to can be proved by a method as in 6. From the assumptions for the coefficients of equations, the assertion of the proposition follows.

15 THE MALLIAVIN CALCULUS FOR SDE 537 Lemma 3.1. For almost all ω, the equality 15 = holds for any and R. Proof. For = 1,..., define processes and by = = for / 1 / =, It is easy to check that = for. We shall consider the following ordnary differential equations: = = Denote solutions of the above equations by and, respectively. From the uniqueness of solutions, it is easy to see that for almost all ω, 16 = for any and R. Note that when process = is replaced by process =, system θ etc. shall be replaced by system θ etc.. Let be a decreasing sequence of positive numbers such that as. Secondly we shall consider the following stochastic differential equations: η = η = θ > η η θ η η θ η

16 538 A. TAKEUCHI θ > η θ θ Solutions will be denoted by η and η, respectively. Then it is easy to check that for almost all ω, 17 η = η η for all and R. Thirdly we shall consider the following stochastic differential equations: = = θ > θ > θ θ θ θ From the assumptions for the coefficients, there exist unique solutions and in the pathwise sense. For a positive integer, set ζ = inf > ; θ > θ = By an arguments as in Chapter VI, Section 7 of 9, we can show that for >, τ ζ µ η τ µ τ µ η τ µ τ τ ζ as for any > 1,, and µ Z. It follows immediately by the Sobolev inequality that µ η τ µ τ τ ζ µ η τ µ τ τ ζ

17 THE MALLIAVIN CALCULUS FOR SDE 539 as for any > 1,, and µ Z. By taking the limits in 17, we can obtain that for almost all ω ω ; ζ >, 18 = for all and R. Finally by the same argument as stated above, we can prove that µ τ µ τ τ ζ µ τ µ τ τ ζ as for any > 1,, and µ Z. By taking the limits in 18, we can obtain that for almost all ω ω ; ζ >, 19 = for all and R. Since ζ as, the assertion of the lemma holds. By Lemma.1, our goal is to find sufficient conditions under which det 1 >1 is satisfied. Since det 1 inf and = 1 λ 1 exp λ it suffices to find conditions under which λ 1 exp λ = λ λ is satisfied for all > 1.

18 54 A. TAKEUCHI First we shall give some notations and remarks. For λ > 1, 1, R, and a vector field, we shall introduce the following notations: Q λ = λ 4 Q λ = λ 4 N λ = λ 4 π φ π φ 1 π θ λ θ where θ = θ. Define an -matrices valued process by = π θ θ θ π Since we see that from Lemma 3.1, we can express as follows: = = Q 1 N 1 In order to give a lower estimate of, we shall prove the following lemma. Though this lemma is elementary, it plays an important role in our argument. For ζ > 1, define random variables and by = ζ = ζ π π θ θ Put ζ = R ; ζ.

19 THE MALLIAVIN CALCULUS FOR SDE 541 Lemma 3.. Let ρ be a positive constant. There exist a positive integer and points in ζ such that inf Q 1 Q 1 ρ ζ inf N 1 N 1 ρ ζ for some 1, and furthermore that ζ Proof. Put σ = σ for σ 1, and, ζ. It is easy to see that Q 1 Q 1 1 σ π σ σ σ σ and N 1 N 1 1 π σ σ θ σ σ θ σ Since ζ is compact in R, there exist a positive integer with ζ/ρ 1 and points in ζ such that ζ = ζ ; ρ Q 1 Hence we obtain Q 1 Q 1 ρ

20 54 A. TAKEUCHI and N 1 N 1 N 1 ρ for some 1. Moreover, from Proposition 3.1 and the Sobolev inequality, we see that for any positive integer µ µ ζ µ ζ vol ; ζ ζ µ ζ ζ µ1 π Similarly we can prove that The proof is complete. µ1 π µ1 π µ1 π ζ 1/ We shall consider estimate. Let < ε < ε /5, where we consider ε = 4 in the case of diffusion processes. At first we see that := exp λ 44ε 1 exp λ = where γ > 4 3ε, 3ε < β < γ 4, and 1 = ω ; 1 > λ ε = ω ; > λ β 3 = ω ; > λ β

21 1 = THE MALLIAVIN CALCULUS FOR SDE 543 ; 1 3 From the Chebyshev inequality, 8 and Lemma 3. with ζ = λ ε, we see that λ ε 1 = λ ε 3 λ β = λ β 3ε 4 λ β = λ β 3ε for any > 1. Moreover, from the Schwarz inequality, inequality 14, and Lemma 3. with ζ = λ ε, ρ = λ γ, we obtain the estimate 1 exp 1 inf Q λ N λ λ ε exp 1 min Q λ N λ λ 4 γβ λ λ λ λ λ λ exp Q λ N λ exp exp 1 exp 1 θ = θ = Q λ 1/ Q λ τ θ λ θ Q λ τ θ λ θ Q λ τ λ θ ν θ 1/ 1 > λ 4ε 1/ exp Q λ τ θ λ θ λ ε θ = 1/ where λ is a certain positive constant satisfying λ λ εγ 1, is a certain positive integer less than λ, and 1 = 1 ; 1 λ 4ε Hence we have the following lemma.

22 544 A. TAKEUCHI Lemma 3.3. If there exists a positive integer less than λ such that λ exp θ = for any > 1, then estimate holds for any > Proof of Theorem 1 Q λ τ θ λ θ = λ In this section, we shall give the proof of Theorem 1. For λ > 1, define a subset λ = 1... of by = λ = > λ ε 1 A 1 = where = π ψ for any vector field = ψ. We shall introduce the fundamental estimate on a certain continuous semimartingale considered in 11. The following lemma plays an important role in our argument. Lemma 4.1. Let ω be a continuous semimartingale such that = = Then there exist a positive random variable λ with λ 1, and positive constants, 1, independent of λ and such that the inequality λ 4 λ 1/8 log λ 1 holds on the complement of the set ω ; = > λ 1/4 λ 1/4 = for sufficiently large λ, where = for any function =. We shall consider estimate 1 in Lemma 3.3. For the sake of simplicity of notations, put Q λ A = A Q λ ζ = ε σ

23 THE MALLIAVIN CALCULUS FOR SDE 545 where < σ < ε 3ε. Then we have 1 exp Q λ A exp Q λ A = 1 3 = λ where ε < δ < ζ ε and = ω ; > λ δ λ ζ = 1 ; λ = From the Chebyshev inequality, the Burkholder inequality and Proposition 3.1, we obtain estimates = A λ ε = = λ ε ε and for all see that 3 λ δ = λ ζ δ ε λ ζ > 1. Finally we shall consider the estimate of 1. From the Ito formula, we π = π π for = A. From Lemma 4.1, there exists a positive random variable with λ 1 such that the inequality λ Q λ ε 1 A 1 λ ε log λ Q λ ε A holds on λ. Put A = = A. By the iterative application of, we have Q λ ε A λ ε 1 log 1 λ Q λ ε 1 A 1

24 546 A. TAKEUCHI λ ε log λ Q λ ε A on = λ. We remark that if then the inequality 1 λ 4ε λ ζ λ δ 3 π φ 1 π φ π λ 4ε δ φ φ holds for any λ ζ, where for any R -valued mapping ψ on R, ψ and ψ denote the best constants satisfying the inequality ψ ψ 1 ψ ψ ψ for, R. From the Jensen inequality, we see λ λ ε λ λ ε 1/ 1 for sufficiently large λ. Hence from and 3, we have 1 λ ε λ exp Q λ ε A exp Q λ ε A 1/ exp Q λ ζ λ ε A 1/ exp λσ inf π φ λ 4ε δ 1 A = exp λ κ Therefore we obtain the estimate λ 1 exp Q λ A = λ 1 for any > 1. From Lemma 3.3, the law of random variable π admits a smooth density.

25 5. Proof of Theorem and Corollary 1 THE MALLIAVIN CALCULUS FOR SDE 547 In this section, we shall discuss the case of general processes with jumps. At first, we prepare the following lemma on a certain semimartingale with jumps considered in 1, which plays an important role in the proof of Theorem. Lemma 5.1. Let ω be a semimartingale such that = = θ 1 θ 1 θ θ >1 θ θ >1 θ θ For < ρ < α 1/4, there exist positive constants, 3 = independent of λ and, and a positive random variable λ with λ 1 such that λ λ 1 λ ρ log 1 λ 1 4ρ 3 λ ρ on the complement of the set ω ; = λ ρ λ λ θ 1 ν θ θ θ ν θ θ > λ ρ θ 1 θ >1 for sufficiently large λ, where = for any function =. Define a subset λ as follows: λ ρ = ω ; η = η >1 η η ν η > λ ρ η 1 where = π φ for any vector field = φ. By using the Ito formula, we see that for =, π = π

26 548 A. TAKEUCHI π π η η From Lemma 5.1, there exists a positive random variable λ satisfying λ 1 such that 4 Q λ λ ρ log λ Q λ 1η η λ η holds on the complement of λ ρ. Set ρ λ = and ρ λ = λ ρ 1 θ 1 θ λ 1θ 1... θ ρ θ λ 1θ 1... θ 1 θ1 λ for 1 and any vector field. For 1, define ρ λ by log ρ λ = λ ρ log λ 1 θ 1 θ λ 1θ 1... θ ρ log λ 1θ 1... θ θ λ 1θ 1... θ 1 θ1 λ From the Jensen inequality, it is easy to see that for any 1 < < λ ρε, ρ λ 1 for sufficiently large λ. Hence by iterating 4, the inequality 5 Q λ log ρ λ Q λ θ 1 θ Q λ 1θ 1... θ θ λ 1θ 1... θ 1 θ1 λ holds on the complement of the set ρ λ for sufficiently large λ.

27 THE MALLIAVIN CALCULUS FOR SDE 549 We shall consider estimate 1 in Lemma exp = 1 exp θ = Q λ τ θ λ θ θ = Q λ τ θ λ θ θ = ρ λ τ θ λ where = 1 ; ρ λ τ λ θ θ = Since =, 1... are bounded smooth mappings and θ satisfies assimption, it is a routine work to show the estimate 6 = θ = θ 1... θ 1 λ θ θ 1 θ λ θ θ 1 θ = η 1 η λ θ θ 1 θ η λ θ θ 1 θ ν θ λ η θ θ 1 θ η >1 where is a constant independent of λ. From the Chebyshev inequality, Proposition 3.1 and 6, we can estimate as follows: = θ = By using the Jensen inequality and 5, we have 7 1 = exp ρ λ τ θ λ = λ ε ε θ = Q λ τ θ λ θ exp log ρ λ τ θ λ θ θ = exp θ θ 1 θ = θ =

28 55 A. TAKEUCHI exp = Q λ τθ 1... θ λ θ θ 1 θ θ = θ θ 1 θ Q λ τθ 1... θ λ θ θ 1 θ where τθ 1... θ = τ for =. Therefore it suffices to estimate λ 8 1 exp Q λ γ for γ > ε σ δ and a vector field. To simplify our argument, we shall consider π = π as a canonical projection from R to R. In fact, it suffices to consider a diffeomorphism π on R such that elements of π is equal to π, that is, π π = Let, be the solutions to following linear stochastic differential equations: = = Then by using the Ito formula, we get = = θ 1 θ θ 1 θ

29 that is, = THE MALLIAVIN CALCULUS FOR SDE 551 θ 1 θ Furthermore, since and, are bounded, we see that 9 < for any and > 1. For any -matrix, we shall decompose to four elements. Let 1 be an -matrix, an -matrix, 3 an -matrix, and 4 an -matrix. Then we shall express as follows: = Denote = 1 = θ 1 1 θ = 3 θ 4 θ = = π = In particular, we can express as follows: = 1 1 θ θ θ θ Before giving an upper estimate of the higher order moment of, we shall introduce the following lemma. It can be easily proved by applying the Burkholder inequality, so we omit the proof cf. 5. Lemma 5.. Let be a positive integer, and θ a predictable process with θ ν θ < θ ν θ < =

30 55 A. TAKEUCHI Then the following estimate holds θ θ θ ν θ θ ν θ where is a positive constant depending on. By using the above lemma, we obtain the following lemma on an upper estimate of the higher order moment of. 3 Lemma 5.3. For > 1, there exists a positive constant such that 3 η / 1 4 1/ 1 η where η is a positive constant and 3 = ; η Proof. In this lemma, we may take = for a positive integer. Under we see from the Hölder inequality that η

31 THE MALLIAVIN CALCULUS FOR SDE θ 1 4 θ 3 θ 1 4 θ 4 θ θ θ 1 4 ν θ 3 3 θ 1 4 ν θ 3 4 θ ν θ θ 1 4 ν θ 3 3 θ 1 4 ν θ 3 4 θ ν θ θν θ θν θ θν θ θ ν θ θ ν θ

32 554 A. TAKEUCHI η θ ν θ 1 θν θ / / 1 θ ν θ 3 θν θ / / 3 θ ν θ θν θ / / 4 θ ν θ 1/ 1 4 1/ 1 η 3 From the Gronwall inequality, we obtain inequality 3.

33 THE MALLIAVIN CALCULUS FOR SDE 555 Let us consider estimate 8. Put ζ = ε σ. Then we have exp Q λ γ exp Q λ ζ λ γ 4 exp Q λ ζ λ γ = 1 3 where < β < ζ ε, ε < δ < ζ 3ε, and = ω > λ δ λ ζ = ω 3 λ 4 4 ζ λ ζ > λ β λ ζ 3 = ; 4 = ; 1 By using the Chebyshev inequality and Lemma 5.3, we get λ δ 3 λ ζ λ δ 3 λ ζ 3 λ ζ 3 1 λ 4 ζ λ ζ λ ζ 1/ 1 λ 4 ζ λ ζ λ ζ λ ζ 1 1/ λ ζβ = λ ζ 3ε δ By using the Chebyshev inequality, we have 3 λ β = λ ζ β ε 3 λ 4 4 ζ λ ζ λ ζ

34 556 A. TAKEUCHI because, are continuous processes. Finally we shall estimate 1. Since inf 1 λ ζ λ δ we have ψ 1 1 ψ ψ 1 1 ψ inf 1 1 λ ζ ψ λ ζ 1 1 ψ ψ λ 4ε δ under Hence we see that 1 λ 4ε λ ζ λ δ λ ζ 1 exp λ 4 γ ψ 1 λ ζ exp 4 λ γ 1 exp ψ exp 4 λ γ λ ζ 1 1 exp ψ 1 λ ζ exp 4 λ γ 1 ψ 1 1 exp λ γ ζ inf inf ψ 1 λ ε 1 Therefore we obtain the estimate 31 1 exp Q λ γ exp λ γ ζ inf inf ψ 1 λ λ ε 1

35 THE MALLIAVIN CALCULUS FOR SDE 557 We now consider estimate 7. From 4 and 31, we have 1 exp θ θ 1 θ = θ = Q λ τθ 1... θ λ θ θ 1 θ exp λ σ = λ inf inf λ ε 1 = θ = θ θ 1 θ φ λ θ θ 1 θ 1 λ Hence we get the estimate λ 1 1 exp θ = Q λ τ θ λ θ = λ for any > 1. From Lemma 3.3, the law of random variable π admits a smooth density. Proof of Corollary 1. It suffices to check that condition 4 in Theorem is satisfied under 5. Let < ε < ε /5 1 σ, and ζ, ζ two constants satisfying ε < ζ < ζ/ < 1 σ, where σ = σθ = α ρ/α θ. Put = π, and ϑθ = θ= 1... θ θ 1... = 1... Let ψ be an R -valued mapping on R. Since θ satisfies assumption, we see that θ λψ = ψ ϑθ λ 1 σ θ = for sufficiently large λ. Hence it is a routine work to check that θ λψ 1 θ ψ 1 = ψ 1 = θ λ ζ θ λ ν θ

36 558 A. TAKEUCHI = θ λ ζ ψ θ ν θ λ ζ α 1 σ = 1 ψ 1 = ψ = 1 λ ζ4 α ψ 1 θ = = θ λ ζ θ ν θ λ ζ α 1 σ Similarly we obtain θ θ 1 θ φ λ θθ1 θ 1 λ ε ζ4 α 1... = 1 1 Choosing σ such that σ > ζε 14 α/, we see that condition 4 in Theorem is satisfied under condition 5. The proof is now complete. ACKNOWLEDGEMENT. The author would like to thank Prof. T. Komatsu for his valuable suggestions, encouragements and helpful comments. The author is grateful to Dr. A. Pilipenko for the careful reading of this article. Finally, the author is also grateful to the anonymous referee for his useful comments and advices. References 1 J. Bertoin: Lévy processes, Cambridge Univ. Press, K. Bichteler, J.B. Gravereaux and J. Jacod: Malliavin calculus for processes with jumps, Stochastic Monographs vol., Gordon and Breach, New York, J.M. Bismut: Martingales, the Malliavin calculus and hypoellipticity under general Hörmander s conditions, Zeit. für Wahr , J.M. Bismut: Calcul des variations stochastique et processus de sauts, Zeit. für Wahr , J.M. Bismut and D. Michel: Diffusions conditionnelles, I. Hypoellipticité partielle, J. Funct. Anal , T. Fujiwara and H. Kunita: Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group, J. Math. Kyoto Univ , S. Hiraba: Existence and smoothness of transition density for jump-type Markov processes: Application of Malliavin calculus, Kodai Math. J , L. Hörmander: Hypoelliptic second order differential equations, Acta Math ,

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