Malliavin Calculus and Anticipative Itô Formulae for Lévy Processes

Dept. of Math. University of Oslo Pure Mathematics ISBN 82 553 1379 6 No. 16 ISSN 86 2439 May 23 Malliavin Calculus and Anticipative Itô Formulae for Lévy Processes Giulia Di Nunno 1, Thilo Meyer-Brandis 1, Bernt Øksendal 1,2 and Frank Proske 1 Oslo, 8th October 24. Abstract We introduce the forward integral with respect to a pure jump Lévy process and we prove an Itô formula for this integral. Then we use Mallivin calculus to establish a relationship between the forward integral and the Skorohod integral and we apply this to obtain an Itô formula for the Skorohod integral. Key words and phrases: Lévy processes, Poisson random measures, white noise, Skorohod integrals, forward integrals, stochastic derivatives, Wick product, anticipative Itô formulae. AMS 2) Classification: primary 6H4; secondary 6G51, 6G57, 6H7. 1 Introduction. The original infinite dimensional calculus developed by Malliavin M] was designed to study the smoothness of the densities of the solutions of stochastic differential equations. Although this technique was developed further by many researchers, this application remained the only one known for several years. This situation changed in 1991, when Karatzas and Ocone KO] showed how the representation theorem that Ocone had formulated some years earlier in terms of the Malliavin derivative could be used in finance. Now this theorem is often known as the Clark-Haussmann-Ocone CHO) formula. More precisely, the CHO theorem gives a method of finding replicating portfolios in complete markets driven by Brownian motion. This discovery led to an enormous increase in the interest in the Malliavin calculus both among mathematicians and finance researchers and since then the theory has been generalized and new applications have been found. In particular, Malliavin calculus for Brownian motion has been applied to compute the greeks in finance, see e.g. AØPU], FLLLT] and FLLL]. Moreover, anticipative stochastic caluclus for Brownian motion involving the forward integral beyond the semimartingale context) has been applied to give a general approach to optimal portfolio and consumption problems for insiders in finance, see e.g. BØ1], BØ2] and KS]. An extension of the Malliavin method to processes with discontinuous trajectories was carried out in 1987 by Bichteler, Gravereaux and Jacod BGJ]. However, their work is focused on the original problem of the smoothness of the densities of the solutions of stochastic differential equations, a question that does not deal with the other more recent aspects of the Malliavin calculus. For related works on stochastic calculus for stochastic measures 1 Centre of Mathematics for Applications CMA), Department of Mathematics, University of Oslo, P.O. Box 153 Blindern, N-316 Oslo, Norway. 2 Norwegian School of Economics and Business Administration, Helleveien 3, N-545 Bergen, Norway. E-mail addresses: giulian@math.uio.no; meyerbr@math.uio.no; oksendal@math.uio.no; proske@math.uio.no 1

generated by a Poisson process on the real line see DKW], K1], K2], NV1], Pi1] and Pi2], for example. ecently two types of Malliavin derivative operators D m) t and D t,z have been introduced for Lévy processes and the corresponding CHO representation theorems have been obtained. See L], LSUV], BDLØP], DØP], and ØP]. Although markets driven by Lévy processes are not in general complete, the corresponding CHO theorem is still important for financial applications. For example, it can be used to find explicitly the minimal variance portfolio see e.g. BDLØP]) and to compute the greeks in certain jump diffucsion market models see e.g. DJ]). There has also been an increased interest in anticipative integration with respect to a Lévy process and this is partly due to its application to insider trading in finance see e.g. DMØP], Ø2] and ØS]). In Section 4 of this paper we introduce the forward integral with respect to a pure jump Lévy process and we prove an Itô formula for such integrals see Theorem 4.7). Then we use a relation between forward integrals and Skorohod integrals see Lemma 4.4) to obtain an Itô formula for Skorohod integrals with respect to a pure jump Lévy process see Theorem 4.9). Since Malliavin calculus plays a crucial role in our achievements, we give a review of the main results of this theory in Section 3. Various versions of those results have already been obtained and are known to the public. Nevertheless we think that it is of intrest to have a unified approach based on white noise theory. For completeness and convenience of the reader we recall the basic theory of white noise for pure jump Lévy processes in Section 2. 2 Framework. In this paper we deal with pure jump Lévy processes with no drift defined on a certain probability space Ω, F, P ) and the time horizon + =, ). General information about Lévy processes can be found in Be], P] and Sa], for example. However we recall briefly our framework. Cf. DØP], ØP]. Let Ω = S ) be the Schwartz space of tempered distributions equipped with its Borel σ- algebra F = BΩ). The space S ) is the dual of the Schwartz space S) of test functions, i.e. the rapidly decreasing smooth functions on. We denote the action of ω Ω = S ) applied to f S) by ω, f = ωf). See GV], for example. Thanks to the Bochner-Milnos-Sazonov theorem, the white noise probability measure P can be defined by the relation e i ω,f dp ω) = e ψfx))dx iα fx)dx, f S), where the real constant α and ψu) = Ω e iuz 1 iuz1 { z <1} ) νdz) 2

are the elements of the exponent in the characteristic functional of a pure jump Lévy process with the Lévy measure νdz), z, which, we recall, is such that 2.1) 1 z 2 νdz) <. Assuming that 2.2) M := z 2 νdz) <, we can set α = z1 { z >1}νdz) and then we obtain that E, f ] = and E, f 2] = M fx)dx, Accordingly the pure jump Lévy process with no drift η = ηω, t), ω Ω, t +, f S). that we do consider here and in the sequel, is the cadlag modification of ω, χ,t], ω Ω, t >, where { 1, < x t 2.3) χ,t] x) =, otherwise, x, with ηω, ) :=, ω Ω. We remark that, for all t +, the values ηt) belong to L 2 P ) := L 2 Ω, F, P ). The Lévy process η can be expressed by 2.4) ηt) = zñdt, dz), t +, where Ñdt, dz) := Ndt, dz) νdz)dt is the compensated Poisson random measure associated with η - cf. I1], for example. ecall that the Poisson process is the most important representative among the pure jump Lévy processes and it corresponds to the specific case in which the measure ν is a point mass at 1. Let F t, t +, be the completed filtration generated by the Lévy process in 2.4). We fix F = F. Aiming to treat the Malliavin calculus by means of chaos expansions, we now recall the required spaces and the corresponding complete orthonormal systems. In the space L 2 λ) = L 2 +, B + ), λ) of the square integrable functions on + equipped with the Borel σ-algebra and the standard Lebesgue measure λdt), t +, we consider the complete orthonormal system ξ j j = 1, 2,...) of the Laguerre functions of order 1/2, i.e. Γj) ) 1/2e 2.5) ξ j t) = t t 1/4 L 1/2 j 1 Γj + 1/2) t) 1, )t), t + j = 1, 2,...), where Γ is the Gamma functions and L 1/2 j are the Laguerre polynomials of order 1/2 defined by d j e t t 1/2 L 1/2 j t) = 1 j! dt j e t t j+1/2 ) j =, 1,...). 3

Cf. T], for example. In the space L 2 ν) := L 2, B), ν) of the square integrable functions on equipped with the Borel σ-algebra and the Lévy measure ν, we fix a complete orthonormal system ψ i i = 1, 2,...). In particular we can choose a complete system of polynomials as it was suggested in NS] and DØP], provided that the moments of order grater than or equal to 2 of the measure ν are finite. To simplify the notation we call J the set of multi-indexes α = α 1, α 2,...) which have only finitely many non-zero values. We denote Index α) = max{n : α n } and α = n α n, for α J. By δ k k = 1, 2,...) we identify the product 2.6) δ k t, z) := ξ j t)ψ i z), t +, z, where k = γi, j), and γ : N N N is a bijective map. Note that any bijective map can be applied, e.g. we could consider the so-called Cantor diagonalization of the Cartesian product N N. We set 2.7) ɛ k n) = { 1, n = k, otherwise k = 1, 2,...). Now, for any α J with Index α) = n and α = m we define the tensor product δ α as δ α = δ α 1 1... δn αn t1, x 1 ),..., t m, x m ) ) := δ 1 t 1, x 1 )... δ 1 t α1, x α1 )... δ n t α1 +...+α n 1 +1, x α1 +...+α n 1 +1)... δ n t m, x m ), with δ k := 1. Then we denote δ α b the symmetrization of the functions δ α k. In the space L 2 P ) := L 2 Ω, F, P ) of the square integrable random variables we consider the following complete orthogonal system K α α J ): 2.8) K α := I α δ b α ), α J, where I m f) := m! t2... ft 1, x 1,..., t m, x m )Ñdt 1, dx 1 )...Ñdt m, dx m ) m = 1, 2,...) for the symmetric function f L 2 λ ν) m) m = 1, 2,...) and I f) := f for f. Cf. DØP] and ØP]. In particular K ɛ k = I 1 δ k ). Note that K α 2 L 2 P ) = α! = α 1!α 2!... α J ). The following result is given in DØP], see also ØP]. Theorem 2.1 Chaos expansion I). Every F L 2 P ) admits the unique representation in the form 2.9) F = α J c α K α where c α for all α J, and c = EF. Moreover we have F 2 L 2 P ) = α J c α α! 4

If we consider the symmetric functions 2.1) f m = α c α δ b α: α =m m = 1, 2,...) then we obtain c α K α = α J m= α: α =m c α I m δ b α ) = I m f m ). The expansion here above is actually a result which was first proved in I2]. We can state it as follows. Theorem 2.2 Chaos expansion II). Every F L 2 P ) admits the unique) representation 2.11) F = I m f m ) via the unique sequence of symmetric functions f m L 2 λ ν) m), m =, 1,... m= m= For any formal expansion f = α J c αk α we define the norm f 2,k := α J α!c 2 α2in) kα k =, 1,...) where 2IN) kα = 2 1) kα 1 2 2) kα 2 2 3) kα 3... Now setting S),k := {f : f,k < } we define with the projective topology, and S) := k= S),k S) := k= S),k with the inductive topology. The space S) is the dual of S) and the action of G = α J a αk α S) applied to f = α J b αk α S) is G, f = α J a α b α α! Note that S) L 2 P ) S). We refer to HKPS] and HoØUZ], for example, for the above definitions in the setting of the Gaussian and Poissonian white noise. See also PT], BG], HoØ] and references therein. For the Lévy case, we refer to DØP]. 5

Definition 2.3 The white noise Ñt, x) of the Poisson random measure by the following formal expansion 2.12) Ñt, z) = ξ i t)ψ j z) K ɛ γi,j). i,j=1 Ñdt, dz) is defined It can be proved that the white noise takes values in S), λ ν)-a.e. See ØP]. The justification of the name white noise comes from the fact that, for any B B) such that its closure does not contain, we have I 1 χ,t] 1 B ) = i,j=1 c γi,j) K ɛ γi,j) with c γi,j) = t B ξ is)ψ j z)νdz)ds K ɛ γi,j). Cf. 2.8) and 2.1). Then So formally we have Ñt, B) = t B ) ξ i s)ψ j z)k ɛ γi,j) νdz)ds. i,j=1 Ñdt, dz) Ñt, z) = dtνdz) which is the analog of the adon-nikodym derivative in S). Definition 2.4 The white noise ηt) for the Lévy process is defined by the following formal expansion 2.13) ηt) = ξ i t)k ɛ γi,1), for a specific choice of the basis ψ i, i = 1, 2,..., in 2.6) - cf. DØP]. i=1 The Lévy white noise takes values in S) for all t +. Here the boundedness of the Laguerre functions can be exploited, cf. T]. Note that the Lévy white noise 2.13) is related to the white noise for the Poisson random measure 2.12) by the following formula which involves Bochner integrals with respect to ν: 2.14) ηt) = I zñt, z)νdz). Definition 2.5 The Wick product F G of two elements F = α J a αk α and G = β J b βk β in S) is defined by 2.15) F G = a α b β K α+β. α,β J The spaces S) and S) are topological algebras with respect to the Wick product. 6

3 Some anticipative calculus formulae. In this section we present some known formulae for the Malliavin calculus in the case of pure jump Lévy processes. We will need these results in Section 4. These formulae generalize the known results for the Malliavin calculus in the case of Brownian motion, cf. NP], M], N], Ø], for example. First we recall the Skorohod integration and the Malliavin type stochastic derivative we are dealing with. Let Xt, z), t +, z, be a random field taking values in L 2 P ). Then, for all t + and z, Theorem 2.2 provides the chaos expansion via symmetric functions Xt, z) = I m fm t 1, z 1,..., t m, z m ; t, z) ) m= Let f m = f m t 1, z 1,..., t m+1, z m+1 ) be the symmetrization of f m t 1, z 1,..., t m, z m ; t, z) as a function of the m + 1 variables t 1, z 1 ),..., t m+1, z m+1 ) with t m+1 = t and z m+1 = z. The following concept was first introduced by Y. Kabanov - see Ka1] and Ka2], for example. Definition 3.1 The random field Xt, z), t +, z, is Skorohod integrable if m= m + 1)! f m 2 L 2 λ ν) m+1 ) <. Then its Skorohod integral with respect to Ñ, i.e. IX) := Xt, z)ñδt, dz), is defined by I + 3.1) IX) := I I m+1 f m ). The Skorohod integral is an element of L 2 P ) and 2 3.2) Xt, z)ñδt, dz) = m + 1)! f m 2 L 2 λ ν) m+1 ). I + Moreover, 3.3) E I + m= L 2 P ) m= Xt, z)ñδt, dz) =. The Skorohod integral can be regarded as an extension of the Itô integral to anticipative integrands. In fact, the following result can be proved. Cf. NV2]. See also BL], DØP] and ØP]. Proposition 3.2 Let Xt, z), t +, z, be a non-anticipative adapted) integrand. Then the Skorohod integral and the Itô integral coincide in L 2 P ), i.e. Xt, z)ñδt, dz) = Xt, z)ñdt, dz). I + I I + I 7

Another remarkable property of the Skorohod integral is given in connection to the Wick product. See ØP] for the details and the proof. Before stating the result, we remind that a random field Y taking values in S) is said to be S) -integrable whenever Y, f L 1 λ ν) for all f S). Definition 3.3 For any S) -integrable random field Y t, z) the S) -integral is the unique element in S) such that 3.4) Y t, z)νdz)dt, f = Y t, z), f νdz)dt, f S). I+ I I I + Theorem 3.4 Let Y t, z), t +, z, be Skorohod integrable and b a I EY t, z)2 ]νdz)dt < for some a < b. Then Y Ñ is S) integrable over a, b] I and we obtain the following relationship 3.5) b a I b Y t, z)ñδt, dz) = a Y t, z) Ñt, z)νdz)dt. I Thanks to the relation 2.4), we can easily recognize the Skorohod integral with respect to the very Lévy process ηt), t +, as a particular case of the Skorohod integration with respect to the compensated Poisson random measure Ñ. See DØP]. In fact, for the integrands Xt, z) = z ϕt): Xt, z) = m= I mz ft 1, z 1,..., t m, z m ; t)), we have 3.6) Xt, z)ñδt, dz) = ϕt)δηt). Now we consider the definition of the Malliavin type derivative D t,z for compensated Poisson random measures which was initially given in L]. Other definitions have also been studied by several authors for the same case of pure jump Lévy processes, the particular case of Poisson random processes and for the case of the general Lévy process with no drift, see for instance BC], Pi1], Pi2], NV1], NV2], NS], LSUV], DØP] and BDLØP]. Definition 3.5 The space ID 1,2 is the set of all the elements F L 2 P ) admitting the chaos expansion 2.11): F = EF ] + m=1 I mf m ), such that F 2 ID 1,2 := m m! f m 2 L 2 λ ν) m ) <. m=1 The Malliavin derivative D t,z is an operator defined on ID 1,2 with values in the standard L 2 -space L 2 P λ ν) given by 3.7) D t,z F := mi m 1 f m, t, z)), m=1 where f m, t, z) = f m t 1, z 1,..., t m 1, z m 1 ; t, z). 8

Note that if F ID 1,2, then E I ] D t,z F ) 2 νdz)dt = F 2 ID 1,2. The operator D t,z is proved to be closed and to coincide with a certain difference operator defined in Pi1]. The above operator can be extended to the whole space S) thanks to the chaos expansion 2.9). Definition 3.6 For any F = α J c αk α S) the Malliavin derivative D t,z F is defined as 3.8) D t,z F := α J c α i,j=1 α γi,j) K α ɛ γi,j) ξ i t)ψ j z). It can be proved that D t,z F S), λ ν-a.e., for all F S). Moreover it can also be shown that if F = lim n F n in S), then there exists a subsequence F nk S) such that D t,z F = lim n D t,z F nk in S), λ ν-a.e. See ØP]. We remark that in general the stochastic derivative D t,z, being essentially a difference operator, does not satisfy a chain rule as in the case of the Malliavin derivative for the Brownian motion setting. Cf. NP], N], Ø], for example. Nevertheless a chain rule can still be formulated in terms of the Wick product. Proposition 3.7 Chain rule via Wick product). Let F S) and let gz) = n a nz n be an analytic function in the whole complex plane. Then n a nf n is convergent in S). Furthermore, for g F ) = n a nf n, the following Wick chain rule is valid 3.9) D s,x g F ) = ) d dz g F ) D s,xf. Proof. The first statement can be derived following similar proofs as in Theorem 2.6.12 and Theorem 2.8.1 in HØUZ]. For what concerns the chain rule, it can be easily shown that it holds for polynomials. Then the result follows by the closeness of D t,z and the continuity of the Wick product. Now we turn our attention more deeply to the calculus and we present some basic explicit formulae. First of all we recall the following result proved in BL]. See also DJ] Theorem 2.6. Theorem 3.8 Duality formula). Let Xt, z), t +, z, be Skorohod integrable and F ID 1,2. Then ] ] 3.1) E Xt, z)d t,z F νdz)dt = E F Xt, z)ñδt, dz). I I 9

Corollary 3.9 Closability of Skorohod integral). Suppose that X n t, z), t +, z, is a sequence of Skorohod integrable random fields and that the corresponding sequence of integrals IX n ) := X n t, z)ñδt, dz), n = 1, 2,... converges in L 2 P ). Moreover suppose that I lim X n = in L 2 P λ ν). n Then we have lim IX n) = in L 2 P ). n Proof. By Theorem 3.8) we have that IXn ), F ) L 2 P ) = X n, D t,z F ), n, L 2 P λ ν) for all F D 1,2. Then we conclude that lim n IX n ) = weakly in L 2 P ). And since the sequence IX n ), n = 1, 2,..., is convergent in L 2 P ), then the result follows. In view of Corollary 3.9 we can extend the definition of Skorohod integral as follows. Definition 3.1 Let X n, n = 1, 2,..., be a sequence of Skorohod integrable random fields such that X = lim n X n in L 2 P λ ν). Then we define the Skorohod integral of X as IX) := Xt, z)ñδt, dz) = lim n provided that this limit exists in L 2 P ). I I X n t, z)ñδt, dz) =: lim IX n), n The following result is Lemma 6.1 in NV1] there obtained in a more general setting). Lemma 3.11 Let F, G ID 1,2 with G bounded. Then F G ID 1,2 and we have 3.11) D t,z F G) = F D t,z G + G D t,z F + D t,z F D t,z G λ ν a.e. Proof. With the help of Lemma 9 in L] the result can be verified for F and G of the form gηt 1 ),..., ηt k )), where g is a smooth function with compact support. Then, by using a limit argument the proof follows from the closedness of D t,z. emark 3.12 For an extension of this result to normal martingales, see for example Proposition 1 in Pr2] or Proposition 5 in PSV]. The following result is Theorem 7.1 in NV1] there obtained in a more general setting). 1

Theorem 3.13 Integration by parts). Let Xt, z), t +, z, be a Skorohod integrable stochastic process and F ID 1,2 such that the product Xt, z) F + D t,z F ), t +, z, is Skorohod integrable. Then 3.12) F Xt, z)ñδt, dz) = I I Xt, z) F + D t,z F ) Ñδt, dz) + I Xt, z)d t,z F νdz)dt. Proof. Let G ID 1,2 be bounded. Then we obtain by Theorem 3.8 and Lemma 3.11 E G = E GF I = E G F ] F Xt, z)ñδt, dz) = E I Xt, z)ñδt, dz) ] E E G I I I G ] F Xt, z)d t,z Gνdz)dt I Xt, z)d t,z F Ñδt, dz) ] Xt, z)ñδt, dz) I I Xt, z)d t,z F Ñδt, dz) )]. The proof then follows by a density argument applied to G. ] Xt, z)d t,z F νdz)dt Xt, z)d t,z F νdz)dt emark 3.14 Using the Poisson interpretation of Fock space, the formula 3.12) has been shown to be an expression of the multiplication formula for Poisson stochastic integrals. See Ka2], Su], Proposition 2 and elation 6) of Pr3], Definition 7 and Proposition 6 of PW], Proposition 2 of PSV] and Proposition 1 of Pr1]. Formula 3.12) has been known for some time to quantum probabilitsts in identical or close formulations. See Proposition 21.6 and Proposition 21.8 in Pa], Proposition 18 in B] and elation 5.6) in A]. The following result is Theorem 4.2 in NV1] there obtained in a more general setting). Theorem 3.15 Fundamental theorem of calculus). Let X L 2 P λ ν). Assume that Xs, y) ID 1,2 for all s, y), and D t,z Xs, y), s +, y, for t, z) λ ν-a.e. is Skorohod integrable and that ) ] 2 E D t,z Xs, y)ñδs, dy) νdz)dt <. Then I Xs, y)ñδs, dy) ID 1,2 and ) 3.13) D t,z Xs, y)ñδs, dy) = I I I I D t,z Xs, y)ñδs, dy) + Xt, z). 11

Proof. First suppose that Xs, y) = I n f n, s, y)), where f n t 1, z 1,..., t n, z n, s, y) is symmetric with respect to t 1, z 1 ),..., t n, z n ). By Definition 3.1 we have 3.14) Xs, y)ñδs, dy) = I n+1 f n ) where I f n t 1, z 1,..., t n, z n, t n+1, z n+1 ) = 1 n + 1 f nt n+1, z n+1,, t 1, z 1 ) +... + f n t n+1, z n+1,, t n, z n ) + f n t 1, z 1,, t n+1, z n+1 )] is the symmetrization of f n with respect to the variables t 1, z 1 ),..., t n, z n ), t n+1, z n+1 ) = s, y). Therefore we get ) D t,z Xs, y)ñδs, dy) = I n f n t, z,, t 1, z 1 ) +... + f n t, z,, t n, z n ) + f n, t, z)). I On the other hand we see that 3.15) where = I I D t,z Xs, y)ñδs, dy) ni n 1 f n, t, z, s, y))ñδs, dy) = ni n f n, t, z, )), f n t 1, z 1,..., t n 1, t n 1, t, z, t n, z n ) = 1 n f nt, z,, t 1, z 1 ) +... + f n t, z,, t n, z n )] is the symmetrization of f n t 1, z n,..., t n 1, z n 1, t, z, t n, z n ) with respect to t 1, z 1 ),..., t n 1, z n 1 ), t n, z n ) = s, y). A comparison of 3.14) and 3.15) yields formula 3.13). Next consider the general case Xs, y) = n I n f n, s, y)). Define m X m s, y) = I n f n, s, y)), m = 1, 2,... n= Then 3.13) holds for X m. Since D t,z X m s, y)ñδs, dy) I = n m+1 n 2 2 n! f n I L 2 λ ν) n+1 ) D t,z Xs, y)ñδs, dy) 2, m L 2 P λ ν) the proof follows by the closedness of D t,z. The following result is Theorem 4.1 in NV1] there obtatined in a more general setting). 12

Theorem 3.16 The Itô-Lévy-Skorohod isometry). Let X L 2 P λ ν) and DX L 2 P λ ν) 2 ). Then the following isometry holds ) ] 2 3.16) E Xt, z)ñδt, dz) = E I ] X 2 t, z)νdz)dt + E I I I ] D t,z Xs, y)d s,y Xt, z)νdy)dsνdz)dt. Proof. Consider Xt, z) = α J c α t, z)k α. Define and S 3 = S 1 = α J α! c α 2 L 2 λ ν), S 2 = α,β J,i,j,k,l IN i,j) k,l) α J,i,j IN α γi,j) α!c α, ξ j ψ i ) 2 α γi,j) + 1)α!c α, ξ j ψ i )c β, ξ k ψ l ) 1 {α+ɛ γi,j) =ɛ γk,l) }, where, ) =, ) L 2 λ ν). Note that by the assumption and Lemma 3.12 in ØP] the sums above are convergent. First it follows that ) ] 2 ) 2 E Xt, z)ñδt, dz) = E Xt, z) Ñt, z)νdz)dt = E = E = I α J,i,j I α,β J,i,j,k,l IN i,j) k,l) ) 2 c α t, z)k α ξ j t)ψ i z)k ɛ γi,j) νdz)dt α J i,j 2 c α, ξ j ψ i )K α+ɛ γi,j) since α + ɛ γi,j) )! = α γi,j) + 1)α!. Next, we have E I α + ɛ γi,j) )!c α, ξ j ψ i )c β, ξ k ψ l ) 1 {α+ɛ γi,j) =ɛ γk,l) } = S 1 + S 2 + S 3, ] X 2 t, z)νdz)dt = α I = E I I I ) 2 c α t, z)k α νdz)dt α J c 2 αt, z)α!νdz)dt = S 1. 13

Finally, we get E = E I I I ] D t,z Xs, y)d s,y Xt, z)νdy)dsνdz)dt I α,k,l c α s, y)ξ k t)ψ l z)α ɛ γk,l)k α ɛ γk,l)) c α t, z)ξ k s)ψ l y)α ɛ γi,j)k β ɛ γi,j))νdy)dsνdz)dt β,i,j = α,β J,i,j,k,l IN c α, ξ j ψ i )c β, ξ k ψ l )β γi,j) α! 1 {α+ɛ γi,j) =ɛ γk,l) } = S 2 + S 3. Combining the three steps of the proof the desired result follows. emark 3.17 Formula 3.16) can also be obtained as a consequence of the Poisson interpretation of Fock space. See Proposition 17 in B] and Proposition 1 in PW]. For an isometry of this type which is not based on Fock space, see Proposition 3.3 in Pr4]. 4 Forward integrals and generalized Itô formulae. In this section we introduce the forward integral with respect to the Poisson random measure Ñ. Then we prove an Itô formula for the corresponding forward processes and we apply this to obtain an Itô formula for processes driven by Skorohod integrals. Here we can refer to NP], V] and HuØ], for example, where these topics are developed for the Brownian motion. Definition 4.1 The forward integral Jθ) := T θt, z)ñd t, dz) with respect to the Poisson random measure Ñ, of a caglad stochastic function θt, z), t +, z, with θt, z) := θω, t, z), ω Ω, is defined as T Jθ) = lim θt, z)1 Um Ñdt, dz) m if the limit exists in L 2 P ). Here U m, m = 1, 2,..., is an increasing sequence of compact sets U m \ {} with νu m ) < such that lim m U m = \ {}. emark 4.2 Note that if G is a random variable then T T 4.1) G θt, z)ñd t, dz) = G θt, z)ñd t, dz), a property that does not hold for the Skorohod integrals. 14

Definition 4.3 In the sequel we let M denote the set of the stochastic functions θt, z), t +, z, such that i) θω, t, z) = θ 1 ω, t)θ 2 ω, t, z) where θ 1 ω, t) D 1,2 is caglag and θ 2 ω, t, z) is adapted and such that T ] E θ2t, 2 z)νdz)dt <, ii) D t +,zξ = lim s t + D s,z ξ exists in L 2 P λ ν), iii) θt, z) + D t +,zθt, z) is Skorohod integrable. We let M 1,2 be the closure of the linear span of M with respect to the norm given by θ 2 M 1,2 := θ 2 L 2 P λ ν) + D t +,zθt, z) 2 L 2 P λ ν). We can now show the relation between the forward integral and the Skorohod integral. Lemma 4.4 If θ M 1,2 then its forward integral exists and T T T θt, z)ñd t, dz) = D t +,zθt, z)νdz)dt+ θt, z)+dt +,zθt, z) ) Ñδt, dz). Proof. First consider the case when θω, t, z) = θ 1 ω, t)θ 2 ω, t, z). Let us take a sequence of partitions of, T ] of the form = t n < tn 1 <... < tn J n = T with t := maxt n j tn j 1 ), for n, into account. By Theorem 3.13 we have t n i t n F θt, z)ñδt, dz) = i F θt, z)ñδt, dz) Hence = T T t n i + t n i 1 t n i 1 θt, z)d t,z F νdz)dt + θt, z)ñd t, dz) = = lim t = lim t J n t n i i=1 + lim t J n i=1 t n i 1 J n t n i i=1 T θt, z)ñδt, dz) + lim t J n i=1 t n i t n θ 1 t n i i 1) t n i 1 t n i 1 t n i 1 θt, z)d t,z F Ñδt, dz). t n θ 1 t n i i 1) θ 2 t, z)ñdt, dz) t n i 1 θ 2 t, z)ñδt, dz) θ1 t n i 1) + D t,z θ 1 t n i 1) ] θ 2 t, z)ñδt, dz) t n i 1 D t,z θ 1 t n i 1) θ 2 t, z)νdz)dt D t +,zθt, z)νdz)dt + T D t +,zθt, z)ñδt, dz). The proof is then completed by a limit argument in view of Definition 3.1 and Definition 4.3. 15

Corollary 4.5 If the forward integral exists in L 2 P ) then T T 4.2) E θt, z)ñd t, dz) = E D t +,zθt, z)νdz)dt. Proof. This follows from 3.3) and Lemma 4.4. Definition 4.6 A forward process is a measurable stochastic function Xt) = Xt, ω), t +, ω Ω, that admits the representation 4.3) Xt) = x + t θs, z)ñd s, dz) + t αs)ds, where x = X) is a constant. A shorthand notation for 4.3) is 4.4) d Xt) = θt, z)ñd t, dz) + αt)dt; X) = x. We call d Xt) the forward differential of Xt), t +. Theorem 4.7 Itô formula for forward integrals). Let Xt), t +, be a forward process of the form 4.3) where θt, z), t +, z, is locally bounded in z near z = P λ- a.e. and such that T θt, z) 2 νdz)dt < P a.s. Suppose also that θt, z), t +, z, is forward integrable. For any function f C 2 ), the forward differential of Y t) = f Xt) ), t +, is given by the following formula: 4.5) d Y t) = f Xt) ) αt)dt + f Xt ) + θt, z) ) f Xt ) ) f Xt ) ) ) θt, z) νdz)dt + f Xt ) + θt, z) ) f Xt ) )) Ñd t, dz). Proof. The proof follows the same line of the one in the classical Itô formula see IW] Chapter 2, Section 5). For simplicity we assume x = and α. We can write X m t) := t θs, z)1 Um z)nds, dz) t θs, z)1 Um z)νdz)ds. We denote by = σ < σ 1 <... the stopping times for which the jumps of the Lévy process occur. Thus we obtain fx m t)) fx m )) = fxm σ i t)) fx m σ i t )) ] i 16

+ i fxm σ i t )) fx m σ i 1 t)) ] =: J 1 t) + J 2 t), with fx m σ i t )) = { fxm σi )), σ i t, fx m t)), σ i > t. By the change of variable formula for finite variation processes, it follows that t J 2 t) = f X m s))θs, z)1 Um z)νdz)ds. Moreover it is J 1 t) = i fxm )σ i ) fx m )σ i )] 1 {σi t,θσ i,ησ i )) } = = + t t t By letting m, formula 4.6) follows. f Xm s ) + θs, z)1 Um z) ) fx m )s ) ] Nds, dz) f Xm s ) + θs, z)1 Um z) ) fx m )s ) ] Ñd s, dz) f Xm,n s ) + θs, z)1 Um z) ) fx m )s ) ] νdz)ds. In order to state an Itô formula for Skorohod integrals we need to combine Lemma 4.4 and Theorem 4.7. To this end we go into the technical step of solving equations of the following type: given a random variable G find the stochastic function F t, z), t +, z, such that 4.6) F t, z) + D t +,zf t, z) = G, for almost all t, z) +. For example, if G = g ηt ) ), for some measurable function g : and t ηt) = zñdt, dz), t, T ], then F t, z) := g ηt ) zχ,t ) t) ) does the job. In fact, with this choice of F t, z), t +, z, we have F t, z) + D t +,zf t, z) = g ηt ) zχ,t ) ) + g ηt ) ) g ηt ) zχ,t ) ) = G. The above observation motivates the following definition. Definition 4.8 The linear operator S is defined on the space of all F T -measurable random variables G as follows. If G = k i=1 g i ηti ) ), for some t i, T ], i = 1,..., k, we define k 4.7) S t,z g i ηti ) )) = i=1 k g i ηti ) zχ,ti )t) ). i=1 17

Note that via this definition the solution of equation 4.6) can be written as F t, z) = S t,z G, i.e. 4.8) S t,z G + D t +,z St,z G ) = G. Combining the above facts with Lemma 4.4 and Theorem 4.7, we obtain the following result. Theorem 4.9 Itô formula for Skorohod integrals). Let Xt) = t or, in shorthand notation, δxt) = Let f C 2 ) and let Y t) = f Xt) ). Set t γs, z)ñδs, dz) + αs)ds, t, T ], γt, z)ñδt, dz) + αt)dt, t, T ]. 4.9) θt, z) := S t,z γt, z) for all t, T ], z, and assume θ M 1,2. Then 4.1) δy t) = f Xt))αt)dt + {f Xt ) + θt, z) ) f Xt ) ) +D t +,z + +D t +,z f Xt ) + θt, z) ) f Xt ) )]} Ñδt, dz) { f Xt ) + θt, z) ) f Xt ) ) f Xt ) ) θt, z) f Xt ) + θt, z) ) f Xt ) )] f Xt ) ) } D t +,zθt, z) νdz)dt. emark 4.1 Note that if γ and α are adapted, then θt, z) = γt, z), t +, z, and D t +,zθt, z) = D t +,z f Xt ) + θt, z) ) f Xt ) )]. Therefore Theorem 4.9 reduces to the classical adapted Itô formula. Proof. For simplicity we assume α. By 4.8) we have Hence by Lemma 4.4 we have Xt) = t θt, z) + D t +,zθt, z) = γt, z). θs, z)ñd s, dz) t D s +,zθs, z)νdz)ds. 18

We can therefore apply Theorem 4.7 and get + + t t + + t t Y t) Y ) = t f Xs) ) ) D s +,zθs, z)νdz) ds { f Xs ) + θs, z) ) f Xs ) ) f Xs ) ) } θs, z) νdz)ds + t { f Xs ) + θs, z) ) f Xs ) )} Ñd s, dz) t f Xs ) ) D s +,zθs, z)νdz)ds { f Xs ) + θs, z) ) f Xs ) ) f Xs ) ) } θs, z) νdz)ds + t D s +,z{ f Xs ) + θs, z) ) f Xs ) )} νdz)dt { f Xs )+θs, z) ) f Xs ) ) +D s +,z{ f Xs )+θs, z) ) f Xs ) )}} Ñδs, dz) = t { f Xs ) + θs, z) ) f Xs ) ) f Xs ) ) θs, z) +D s,z + f Xs ) + θs, z) ) f Xs ) )] f Xs ) ) } D s +,zθs, z) νdz)ds { f Xs )+θs, z) ) f Xs ) ) +D s +,z f Xs )+θs, z) ) f Xs ) )]} Ñδs, dz). This completes the proof. emark 4.11 In Pr1] a different anticipative Itô formula is obtained, valid for polynomials f. emark 4.12 The Itô formula can be extended to cover the mixed case, involving a combination of Gaussian and compensated Poisson random measures. eferences A] Applebaum, D.: Covariant Poisson fields in Fock space. In Analysis, Geometry and Probability. Hindustan Book Agency, Delhi 1996, pp. 1 15. AØPU] Aase, K., Øksendal, B., Privault, N., Ubøe, J.: White noise generalizations of the Clark- Haussmann-Ocone theorem with application to mathematical finance. Finance and Stochastics 4 2), pp. 465 496. B] Biane, P.:Calcul Stochastic non-commutatif. Lectures on Probability Theory Saint-Flour, 1993). Lect. Notes in Math. 168. Springer 1995, pp. 1 96. 19

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