Malliavin Calculus and Anticipative Itô Formulae for Lévy Processes
|
|
- Coral Bruce
- 5 years ago
- Views:
Transcription
1 Dept. of Math. University of Oslo Pure Mathematics ISBN No. 16 ISSN 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. addresses: giulian@math.uio.no; meyerbr@math.uio.no; oksendal@math.uio.no; proske@math.uio.no 1
2 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
3 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
4 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 δn αn t1, x 1 ),..., t m, x m ) ) := δ 1 t 1, x 1 )... δ 1 t α1, x α1 )... δ n t α α n 1 +1, x α 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
5 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
6 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
7 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
8 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
9 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 and Theorem 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
10 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
11 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
12 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
13 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 ) ] ) 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
14 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
15 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
16 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
17 + 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
18 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
19 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 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 B] Biane, P.:Calcul Stochastic non-commutatif. Lectures on Probability Theory Saint-Flour, 1993). Lect. Notes in Math Springer 1995, pp
20 BC] Bass,.F., Cranston, M.: The Malliavin calculus for pure jump processes and applications to local time. Annals. Prob ), pp BG] Benth, F.E., Gjerde, J.: A remark on the equivalence between Poisson and Gaussian stochastic partial differential equations. Pot. Anal ), pp BL] Benth, F.E., Løkka, A.: Anticipative calculus for Lévy processes and stochastic differential equations. Preprint Series in Pure Mathematics, University of Oslo, 6, 22. BDLØP] Benth, F.E., di Nunno, G., Løkka, A., Øksendal, B., Proske, F.: Explicit representation of the minimal variance portfolio in markets driven by Lévy processes. Math. Finance 13 23), pp Be] Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge BØ1] Biagini, F., Øksendal, B.: A general stochastic caluculus approach to insider trading. Preprint Series in Pure Mathematics, University of Oslo, 17, 22. BØ2] Biagini, F., Øksendal, B.: Minimal variance hedging for insider trading. Preprint Series in Pure Mathematics, University of Oslo, 9, 24. BGJ] Bichteler, K., Gravereaux, J.B., Jacod, J.: Malliavin Calculus for Processes with Jumps. Gordon and Breach Science Publisher, New York, DJ] Davis, M.H.A., Johansson, M.P.: Malliavin Monte Carlo greeks for jump diffusions. Preprint Imperial College, June 24. DKW] Dermoune, A., Kree, P., Wu, L.: Calcul stochastique non adapté par rapport à la mesure aléatoire de Poisson. Séminaire de Probabilités XXII, Lect. Notes. Math 1321, pp , Springer, Berlin, DMØP] di Nunno, G., Meyer-Brandis, T., Øksendal, B., Proske, F.: Optimal portfolio for an insider in a market driven by Lévy processes. Preprint Series in Pure Mathematics, University of Oslo, 36, 23. DØP] di Nunno, G., Øksendal, B., Proske, F.: White noise analysis for Lévy proceses. Journal of Functional Analysis 26 24), pp FLLLT] Fournié, E., Larry, J.M., Lebuchoux, J., Lions, P.L., Touzi, N.: Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stoch ), pp FLLL] Fournié, E., Larry, J.M., Lebuchoux, J., Lions, P.L.: Applications of Malliavin calculus to Monte Carlo methods in finance II. Finance Stoch. 5 21), pp GV] Gelfand, I. M., Vilenkin, N. Y.: Generalized Functions, Vol. 4: Applications of Harmonic Analysis. Academic Press English translation), New York-London HKPS] Hida, T., Kuo, H.-H, Potthoff, J., Streit, L.: White Noise. Kluwer, Dordrecht, HuØ] Hu, Y., Øksendal, B.: Optimal smooth portfolio selection for an insider. Preprint Series in Pure Mathematics, University of Oslo, 12, 23. HoØ] Holden, H., Øksendal, B.: A white noise approach to stochastic differential equations driven by Wiener and Poisson processes. In M. Grosser et al. editors): Nonlinear Theory of Generalized Functions, Chapman & Hall/CC 1999, pp
21 HoØUZ] Holden, H., Øksendal, B., Ubøe, J., Zhang T.-S.: Stochastic Partial Differential Equations:A Modeling, White Noise Functional Approach. Birkhäuser, Boston IW] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North- Holland/Kondansha LTD., Tokyo I1] Itô, K.: On stochastic processes I. Infinitely divisible laws of probability. Jap. J. Math ), pp I2] Itô, K.: Spectral type of the shift transformation of differential processes with stationary increments, Trans. Am. Math. Soc ), pp Ka1] Kabanov, Y.: A generalized Itô formula for an extended stochastic integral with respect to a Poisson random measure In ussian), Usp. Mat. Nauk ), No ), pp Ka2] Kabanov, Y.: On extended stochastic integrals. Theory of Probability and its Applications ), pp K1] Kaminsky, A.D.: Extended stochastic clalculus for the Poisson random measures. Nats. Akad. Nauk Ukrain, Inst. Mat. Preprint, ). K2] Kaminsky, A.D.: A white noise approach to stochastic integration for a Poisson random measure. Theory Probab. Math. Statist ), pp KO] Karatzas, I., Ocone, D.: A generalized Clark representation formula, with application to optimal portfolios. Stochastics and Stochastic eports ), pp KS] Kohatsu-Higa, A., Sulem, A.: Utility maximization in an insider influenced market. Preprint INIA, September 24. LSUV] Léon, J.A., Solé, J.L., Utzet, F., Vives, J.: On Lévy processes, Malliavin calculus and market models with jumps. Prepublications 21, U.A.B., Barcelona. To appear in Finance and Stochastics. L] Løkka, A.: Martingale representation and functionals of Lévy processes. Preprint Series in Pure Mathematics, University of Oslo, 21, 21. M] Malliavin, P.: Stochastic Analysis. Springer-Verlag, New York N] Nualart, D.: The Malliavin Calculus and elated Topics. Springer, Berlin Heidelberg New York NP] Nualart, D., Pardoux E.: Stochastic calculus with anticipating integrands. Probab. Th. el. Fields ), pp NS] Nualart, D., Schoutens, W.: Chaotic and predictable representations for Lévy processes. Stochastic Process. Appl. 9 2), pp NV1] Nualart, D., Vives, J.: Anticipative calculus for the Poisson process based on the Fock space, Séminaire de Probabilités XXIV, LNM 1426, 199. NV2] Nualart, D., Vives, J.: A duality formula on the Poisson space and some applications. Seminar on Stochastic Analysis, andom Fields and Applications Ascona, 1993), pp , Progr. Probab., 36, Birkhuser, Basel,
22 Ø] Øksendal, B.: An introduction to Malliavin calculus with applications to economics. Working paper, No 3/96, Norwegian School of Economics and Business Administration, Ø2] Øksendal, B.: A universal optimal consumption rate for an insider. Preprint Series in Pure Mathematics, University of Oslo, 27, 24. ØP] Øksendal, B., Proske, F.: White noise for Poisson random measures. Preprint Series in Pure Mathematics, University of Oslo, 12, 22. ØS] Øksendal, B., Sulem, A.: Partial observation control in an anticipating environment. Preprint Series in Pure Mathematics, University of Oslo, 31, 23. Pi1] Picard, J.: On the existence of smooth densities for jump processes. Prob. Th. el. Fields ), pp Pi2] Picard, J.: Formules de dualité sur l espace de Poisson. Ann. IHP, Série Proba. et Stat ), pp Pa] Parthasarathy, K..: An Introduction to Quantum Stochastic Calculus. Birkäuser Pr1] Privault, N.: An extension of stochastic calculus to certain non-markovian processes. Manuscript Pr2] Privault, N.: Independence of a class of multiple stochastic integrals. In Seminar on Stochastic Analysis, andom Fields and Applications Ascona 1996). Birkäuser 1999, pp Pr3] Privault, N.: Equivalence of gradients on configuration spaces. andom Oper. Stoch. Equations ), pp Pr4] Privault, N.: Connections and curvature in the iemannian geometry of configuration spaces. Journal of Functional Analysis ), pp PSV] Privault, N., Solé, J.L. and Vives, J.: Chaotic Kabanov formula for the Azma martingales. Bernoulli 6 2), pp PW] Privault, N. and Wu, J.: Poisson stochastic integration in Hilbert spaces. Ann. Math. Blaise Pascal ), pp PT] Potthoff, J., Timpel, M.: On a dual pair of smooth and generalized random variables. Potential Analysis ), pp P] Protter, P.: Stochastic Integration and Differential Equations. Springer-Verlag, Berlin 199. V] usso, F., Vallois, P.: Forward, backward and symmetric stochastic integration. Prob. Th. el. Fields ), pp Sa] Sato, K.: Lévy Processes and Infinitely Divisible Distributions, Cambridge University Studies in Advanced Mathematics, Vol. 68, Cambridge University Press, Cambridge Su] Surgailis, D.:On multiple Poisson stochastic integrals and associated Markov semigroups. Probab. Math. Statist ), pp T] Thangavelu, S.: Lectures on Hermite and Laguerre Expansions. Mathematical Notes 42, Princeton University Press
Random Fields: Skorohod integral and Malliavin derivative
Dept. of Math. University of Oslo Pure Mathematics No. 36 ISSN 0806 2439 November 2004 Random Fields: Skorohod integral and Malliavin derivative Giulia Di Nunno 1 Oslo, 15th November 2004. Abstract We
More informationIntroduction to Malliavin calculus and Applications to Finance
Introduction to Malliavin calculus and Applications to Finance Part II Giulia Di Nunno Finance and Insurance, Stochastic Analysis and Practical Methods Spring School Marie Curie ITN - Jena 29 PART II 1.
More informationWhite noise generalization of the Clark-Ocone formula under change of measure
White noise generalization of the Clark-Ocone formula under change of measure Yeliz Yolcu Okur Supervisor: Prof. Bernt Øksendal Co-Advisor: Ass. Prof. Giulia Di Nunno Centre of Mathematics for Applications
More informationChaos expansions and Malliavin calculus for Lévy processes
Chaos expansions and Malliavin calculus for Lévy processes Josep Lluís Solé, Frederic Utzet, and Josep Vives Departament de Matemàtiques, Facultat de Ciències, Universitat Autónoma de Barcelona, 8193 Bellaterra
More informationAnticipative stochastic control for Lévy processes with application to insider trading
Anticipative stochastic control for Lévy processes with application to insider trading Giulia Di Nunno 1, Arturo Kohatsu-Higa 2, Thilo Meyer-Brandis 1, Bernt Øksendal 1,3, Frank Proske 1 and Agnès Sulem
More informationSTOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY MULTI-PARAMETER WHITE NOISE OF LÉVY PROCESSES
Dept. of Math./CMA Univ. of Oslo Pure Mathematics No 7 ISSN 86 2439 June 27 STOCHASTIC PATIAL DIFFEENTIAL EQUATIONS DIVEN BY MULTI-PAAMETE WHITE NOISE OF LÉVY POCESSES BENT ØKSENDAL Abstract. We give a
More informationIndependence of some multiple Poisson stochastic integrals with variable-sign kernels
Independence of some multiple Poisson stochastic integrals with variable-sign kernels Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological
More informationPoisson stochastic integration in Hilbert spaces
Poisson stochastic integration in Hilbert spaces Nicolas Privault Jiang-Lun Wu Département de Mathématiques Institut für Mathematik Université de La Rochelle Ruhr-Universität Bochum 7042 La Rochelle D-44780
More informationThe Wiener Itô Chaos Expansion
1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in
More informationAn Introduction to Malliavin calculus and its applications
An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart
More informationOn Reflecting Brownian Motion with Drift
Proc. Symp. Stoch. Syst. Osaka, 25), ISCIE Kyoto, 26, 1-5) On Reflecting Brownian Motion with Drift Goran Peskir This version: 12 June 26 First version: 1 September 25 Research Report No. 3, 25, Probability
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationCOVARIANCE IDENTITIES AND MIXING OF RANDOM TRANSFORMATIONS ON THE WIENER SPACE
Communications on Stochastic Analysis Vol. 4, No. 3 (21) 299-39 Serials Publications www.serialspublications.com COVARIANCE IDENTITIES AND MIXING OF RANDOM TRANSFORMATIONS ON THE WIENER SPACE NICOLAS PRIVAULT
More informationA NOTE ON STOCHASTIC INTEGRALS AS L 2 -CURVES
A NOTE ON STOCHASTIC INTEGRALS AS L 2 -CURVES STEFAN TAPPE Abstract. In a work of van Gaans (25a) stochastic integrals are regarded as L 2 -curves. In Filipović and Tappe (28) we have shown the connection
More informationMan Kyu Im*, Un Cig Ji **, and Jae Hee Kim ***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No. 4, December 26 GIRSANOV THEOREM FOR GAUSSIAN PROCESS WITH INDEPENDENT INCREMENTS Man Kyu Im*, Un Cig Ji **, and Jae Hee Kim *** Abstract.
More informationRegularity of the density for the stochastic heat equation
Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department
More informationfor all f satisfying E[ f(x) ] <.
. Let (Ω, F, P ) be a probability space and D be a sub-σ-algebra of F. An (H, H)-valued random variable X is independent of D if and only if P ({X Γ} D) = P {X Γ}P (D) for all Γ H and D D. Prove that if
More informationCLARK-OCONE FORMULA BY THE S-TRANSFORM ON THE POISSON WHITE NOISE SPACE
Communications on Stochastic Analysis Vol. 6, No. 4 (2012 513-524 Serials Publications www.serialspublications.com CLARK-OCONE FORMULA BY THE S-TRANSFORM ON THE POISSON WHITE NOISE SPACE YUH-JIA LEE*,
More informationFiltrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition
Filtrations, Markov Processes and Martingales Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition David pplebaum Probability and Statistics Department,
More informationA note on the σ-algebra of cylinder sets and all that
A note on the σ-algebra of cylinder sets and all that José Luis Silva CCM, Univ. da Madeira, P-9000 Funchal Madeira BiBoS, Univ. of Bielefeld, Germany (luis@dragoeiro.uma.pt) September 1999 Abstract In
More informationA Concise Course on Stochastic Partial Differential Equations
A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original
More informationNormal approximation of Poisson functionals in Kolmogorov distance
Normal approximation of Poisson functionals in Kolmogorov distance Matthias Schulte Abstract Peccati, Solè, Taqqu, and Utzet recently combined Stein s method and Malliavin calculus to obtain a bound for
More informationDOOB S DECOMPOSITION THEOREM FOR NEAR-SUBMARTINGALES
Communications on Stochastic Analysis Vol. 9, No. 4 (215) 467-476 Serials Publications www.serialspublications.com DOOB S DECOMPOSITION THEOREM FOR NEAR-SUBMARTINGALES HUI-HSIUNG KUO AND KIMIAKI SAITÔ*
More informationLAN property for ergodic jump-diffusion processes with discrete observations
LAN property for ergodic jump-diffusion processes with discrete observations Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Arturo Kohatsu-Higa (Ritsumeikan University, Japan) &
More informationMartingale Representation, Chaos Expansion and Clark-Ocone Formulas
Martingale Representation, Chaos Expansion and Clark-Ocone Formulas ISSN 1398-599X MPS-RR 1999-22 www.maphysto.dk Arne Løkka Department of Mathematics University of Oslo P.O. Box 153 Blindern N-316 Oslo
More informationBilinear Stochastic Elliptic Equations
Bilinear Stochastic Elliptic Equations S. V. Lototsky and B. L. Rozovskii Contents 1. Introduction (209. 2. Weighted Chaos Spaces (210. 3. Abstract Elliptic Equations (212. 4. Elliptic SPDEs of the Full
More informationBOOK REVIEW. Review by Denis Bell. University of North Florida
BOOK REVIEW By Paul Malliavin, Stochastic Analysis. Springer, New York, 1997, 370 pages, $125.00. Review by Denis Bell University of North Florida This book is an exposition of some important topics in
More informationThe multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications
The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications
More informationTHE LENT PARTICLE FORMULA
THE LENT PARTICLE FORMULA Nicolas BOULEAU, Laurent DENIS, Paris. Workshop on Stochastic Analysis and Finance, Hong-Kong, June-July 2009 This is part of a joint work with Laurent Denis, concerning the approach
More informationLévy Processes and Infinitely Divisible Measures in the Dual of afebruary Nuclear2017 Space 1 / 32
Lévy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space David Applebaum School of Mathematics and Statistics, University of Sheffield, UK Talk at "Workshop on Infinite Dimensional
More informationMartingale representation for Poisson processes with applications to minimal variance hedging
Martingale representation for Poisson processes with applications to minimal variance hedging Günter Last and Mathew D. Penrose January 14, 2010 Abstract We consider a Poisson process η on a measurable
More informationDiscrete approximation of stochastic integrals with respect to fractional Brownian motion of Hurst index H > 1 2
Discrete approximation of stochastic integrals with respect to fractional Brownian motion of urst index > 1 2 Francesca Biagini 1), Massimo Campanino 2), Serena Fuschini 2) 11th March 28 1) 2) Department
More informationSupermodular ordering of Poisson arrays
Supermodular ordering of Poisson arrays Bünyamin Kızıldemir Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 637371 Singapore
More informationIntroduction to Infinite Dimensional Stochastic Analysis
Introduction to Infinite Dimensional Stochastic Analysis By Zhi yuan Huang Department of Mathematics, Huazhong University of Science and Technology, Wuhan P. R. China and Jia an Yan Institute of Applied
More informationDIFFERENTIAL EQUATIONS DRIVEN BY LÉVY WHITE NOISE IN SPACES OF HILBERT SPACE VALUED STOCHASTIC DISTRIBUTIONS
Dept. of Math. Univ. of Oslo Pure Mathematics No. 9 ISSN 86 2439 March 25 DIFFEENTIAL EQUATIONS DIVEN BY LÉVY WHITE NOISE IN SPACES OF HILBET SPACE VALUED STOCHASTIC DISTIBUTIONS THILO MEYE-BANDIS Abstract.
More informationA connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand
A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand Carl Mueller 1 and Zhixin Wu Abstract We give a new representation of fractional
More informationWHITE NOISE APPROACH TO THE ITÔ FORMULA FOR THE STOCHASTIC HEAT EQUATION
Communications on Stochastic Analysis Vol. 1, No. 2 (27) 311-32 WHITE NOISE APPROACH TO THE ITÔ FORMULA FOR THE STOCHASTIC HEAT EQUATION ALBERTO LANCONELLI Abstract. We derive an Itô s-type formula for
More informationCLARK-OCONE FORMULA BY THE S-TRANSFORM ON THE POISSON WHITE NOISE SPACE
CLARK-OCONE FORMULA BY THE S-TRANSFORM ON THE POISSON WHITE NOISE SPACE YUH-JIA LEE*, NICOLAS PRIVAULT, AND HSIN-HUNG SHIH* Abstract. Given ϕ a square-integrable Poisson white noise functionals we show
More informationLOCAL TIMES OF RANKED CONTINUOUS SEMIMARTINGALES
LOCAL TIMES OF RANKED CONTINUOUS SEMIMARTINGALES ADRIAN D. BANNER INTECH One Palmer Square Princeton, NJ 8542, USA adrian@enhanced.com RAOUF GHOMRASNI Fakultät II, Institut für Mathematik Sekr. MA 7-5,
More informationTHE MALLIAVIN CALCULUS FOR SDE WITH JUMPS AND THE PARTIALLY HYPOELLIPTIC PROBLEM
Takeuchi, A. Osaka J. Math. 39, 53 559 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
More informationMalliavin differentiability of solutions of SPDEs with Lévy white noise
Malliavin differentiability of solutions of SPDEs with Lévy white noise aluca M. Balan Cheikh B. Ndongo February 2, 217 Abstract In this article, we consider a stochastic partial differential equation
More informationSTOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY MULTI-PARAMETER WHITE NOISE OF LÉVY PROCESSES
QUATELY OF APPLIED MATHEMATICS VOLUME LXVI, NUMBE 3 SEPTEMBE 28, PAGES 521 537 S 33-569X(8)19-5 Article electronically published on July 3, 28 STOCHASTIC PATIAL DIFFEENTIAL EQUATIONS DIVEN BY MULTI-PAAMETE
More informationOn pathwise stochastic integration
On pathwise stochastic integration Rafa l Marcin Lochowski Afican Institute for Mathematical Sciences, Warsaw School of Economics UWC seminar Rafa l Marcin Lochowski (AIMS, WSE) On pathwise stochastic
More informationOptimal stopping of stochastic differential equations with delay driven by Lévy noise
Dept. of Math./CMA University of Oslo Pure Mathematics No 18 ISSN 86 2439 August 29 Optimal stopping of stochastic differential equations with delay driven by Lévy noise Salvatore Federico 1 Bernt Øksendal
More informationOn chaos representation and orthogonal polynomials for the doubly stochastic Poisson process
dept. of math./cma univ. of oslo pure mathematics No 1 ISSN 86 2439 March 212 On chaos representation and orthogonal polynomials for the doubly stochastic Poisson process Giulia Di Nunno and Steffen Sjursen
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationA Class of Fractional Stochastic Differential Equations
Vietnam Journal of Mathematics 36:38) 71 79 Vietnam Journal of MATHEMATICS VAST 8 A Class of Fractional Stochastic Differential Equations Nguyen Tien Dung Department of Mathematics, Vietnam National University,
More informationOPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS
APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze
More informationPathwise Construction of Stochastic Integrals
Pathwise Construction of Stochastic Integrals Marcel Nutz First version: August 14, 211. This version: June 12, 212. Abstract We propose a method to construct the stochastic integral simultaneously under
More informationThe Poisson Process in Quantum Stochastic Calculus
The Poisson Process in Quantum Stochastic Calculus Shayanthan Pathmanathan Exeter College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity Term 22 The Poisson Process
More informationThe Azéma-Yor Embedding in Non-Singular Diffusions
Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let
More informationVITA of Yaozhong HU. A. List of submitted papers
VITA of Yaozhong HU A. List of submitted papers 1. (with G. Rang) Parameter Estimation For Stochastic Hamiltonian Systems Driven By Fractional Brownian Motions. 2. (with G. Rang) Identification of the
More information(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS
(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS Svetlana Janković and Miljana Jovanović Faculty of Science, Department of Mathematics, University
More informationEstimates for the density of functionals of SDE s with irregular drift
Estimates for the density of functionals of SDE s with irregular drift Arturo KOHATSU-HIGA a, Azmi MAKHLOUF a, a Ritsumeikan University and Japan Science and Technology Agency, Japan Abstract We obtain
More information[Ahmed*, 5(3): March, 2016] ISSN: (I2OR), Publication Impact Factor: 3.785
IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY DENSITIES OF DISTRIBUTIONS OF SOLUTIONS TO DELAY STOCHASTIC DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS INITIAL DATA ( PART II)
More informationITÔ S ONE POINT EXTENSIONS OF MARKOV PROCESSES. Masatoshi Fukushima
ON ITÔ S ONE POINT EXTENSIONS OF MARKOV PROCESSES Masatoshi Fukushima Symposium in Honor of Kiyosi Itô: Stocastic Analysis and Its Impact in Mathematics and Science, IMS, NUS July 10, 2008 1 1. Itô s point
More information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More information(B(t i+1 ) B(t i )) 2
ltcc5.tex Week 5 29 October 213 Ch. V. ITÔ (STOCHASTIC) CALCULUS. WEAK CONVERGENCE. 1. Quadratic Variation. A partition π n of [, t] is a finite set of points t ni such that = t n < t n1
More informationProbability approximation by Clark-Ocone covariance representation
Probability approximation by Clark-Ocone covariance representation Nicolas Privault Giovanni Luca Torrisi October 19, 13 Abstract Based on the Stein method and a general integration by parts framework
More informationDirichlet Forms for Poisson Measures and Lévy Processes: The Lent Particle Method
Dirichlet Forms for Poisson Measures and Lévy Processes: The Lent Particle Method Nicolas BOULEAU and Laurent DENIS Abstract We present a new approach to absolute continuity of laws of Poisson functionals.
More informationOn the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family
More informationMalliavin calculus and central limit theorems
Malliavin calculus and central limit theorems David Nualart Department of Mathematics Kansas University Seminar on Stochastic Processes 2017 University of Virginia March 8-11 2017 David Nualart (Kansas
More informationThe Lévy-Itô decomposition and the Lévy-Khintchine formula in31 themarch dual of 2014 a nuclear 1 space. / 20
The Lévy-Itô decomposition and the Lévy-Khintchine formula in the dual of a nuclear space. Christian Fonseca-Mora School of Mathematics and Statistics, University of Sheffield, UK Talk at "Stochastic Processes
More informationWHITE NOISE APPROACH TO FEYNMAN INTEGRALS. Takeyuki Hida
J. Korean Math. Soc. 38 (21), No. 2, pp. 275 281 WHITE NOISE APPROACH TO FEYNMAN INTEGRALS Takeyuki Hida Abstract. The trajectory of a classical dynamics is detrmined by the least action principle. As
More informationResearch Article Soliton Solutions for the Wick-Type Stochastic KP Equation
Abstract and Applied Analysis Volume 212, Article ID 327682, 9 pages doi:1.1155/212/327682 Research Article Soliton Solutions for the Wick-Type Stochastic KP Equation Y. F. Guo, 1, 2 L. M. Ling, 2 and
More informationEULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS
Qiao, H. Osaka J. Math. 51 (14), 47 66 EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS HUIJIE QIAO (Received May 6, 11, revised May 1, 1) Abstract In this paper we show
More informationFunctional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals
Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico
More informationSome SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen
Title Author(s) Some SDEs with distributional drift Part I : General calculus Flandoli, Franco; Russo, Francesco; Wolf, Jochen Citation Osaka Journal of Mathematics. 4() P.493-P.54 Issue Date 3-6 Text
More informationCHARACTERIZATION THEOREMS FOR DIFFERENTIAL OPERATORS ON WHITE NOISE SPACES
Communications on Stochastic Analysis Vol. 7, No. 1 (013) 1-15 Serials Publications www.serialspublications.com CHARACTERIZATION THEOREMS FOR DIFFERENTIAL OPERATORS ON WHITE NOISE SPACES ABDESSATAR BARHOUMI
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationProperties of delta functions of a class of observables on white noise functionals
J. Math. Anal. Appl. 39 7) 93 9 www.elsevier.com/locate/jmaa Properties of delta functions of a class of observables on white noise functionals aishi Wang School of Mathematics and Information Science,
More informationOptimal investment strategies for an index-linked insurance payment process with stochastic intensity
for an index-linked insurance payment process with stochastic intensity Warsaw School of Economics Division of Probabilistic Methods Probability space ( Ω, F, P ) Filtration F = (F(t)) 0 t T satisfies
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationMalliavin Calculus: Analysis on Gaussian spaces
Malliavin Calculus: Analysis on Gaussian spaces Josef Teichmann ETH Zürich Oxford 2011 Isonormal Gaussian process A Gaussian space is a (complete) probability space together with a Hilbert space of centered
More informationON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WITH MONOTONE DRIFT
Elect. Comm. in Probab. 8 23122 134 ELECTRONIC COMMUNICATIONS in PROBABILITY ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WIT MONOTONE DRIFT BRAIM BOUFOUSSI 1 Cadi Ayyad University FSSM, Department
More informationContents. 1 Preliminaries 3. Martingales
Table of Preface PART I THE FUNDAMENTAL PRINCIPLES page xv 1 Preliminaries 3 2 Martingales 9 2.1 Martingales and examples 9 2.2 Stopping times 12 2.3 The maximum inequality 13 2.4 Doob s inequality 14
More informationJump-type Levy Processes
Jump-type Levy Processes Ernst Eberlein Handbook of Financial Time Series Outline Table of contents Probabilistic Structure of Levy Processes Levy process Levy-Ito decomposition Jump part Probabilistic
More informationConvergence at first and second order of some approximations of stochastic integrals
Convergence at first and second order of some approximations of stochastic integrals Bérard Bergery Blandine, Vallois Pierre IECN, Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes B.P. 239 F-5456
More informationThe Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes1
Journal of Theoretical Probability. Vol. 10, No. 1, 1997 The Equivalence of Ergodicity and Weak Mixing for Infinitely Divisible Processes1 Jan Rosinski2 and Tomasz Zak Received June 20, 1995: revised September
More informationExplicit Representation of Strong Solutions of SDE s driven by Infinite Dimensional Lévy Processes
Dept. of Math./CMA Univ. of Oslo Pure Mathematics No 27 ISSN 86 2439 December 28 Explicit Representation of Strong Solutions of SDE s driven by Infinite Dimensional Lévy Processes Thilo Meyer-Brandis and
More informationMULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES
MULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES D. NUALART Department of Mathematics, University of Kansas Lawrence, KS 6645, USA E-mail: nualart@math.ku.edu S. ORTIZ-LATORRE Departament de Probabilitat,
More informationIntroduction to Infinite Dimensional Stochastic Analysis
Introduction to Infinite Dimensional Stochastic Analysis Mathematics and Its Applications Managing Editor M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 502
More informationAn Itō formula in S via Itō s Regularization
isibang/ms/214/3 January 31st, 214 http://www.isibang.ac.in/ statmath/eprints An Itō formula in S via Itō s Regularization Suprio Bhar Indian Statistical Institute, Bangalore Centre 8th Mile Mysore Road,
More informationOptimal control for an insider
Optimal control for an insider Bernt Øksendal - Department of Mathematics, University of Oslo - Norwegian School of Economics (NHH),Bergen - Stochastics in Environmental and Financial Economics (SEFE),
More informationA REPRESENTATION FOR THE KANTOROVICH RUBINSTEIN DISTANCE DEFINED BY THE CAMERON MARTIN NORM OF A GAUSSIAN MEASURE ON A BANACH SPACE
Theory of Stochastic Processes Vol. 21 (37), no. 2, 2016, pp. 84 90 G. V. RIABOV A REPRESENTATION FOR THE KANTOROVICH RUBINSTEIN DISTANCE DEFINED BY THE CAMERON MARTIN NORM OF A GAUSSIAN MEASURE ON A BANACH
More informationSet-Indexed Processes with Independent Increments
Set-Indexed Processes with Independent Increments R.M. Balan May 13, 2002 Abstract Set-indexed process with independent increments are described by convolution systems ; the construction of such a process
More informationWe denote the space of distributions on Ω by D ( Ω) 2.
Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationApplications of Ito s Formula
CHAPTER 4 Applications of Ito s Formula In this chapter, we discuss several basic theorems in stochastic analysis. Their proofs are good examples of applications of Itô s formula. 1. Lévy s martingale
More informationCONDITIONAL FULL SUPPORT OF GAUSSIAN PROCESSES WITH STATIONARY INCREMENTS
J. Appl. Prob. 48, 561 568 (2011) Printed in England Applied Probability Trust 2011 CONDITIONAL FULL SUPPOT OF GAUSSIAN POCESSES WITH STATIONAY INCEMENTS DAIO GASBAA, University of Helsinki TOMMI SOTTINEN,
More informationUNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.
More informationChange of Measure formula and the Hellinger Distance of two Lévy Processes
Change of Measure formula and the Hellinger Distance of two Lévy Processes Erika Hausenblas University of Salzburg, Austria Change of Measure formula and the Hellinger Distance of two Lévy Processes p.1
More informationGENERALIZED COVARIATION FOR BANACH SPACE VALUED PROCESSES, ITÔ FORMULA AND APPLICATIONS
Di Girolami, C. and Russo, F. Osaka J. Math. 51 (214), 729 783 GENERALIZED COVARIATION FOR BANACH SPACE VALUED PROCESSES, ITÔ FORMULA AND APPLICATIONS CRISTINA DI GIROLAMI and FRANCESCO RUSSO (Received
More informationOn Optimal Stopping Problems with Power Function of Lévy Processes
On Optimal Stopping Problems with Power Function of Lévy Processes Budhi Arta Surya Department of Mathematics University of Utrecht 31 August 2006 This talk is based on the joint paper with A.E. Kyprianou:
More informationA Short Introduction to Diffusion Processes and Ito Calculus
A Short Introduction to Diffusion Processes and Ito Calculus Cédric Archambeau University College, London Center for Computational Statistics and Machine Learning c.archambeau@cs.ucl.ac.uk January 24,
More informationDensities for the Navier Stokes equations with noise
Densities for the Navier Stokes equations with noise Marco Romito Università di Pisa Universitat de Barcelona March 25, 2015 Summary 1 Introduction & motivations 2 Malliavin calculus 3 Besov bounds 4 Other
More informationA BAYES FORMULA FOR NON-LINEAR FILTERING WITH GAUSSIAN AND COX NOISE
A BAYES FOMULA FO NON-LINEA FILTEING WITH GAUSSIAN AND COX NOISE V. MANDEKA, T. MEYE-BANDIS, AND F. POSKE Abstract. A Bayes type formula is derived for the non-linear filter where the observation contains
More informationA New Wavelet-based Expansion of a Random Process
Columbia International Publishing Journal of Applied Mathematics and Statistics doi:10.7726/jams.2016.1011 esearch Article A New Wavelet-based Expansion of a andom Process Ievgen Turchyn 1* eceived: 28
More informationStochastic Processes. Winter Term Paolo Di Tella Technische Universität Dresden Institut für Stochastik
Stochastic Processes Winter Term 2016-2017 Paolo Di Tella Technische Universität Dresden Institut für Stochastik Contents 1 Preliminaries 5 1.1 Uniform integrability.............................. 5 1.2
More informationThe Cameron-Martin-Girsanov (CMG) Theorem
The Cameron-Martin-Girsanov (CMG) Theorem There are many versions of the CMG Theorem. In some sense, there are many CMG Theorems. The first version appeared in ] in 944. Here we present a standard version,
More information