Statistical inference and Malliavin calculus

Transcription

1 Statistical inference and Malliavin calculus José M. Corcuera and Arturo Kohatsu-Higa Abstract. The derivative of the log-likelihood function, known as score function, plays a central role in parametric statistical inference. It can be used to study the asymptotic behavior of likelihood and pseudo-likelihood estimators. For instance, one can deduce the local asymptotic normality property which leads to various asymptotic properties of these estimators. In this article we apply Malliavin Calculus to obtain the score function as a conditional expectation. We then show, through different examples, how this idea can be useful for asymptotic inference of stochastic processes. In particular, we consider situations where there are jumps driving the data process. Mathematics Subject Classification 2. Primary 62Mxx, 6Hxx; Secondary 6Fxx, 62Fxx. Keywords. Diffusion processes, Malliavin calculus, parametric estimation, Cramer- Rao lower bound, LAN property, LAMN property, jump-diffusion processes, score function.. Introduction In classical statistical theory, the Cramer-Rao lower bound is obtained by using two steps: an integration by parts and the Cauchy-Schwarz inequality. Therefore it seems natural that the integration by parts formulas of Malliavin calculus will play a role in this context. In recent times, the theory of Malliavin Calculus has attracted attention from Computational Finance to derive expressions for the calculation of the greeks which measure the sensitivity of option prices conditional expectations with respect to certain parameters, see e.g. [3],[4]. We show in this article, that similar techniques can be used to derive expressions for the score function in a parametric statistical model and This work is supported by the MEC Grant No. MTM26-32.

2 2 J.M. Corcuera and A. Kohatsu-Higa consequently we obtain the Fisher information and the Cramer-Rao s bounds. The advantage of these expressions is that they do not require the explicit expression of the likelihood function directly and that its form is appropriate to study asymptotic properties of the model and some estimators. Gobet [5] was the first contribution in this direction. He uses the duality property in a Gaussian space to study the local asymptotic mixed normality of parametric diffusion models when the observations are discretely observed. Recently, duality properties have been used to obtain a Stein s type estimator in the context of a Gaussian space see [9] and [], and in the context of a Poisson process see []. An interesting discussion of different Cramer-Rao s bounds is given in [2]. In this paper we use Malliavin calculus with the aim of giving alternative expressions for the score function as conditional expectations of certain expressions involving Skorohod s integrals and we show how to use them to study the asymptotic properties of the statistical model, to derive Cramer-Rao lower bounds and expressions for the maximum likelihood estimator. The paper is organized as follows. In the first section we formulate the statistical model in a way that appears clearly that the Cramer- Rao lower bound can be obtained after integrating by parts and using the Cauchy-Schwarz inequality. In the second section, we consider the parametric models associated with diffusion processes where the driving process is a Wiener process. In the last section we consider the case of diffusions with jumps. The basic reference on Malliavin calculus is [7]. In particular, we refer the reader to this textbook for definitions and notation. As our goal is to concentrate in general principles rather than technical details, we briefly sketch the mathematical framework required and explain in the examples the procedure. 2. The Cramer-Rao lower bound A parametric statistical model is defined as the triplet X, F, {P, Θ} where X is the sample space that corresponds to the possible values of certain n-dimensional random vector X X,X 2,...,X n, F is the σ-field of observable events and {P, Θ} is the family

3 Inference and Malliavin calculus 3 of possible probability laws of X. However, when the vector X corresponds to observations of a random process where certain parameter is involved, it is better to assume that X itself depends explicitly. Then we define a parametric statistical model as a triplet consisting of a probability space Ω, F,P, a parameter space Θ, an open set in R d, and a measurable map X : Ω Θ ω, X R n Xω,, where in Θ we consider its Borelian σ-field. As usual, a statistic is a measurable map T : X R m x Tx y. For simplicity, take m d and Θ an open interval in R. Let us denote g : E T E TX, then, under smoothness conditions on g, we can evaluate where we write. ETX, Definition 2.. A square integrable statistic T C is said to be regular if ETX E TX and V artx <. Suppose that the family of random variables {X,, Θ} is also regular in the following sense: i. X has a density p ; C for all Θ with support, suppx, independent of. ii. Xω, C as a function of, ω Ω. Furthermore, X j L 2 Ω and E X j X x C as a function of x, Θ and j,...,n. iii. x j E X j XpX; px; L 2 Ω, j,...,n; where, for any smooth function h, we denote xj hx : xj hx xx iv. Any statistic T C with compact support in the interior of suppx is regular.

4 4 J.M. Corcuera and A. Kohatsu-Higa Remark 2.2. Note that if T C has compact support in the interior of suppx and the family {X,, Θ} is regular then ETX E TX E R n Tx R n xj TX X j xj TxE X j X xpx;dx xj E X j X xpx; dx, since lim E X j X xpx;tx,j,...,n, x x Θ and x suppx, where suppx is the boundary of the support of X. So, xj E X j XpX; ETX E TX. 2. px; Here the quotient is defined as if px;. The following result is a statement of Cramer-Rao s inequality. Proposition 2.3. Let T be a regular statistic satisfying i-iii and lim E X j X xpx;,j,...,n, 2.2 x x Θ and x suppx, including x in suppx if the suppx is not compact, then VarTX E ETX 2 n xj E X j XpX; px; provided that the denominator is not zero. Proof. By the boundary condition 2.2 we have that xj E X j XpX; E. px; 2, 2.3

5 Inference and Malliavin calculus 5 Then, by condition iii and since X j L 2 and T is regular we also have that E[ X j TX ] < and therefore lim E X j X xpx;tx,j,...,n, x x Θ and x suppx, then ETX E TX xj E X j XpX; px; 2.4 and we can write ETX E TX g xj E X j XpX;. px; The result follows from the Cauchy-Schwarz inequality. In most classical models, the above calculation can be straightforward, as the explicit form of the density function p is available. This is carried out in the following examples. Our goal later is to show that in some cases where the explicit density is not known, the above bound can also be written without using directly the form of p. This will be the case of elliptic diffusions. Example. Assume that X X,X 2,...,X n where X i U i are i.i.d. r.v. with U i exp. This situation corresponds to a usual parametric model of n independent observations exponentially distributed

6 6 J.M. Corcuera and A. Kohatsu-Higa with parameter. We have that X j X j ; E X j XpX; X j n exp{ xj E X j XpX; px; + X j; xj E X j XpX; X j n px; ; 2 xj E X j XpX; E V ar X j px; X j }; V ar logpx; I, where I is the Fisher information. So we see that, in this case, 2.3 is the classical Cramer-Rao lower bound. If we also assume that v. px; is smooth as function of, for every fixed x. vi. For any smooth statistic T with compact support in the interior of suppx, R n Txpx;dx R n Tx px;dx, Θ, we have the following proposition that shows that 2.3 is just the usual Cramer-Rao inequality. Proposition 2.4. Assume conditions i-vi. Then, a.e. xj E X j X xpx; px; log px;, Θ. Proof. We have seen in 2. that ETX E TX xj E X j XpX;. px;

7 Inference and Malliavin calculus 7 On the other hand, ETX Txpx;dx R n log px;txpx;dx R n ETX log px;. Using a density argument w.r.t. T, we have that xj E X j X xpx; px; log px; a.e. We have assumed in the above proof that the support of X does not depend on. In the following example we suggest a localization method to treat the case where the support depends on. Example 2. Assume that X X,X 2,...,X n where X i U i, with U i Uniform, independent r.v. s for i,...,n and R. Consider a function T TX C and function π : [, ] n R also belonging to C in, n and continuous in [, ] n such that πu

8 8 J.M. Corcuera and A. Kohatsu-Higa if there exists i,...,n with U i {, }. Then ETXπU E TXπU E xj TX X j πu E xj TX X j πu... n xj TxE X j X xπ x [,] [,] dx... Tx n xj E X j X xπ x dx ETX [,] [,] xj X j π X ETX ETX then we have that where V art xj X j π X ETXπU 2 E n x j X j π X 2 xj X j π X u j U j πu. Note that if πu and this could be considered as a limit case then the Cramer-Rao bound is of order n 2 which corresponds to the variance of maxx,x 2,...,X n the maximum likelihood estimator of. 3. Statistical Inference in Gaussian spaces In this section, we give an alternative derivation of the Cramer-Rao bound using the integration by parts formula of Malliavin Calculus. We start explaining some of the basic concepts of Malliavin Calculus. Consider a probability space Ω, F,P and a Gaussian subspace H of

9 Inference and Malliavin calculus 9 L 2 Ω, F,P whose elements are zero-mean Gaussian random variables. Let H be a separable Hilbert space with scalar product denoted by, H and norm H, we will assume that there exists an isometry in the sense that W : H h H Wh EWh Wh 2 h,h 2 H. Let S be the class of smooth random variables TWh,Wh 2,...,Wh n such that T and all its derivatives have polynomial growth. Given T S we can define its differential as DT i TWh,Wh 2,...,Wh n h i. DT can be seen as a random variable with values in H. Then we can define the stochastic derivative operator as D : D,2 L 2 Ω, R T DT. L 2 Ω,H where D,2 is the closure of the class of smooth random variables with respect to the norm T,2 E T 2 + E DT 2 H /2 Let u be an element of L 2 Ω,H and assume there is an element δu belonging to L 2 Ω and such that E DT,u H ETδu for any T D,2, then we say that u belongs to the domain of δ denoted by domδ and that δ is the adjoint operator of D. Proposition 3.. Let h be an element of H, then δh Wh. Proof. Without loss of generality we can assume that h and that T TWh,Wh 2,...,Wh n is in S with h i, i 2,...,n

10 J.M. Corcuera and A. Kohatsu-Higa orthogonal to h. Then E DT,h H E T E Tx,Wh 2,...,Wh n e 2 x2 dx R 2π E x Tx,Wh 2,...,Wh n e 2 x2 dx 2π R ETWh. Proposition 3.2. If u F j h j where F j S and h j are elements of H then δu F j Wh j DF j,h j H. Proof. ETδu ETF j Wh j ET DF j,h j H E DTF j,h j ET DF j,h j H E DTF j TDF j,h j H EF j DT,h j H E DT, F j h j H E DT,u H.

11 Inference and Malliavin calculus In the following results we assume that our observations are expressed as the measurable map X : Ω Θ ω, R n x Xω,, Θ an open subset of R with the Borelian σ-field and the σ-field in Ω is the σ-field generated by H. Theorem 3.3. Let X j D,2,j,...,n and Z be a random variable with values in H, in the domain of δ, such that Z,DX j H X j. 3. If T is a regular unbiased estimator of g then V artxv areδz X g Furthermore, assume that i X has density px; C as function of with support, suppx, independent of, ii Any smooth statistic with compact support in the interior of suppx is regular and R n Txpx;dx R n Tx px;dx, for all Θ, then EδZ X log px;,a.s. and for all Θ. Proof. E TX E xk TX X k. k If we have Z with Z,DX j H X j then ETX E xk TX Z,DX k H. By the chain rule for the derivative operator D, then DTX xk TXDX k, ETX E Z,DTX H ETXδZ.

12 2 J.M. Corcuera and A. Kohatsu-Higa Finally, by the Cauchy-Schwarz inequality Next, we have ETX 2 V artxv areδz X. ETXδZ ETX Txpx;dx R n Tx px;dx R n Tx log px;px;dx R n ETX log px;. Therefore the result follows from a density argument. Remark 3.4. The next goal is to provide a somewhat standard way of finding Z. For example, if there exists U, an n-dimensional random vector with values on H such that U k,dx j H δ kj where δ kj is Kronecker s delta then Z U k X k k verifies condition 3. if Z domδ. In particular, if A kj DX k,dx j H is well defined then we can take U k A kj DX j. The matrix A is called the Malliavin covariance matrix and the property that its inverse is well defined implies see Theorem 2..2 in [7] that the random vector X has a density function px;. In order to understand the role of each of the elements in Theorem 3.3, we treat some classical examples from time series analysis in the present framework.

13 Inference and Malliavin calculus 3 Example 3. Let X j ε j +, j n, where ε j are independent standard normally distributed random variables for j,..., n. Let H be the linear space generated by the sequence ε,ε 2,..., then DX j e j where We j ε j, and e j,e k H Eε j ε k δ jk, so A kj δ kj, and Z X k A kj DX j e j, since X k. Then δz j,k We j ε j X j n. Therefore we obtain the classical Cramer-Rao lower bound V ar T E T 2. n Example 4. Let X j ε j where ε j are independent standard normally distributed r.v. Let H be again the linear space generated by the sequence ε,ε 2,..., then DX j e j where We j ε j,a jk δ 2 jk, and X k ε j, so we can take Z X k A kj DX j ε je j. j,k then using Proposition 3.2, δz ε2 j e j,e j H X 2 j 3 n. Therefore we also obtain the classical Cramer-Rao lower bound V art 2 E T 2 As we mention in remark 3.4, the problem is how to find Z. The following elementary proposition can be useful in this sense. 2n

14 4 J.M. Corcuera and A. Kohatsu-Higa Proposition 3.5. Let X X,X 2,...,X n T and Y Y,Y 2,...,Y n T be two random vectors with components in D,2, where Y h,x with h C, and h, one to one for all. Assume that Z,DY j H Y j. Then, Also we have that Z,DX j H X j. DY r B rl d Y l h l,x r,l DX r A rl X l r,l where B kj DY j,dy k H, A kj DX j,dx k H and d means total derivative with respect to. That is, d Y l h,x. Proof. It is straightforward Next, we give an application of the above proposition. Example 5. Let X j X j + ε j, j,...,n where X is a constant and ε j are independent standard normal distributed. Let H be the linear space generated by the sequence ε,ε 2,..., then define Y h,x, where h j,x x j x j with x X. Therefore Y j ε j and d Y j d ε j, h j,x X j, so we have that DY r A rl d Y l h l,x e r δ rl X l. r,l r,l

15 As We j ε j, then Inference and Malliavin calculus 5 δdy r A rl d Y l h l,x r,l δe r δ rl X l r,l We l X l l ε l X l l DX l,e l H. l X l X l X l l and we obtain the score function. Another way to obtain Z in Theorem 3.3 would be to use the relation so that l X j X j + X j DX j e j + DX j e l X l,dx j X j + e l X l,dx j and therefore by uniqueness of solutions for difference equations, we have that X j e l X l,dx j. So, if we define then δz And consequently Z l l e l X l, l ε l X l l log px; X l X l X l. l X l X l X l. l

16 6 J.M. Corcuera and A. Kohatsu-Higa The continuous version of the previous example is given by the following one. Example 6. Let X n X t,x t2,...,x tn, < t < t 2... < t n, be observations of the Ornstein-Uhlenbeck process Or, by integrating, dx t X t dt + db t,t,x. X t t e t s db s. Here {B t,t } is a standard one dimensional Brownian motion. Let H be the closed linear space generated by the random variables {B t, t T } and H L 2 [,T],dx. Then the map W : H H [,t] W [,t] T [,t] sdb s B t defines a linear isometry, and Wh is the stochastic integral of the function h. Consequently DX t e t [,t]. We have so d X t X t dt X t dt,t, X, X t t T e t s X s ds where we write DX t s D s X t. Then X s D s X t ds X,DX t H, log px n ; EδX X n E T X s db s X n T T E X s dx s + Xsds X 2 n. In particular the maximum likelihood estimator of is given by ˆ E T X sdx s X n E T X2 sds X n.

17 Inference and Malliavin calculus 7 Take now t i n and n observations in such a way that n and n n when n goes to infinity. Then n n log px n ; E X s db s X n. n n n n By ergodicity n n n n X 2 sds P 2, then we have that n n δx L N, see Theorem.9 in [6]. 2 On the other hand n n n n n n δx X s db s X s dx s + Xsds 2 where R i X ti X ti + t i X s X ti db s + Xt 2 i n + R i, t i X ti X s X ti ds. By straightforward calculations one can see that n n n R L 2 i, so n n δx n n X ti X ti + Xt 2 i n and, since the second term is X n measurable, we have that δx E δx n n n n Xn L 2, so finally E δx n n Xn L N, 2. L 2, Note that the asymptotic Fisher information is given by. Also if we 2 consider Z n, + u n n : log px n ; + u n n log px n ;,

18 8 J.M. Corcuera and A. Kohatsu-Higa we have Z n, + where EM 2 n u n n M n E + u n n + u n n M n + u n n E L We can show that M 2 n. In fact + u n n and +2 E i<j + u n n E + u n n u + n n u + n n ER i X n EδX X n R i X n ER i X n ER i X n u n n u n n X n d X ti X ti + X n d. Xt 2 i n d X n d X n d + EERi 2 X n E u n n X n d 2 X n d ERi 2 X n p i p i ER j X n d, where p i is the joint density of n X ti, X ti evaluated at X. Then, since E ERi 2 X n p i p i /2 EERi 2 X n 2 pi E p i C EER 2 i X n 2 C 2 n 2 /2 /2 C ER 4 i /2 X n d,

19 we have that E + u n n Inference and Malliavin calculus 9 ER i X n 2 X n d C u2 n n. Here we have used Burkholder s inequality and, in order to prove that 2 E pi p i is uniformly bounded, we used the explicit form of the Gaussian density of X ti, X ti. Finally the term 3.3 goes to zero since the covariance function of X goes to zero exponentially. Then Z n, + u u X ti X ti + Xt 2 i n n n n n u 2 2 n n Xt 2 in n M n. and, since X is ergodic, we have that Z n, + u n n L un, 2 u2 4. That is, the model satisfies the LAN Local Asymptotic Normality property. 3.. Elliptic diffusions Let X n X t,x t2,...,x tn be the vector of observations. Here, t i i n, with n n where we omit the dependence of the partition on n. X is defined as the solution to X t x + t b s,x s ds + t σ s,x s db s. and B is a standard Brownian motion. Assume that b s, σ s and their derivatives with respect to and x are Cb.3 as functions of t and x and that σ s is uniformly bounded and uniformly elliptic. Sometimes, when the arguments are clear, we will write σ s and b s instead of σ s,x s and b s,x s respectively. Note that the setting here is different from the previous example where the time frame for the data satisfied that n n while in this subsection we have that n n.

20 2 J.M. Corcuera and A. Kohatsu-Higa Define and It can be shown that see [7] DX t x X t x X σ,x [,t]. β t x X t σ t,x t β t x X t X t atβ ti β ti {ti t<t i }, where a L 2 [,T], atdt,i,...,n. t i Then, it is easy to see that In fact β,dx ti H X ti β,dx ti H. x X ti atβ tj β tj {tj t<t j }dt i x X ti β tj β tj tj t j atdt i x X ti β tj β tj x X ti β ti β t x X ti β ti X ti. By the assumption of uniform ellipticity and smoothness of the coefficients, we have that β is in the domain of δ and that the score function is given by Eδ β X n, where we have for at n, n δ β β ti β ti t i x X t σ t,x t db t t i D t β ti x X t σ t,x t dt.

21 Inference and Malliavin calculus 2 Conditions i and ii of Theorem 3.3 are fulfilled since suppx is R n and px; is uniformly bounded with respect to by an integrable function. In addition, and X t x X t + t t x b s x X s ds + b s + x b s X s ds + t t so, by applying the Itô formula we have that β t X t x X t t µ s ds + t x σ s x X s db s, σ s + x σ s X s db s, σ s x X s db s, 3.4 for a certain adapted process µ that can be explicitly calculated. Then, by Theorem 2. in [] and the polarization identity, we have that n β ti β ti x X t σt t i t i σ t σ t dt L 2 db t σ s σ s dw s. Here W is a Brownian motion independent of B. Moreover, for t t i, D t β ti σ t x X t + and therefore we have n L 2. t D t µ s D t X s ds + t i D t β ti x X t σ t dt t σ s x D t X s db s. x X s t i σ t σ t dt Consequently n δ β L 2 σ s σ s dw s. 3.5

22 22 J.M. Corcuera and A. Kohatsu-Higa Moreover, by using the polarization identity and Burkholder s inequality we can see that δ β { } σ n ti X ti 2 σ ti L α, n n σ ti for any α >. So, finally σ 3 t i n Eδ β X n L 2 σ s σ s dw s. In particular this implies that the asymptotic Fisher information for is given by [ 2 σ s 2E ds]. If we define Z n, + u n log px n ; + u n log px n ; Z n, + u n It can be seen that { Eδ β X n σ n ti σ 3 t i σ s +u/ n Eδ β X n X n d. X ti 2 σ ti n σ ti } + R i, X ti, X ti 3.6 where E R i, X ti, X ti α /α O/n, uniformly in i, for any α >. Then, E R i, X ti, X ti E R i, X ti, X ti p i E R i α /α E [ pi p i p i ] β /β, where /α + /β,α >,β >, R i R i, X ti, X ti and p i is the joint density of X ti, X ti evaluated at X. The expectation with respect to this process is denoted by E.

23 Inference and Malliavin calculus 23 Now, since σ C,3 is uniformly elliptic, transition densities can be bounded from above and below uniformly by the Gaussian kernel, and then by continuity there exists β > such that [ ] β pi E < C, p i see Proposition 5. in [5]. So +u/ n E R i, X ti, X ti d C +u/ n E R i α /α d C u n 3/2. Now, we only need to calculate the derivatives, with respect to fixed X n, of the expression 3.6. We obtain, after some calculations, n Eδ β X n { 2 σ ti + σ 3 t i X ti 2 2σ t i n σ ti { 2 σ ti 3 X ti } σ ti n +o p. σ 4 t i } σ 2 t i Finally, by taking into account the continuity of the derivatives with respect to, we have, Z n, + u L σ 2 s u 2 dw s u2 σ s 2 ds. n σ s 2 That is, the model satisfies the LAMN Local Asymptotic Mixed Normality property. σ s 4. Statistical inference for models with jumps In this section, we show how to treat cases where the observed process have jumps. Essentially the question reduces to the use of an appropriate extension of Malliavin s Calculus to processes with jumps. That is, the integration by parts formulas. First, we consider the case when assume that the vector of observations Z n has a Gaussian component c.f. [4]. That is, Z is a sequence of G F H-measurable random

24 24 J.M. Corcuera and A. Kohatsu-Higa variables where F is the σ-field generated by the isonormal Gaussian process and H is a σ-field independent of F. For instance Z k F,X k,y k,k,...,n where X n X,...,X n is a G-measurable random vector and Y n Y,...,Y n is independent of G. We also set Z n Z,...,Z n. Let T be a regular statistic T TZ n and g E T. Then we have E T ETZ n E TZ n E zi TZ n Z i. Now we extend the operator D to the product filtration this is also called partial Malliavin Calculus so that DZ k xk FDX k Then using this duality property we have that DT zi TZ n DZ i zi TZ n xi FDX i. As before, if there exists a random variable V with values in H such that DZ i,v H Z i, we will have that log pz n ; EδV Z n. 4.. Jump diffusions Let X n X t,x t2,...,x tn, t i i n, where n and X t x+ t b s,x s ds+ t σ s,x s db s + t n c s,z,x s Mdz,ds. Where Mdz,ds is a compensated Poisson random measure with intensity νdz,ds ν s dzds and B is an independent standard Brownian motion. Assume that b s, σ s, c s,z and their derivatives with respect to and x are Cb.3 as functions of t and x and that σ s is uniformly bounded below by a positive constant. By using the same arguments that in the continuous case and assuming that x c s,z,x >, we have that DX t x X t x X σ,x [,t].

25 Inference and Malliavin calculus 25 Then, writing and where we have Consequently, where β t x X t σ t,x t δ β β t x X t X t atβ ti β ti {ti t<t i }, a L 2 [,T], atdt,i,...,n, t i X ti β,dx ti H. β ti β ti Also we will have that Z n, + u n L u Eδ β X n, t i at x X t σ t,x t db t t i atd t β ti x X t σ t,x t dt. σ s 2 dw s u2 σ s 2 2 σ s 2 ds. Notice that we would obtain another asymptotic regime if σ s would not depend on as we have seen in example 6. This is also the case in the next subsections The pure jump case In the previous calculation, we have used the Brownian motion in order to use the integration by parts property that will lead to an expression of the score. In this section we use the jump structure and, for the sake of simplicity, we assume that there is no Brownian component. Clearly, this also leads to different expressions for the same models if we consider models with Brownian and jump components. σ s

26 26 J.M. Corcuera and A. Kohatsu-Higa Processes with different intensity. Let Y t be an observation of a compound Poisson process of parameter λ and with jumps given by a random variable X, then λ E {Yt A} λ PY t A e λt λt n λ PX X n A n! n e λt λt n t PX X n A n! n ne λt λt n +t PX X n A n! n tpy t A + e λt λt n npx X n A λ n! n te {Yt A} + λ E {Y t A}N t E {Yt A} N t λ t, where we should understand that X +...+X n : if n. Suppose now that we observe Y t,y t2,..,y tn, t.. t n T. Since the increments are independent, the score function is given by E N T λ T Y n t,y t2,...,y tn λ EN ti+ N ti Y ti+ Y ti T. Here i f k Y t λt k EN t Y t k k k!, i fi Y t λti i! where f i stands for the i-th convolution of the density of the random variable X. We remark that this is related to Proposition 3.6 in [3], where the authors use Girsanov s theorem. The trajectories of a compound Poisson process can be described by indicating the time and amplitude of the jumps. To see this, let Ω n R + R n be the sample space and denote an element ω n R + R n as ω s,x,..., s k,x k for certain natural k. In this space we assume that the canonical random variables s j s j expλ and x j fx are independent sequences and also independent

27 of each other. Next, consider the maps Inference and Malliavin calculus 27 Y t : Ω s,x,..., s k,x k Y t ω R k x i [si, ]t the family Y t t is a compound Poisson process. Then, note that λ E {Yt A} e λt λt n t PX X n+ A PX X n A n! n e λt λt n te PX X n + x A PX X n A xxn+ n! n T T EΨ s,x {Yt A} xxn+ ds E Ψ s,x {Yt A}fxdxds, where for any random variable F, we let Ψ s,x Fω Fω s,x Fω, where ω s,x is a trajectory like ω but where we add a jump of amplitude x at time s. We also can write λ E {Yt A} λ E R T R Ψ s,x {Yt A}νdxds, where νdx λfxdx is the Lévy measure of the compound Poisson process Y. If we denote by δ the adjoint operator of Ψ, we deduce that δ N T λt. It can be seen that if u s,x is a predictable random field on [,T] R belonging to the domain of the operator δ then T δu u s,x Mdx,ds, R where M is the compensated random measure associated with the compound Poisson process we assume here that xνdx <, x > see [8] for the Poisson process and [2] for the general case, note that the authors use the operator Ψs,x instead of Ψ x s,x.

28 28 J.M. Corcuera and A. Kohatsu-Higa Processes with different amplitude parameter. Let L i : Y ti Y ti g,xmdx,ds, t i i n, i,...n, with Mdx,ds t i R δ {τk }dsδ {Xk }dx, k where < τ < τ 2 <... < τ n is a sequence of jump times of a Poisson process with intensity one and X i, i,...,n,... is an iid sequence independent of τ i i and with density f, g,x is a deterministic function. Note that the intensity of the Lévy process is assumed to be known and the inference is on the amplitude of the jumps. The compound Poisson process associated with Mdx,ds is t Z t xmdx,ds. R Note that when Z jumps x units, Y t t g,xmdx,ds jumps R g,x but with same frequency as Z. In fact, Y is a compound Poisson process with random Poisson measure M Y dx,ds δ {τk }dsδ {g,xk }dx k In such a situation we have for a regular statistic T TL,...,L n ET E lj T L j, and L j tj t j g,xmdx,ds R g,x k τk t j,t j }, k

29 Inference and Malliavin calculus 29 so ET E lj T g,x k τk t j,t j } k k E lj T xk L j g,x k x g,x k τ k t j,t j }. Since we can write ET k k xk L j x g,x k τk t j,t j } E lj T xk L j g,x k x g,x k τ k t j,t j } E xk T g,x k x g,x k τ k t j,t j } E k T fx k x g,x k fx k x g,x k τk t j,t j }, where in the last inequality we use the independence between X k k and the jump times and where we assume there are not border effects. That is, Finally ET E E k T g,xfx x g,x E T T T R tn R fx k x tj t j fx x t fx x. x suppf g,xfx x g,x g,xfx x g,x g,xfx x g,x xx k τk t j,t j } Mdx,ds Mdx,ds.

30 3 J.M. Corcuera and A. Kohatsu-Higa So, the score function is given by tn E t fx g,xfx x Mdx,ds x g,x Y t,y t2,...,y tn. R If we define the random variables V : g,x the previous expression can be written as tn v vf V v E Mdv,ds f V v Y t,y t2,...,y tn, R t and we can compare it with expression 2.4. References [] O.E. Barndorff-Nielsen, S.E. Graversen, J. Jacod, M. Podolskij, N. Shephard, A central limit theorem for realised power and bipower variations of continuous semimartingales, in: Yu. Kabanov, R. Liptser and J. Stoyanov Eds., From Stochastic Calculus to Mathematical Finance. Festschrift in Honour of A.N. Shiryaev, Heidelberg: Springer, 26, pp [2] B.Z. Bobrovsky, E. Mayer-Wolf and M. Zakai, Some Classes of Global Cramer-Rao Bounds, Annals of Statistics, 5 987, [3] M.H.A. Davis and P. Johansson, Malliavin Monte Carlo Greeks for jump diffusions. Stoch. Proc. Appl. 6 26, -29. [4] V. Debelley and N. Privault, Sensitivity Analysis of European options in jumpdiffusiom models via Malliavin calculus on the Wiener space. Preprint 26. [5] E. Gobet Local asymptotic mixed normality property for elliptic disffusion: a Malliavin calculus approach. Bernoulli 762, [6] Y. A. Kutoyants, Statistical inference for ergodic diffusion processes, Springer, 24. [7] D. Nualart, The Malliavin calculus an related topics. 2nd edition, Springer-Verlag, 26. [8] D. Nualart and J. Vives, A duality formula on the Poisson space and some applications. In R. Dalang, M. Dozzi, and F. Russo, editors, Seminar on Stochastic Analysis, Random Fields and ApplicationsAscona, 993, volume 36 of Progress in Probability, pages Birkhäuser, 995. [9] N. Privault and A. Réveillac Superefficient estimation on the Wiener space. C.R. Acad. Sci. Paris Sér. I Math.,343 26, [] N. Privault and A. Réveillac, Stein estimation for the drift of Gaussian processes using the Malliavin calculus. Preprint 27. To appear in the Annals of Statistics. [] N. Privault and A. Réveillac, Stein estimation of Poisson process intensities. Preprint 27. [2] J.L. Solé, F. Utzet and J. Vives Canonical Lévy Process and Malliavin Calculus. Stochastic Processes and their Applications, 72,27,65 87.

31 Inference and Malliavin calculus 3 José M. Corcuera Universitat de Barcelona Gran Via de les Corts Catalanes. 585 E-87, Barcelona Spain jmcorcuera@ub.edu Arturo Kohatsu-Higa Osaka University Graduate School of Engineering Sciences Machikaneyama cho. -3 Osaka Japan. kohatsu@sigmath.es.osaka-u.ac.jp