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Obituarywas created and the relationships between these integrals andStratonowich integrals were studied (see, for example, [13–17]). In the case of the Skorokhod integral with an indefiniteupper limit, the quadratic variation remains the same as in theItô case and some interesting relations with martingales canbe obtained [18–20, 32]. One of the main properties of theItô integral is its locality. Namely, if random functions x 1 , x 2on [0; 1] are integrable with respect to the Wiener process wthen 1x 0 1dw − 1x 0 2dw 1 {x1 =x 2 } = 0. This property is veryimportant. For example, it allows the use of the stopping techniquein the consideration of stochastic differential equations.The corresponding property of the Skorokhod integral wasestablished in two different situations. In an initial Skorokhodarticle, the following statement was proved: if the random elementx has a stochastic derivative then x lies in the domain ofI. For stochastically differentiable random functions the Skorokhodintegral has a locality property. This was proved in[19].The other approach was proposed in [5, 2]. Define thesmooth open subset Δ of probability space as Δ = {ω :α(ω) > 0}, where α is a stochastically differentiable randomvariable. Then it can be proved that for x 1 , x 2 from the domainof I (not necessarily stochastically differentiable) the equality(x 1 − x 2 )1 Δ = 0 implies(I(x 1 ) − I(x 2 ))1 Δ = 0. (4)The statements mentioned above give us the possibility ofdefining the Skorokhod integral for the random functions thatdo not have finite moments. Recent results on the descriptionof subsets of probability space that have a locality property (4)can be found in [21]. Stochastic equations with non-Itô integralshave been actively studied since the 1970s. As an example,the boundary value problems or integral equations of thesecond kind can be considered. Such equations were treatedinitially in the article [17, 1] using the algebraic definition ofthe Skorokhod integral. One of the interesting cases where anticipationarises is the Cauchy problem for ordinary stochasticdifferential equations with the initial condition which dependson the future noise. The interesting result was obtainedin [29]. A linear one-dimensional equation was considered⎧⎪⎨ dx(t) = a(t)x(t)dt + b(t)x(t)dw(t)(5)⎪⎩ x(0) = α.Here α is a functional of w and the equation is treated in thesense of the Skorokhod integral. It was proved in [29] that thesolution has the formx(t) = T t α ·E t 0 , (6)where T t α is a transformation of α corresponding to thechange w(·) → w(·) + t∧·b(s)ds, and E t 00is a usual stochasticexponent (related to the case x(0) = 1). The reason forthe appearance of T t α in the solution can easily be seen ifwe recall the definition of the Skorokhod integral as a logarithmicderivative. It can be verified that I has a structureof the infinite-dimensional divergence operator. So (5) canbe treated as a first order differential equation in infinitedimensionalspace. Then (6) becomes the solution obtainedvia the characteristic method. This approach was introducedin [7]. This point of view on equation (6) explains the absenceof the solution in a general non-linear case and the absenceof the good form of a solution in the linear vector case.The last case with the commuting matrix-valued b was consideredin [30]. Surprisingly, the algebraic definition of theSkorokhod integral turns out to be very helpful in the considerationof (6). The following representation of the solutionwas obtained in [8] x(t) = ∞ (−1) kk=0 k!0 Dk α(s 1 ,...,s k )··E t s kb(s k )E s ks k−1b(s k−1 ) ...E s 10 ds 1 ...ds k . In the 1990s a new reasonfor studying the extended stochastic integral arose. Thiswas the development of the models from financial mathematics.In these models, fractional Brownian motion plays therole of the noise process [31]. So, integration with respect toit must be constructed. One of the possible approaches leadsto the extended stochastic integral with respect to Gaussianintegrators [9]. The Gaussian process η on [0; 1] is called theintegrator if there exists such c > 0 that for an arbitrary partition0 = t 0 < t 1 < ... < t n = 1 and real numbers a 1 ,...,a n ,we have t0... ktE n−1a k (η(t k+1 ) − η(t k )) 2 n−1≤ C a 2 k (t k+1 − t k ). (7)k=0In general setup, introduced by A. V. Skorokhod, the noisegenerated by the integrator η can be related to the Gaussiangeneralised element in H of the form Aξ with the certain continuouslinear operator A. Now, the definition of the integralwith respect to η can be I(A ∗ x), where I is the original Skorokhodintegral. This construction is a particular case of theaction of the random map on the random elements proposedin [5]. The corresponding Itô formula for the Skorokhod integralwith respect to the Gaussian integrator was proved in[9]. Note that the integrator does not necessarily have thesemimartingale property. Hence, the related stochastic calculusis purely non-Itô calculus. In various mathematical modelswhere the anticipation arises, the type of construction of thestochastic integral is motivated by external reasons. For example,in the stochastic boundary value problem, physicistsprefer the symmetric stochastic integral [33]. But the sameproblem can be treated with the Skorokhod integral (with adifferent solution of course) [1]. In spite of this, there existmathematical problems where the extended stochastic integralarises naturally by necessity. These are the problems offiltration theory. The main property of the Skorokhod integralhere is the followingΓ(A) T0x(t)dw(t) = T0k=0Γ(A)x(t)dγ(t), (8)where Γ(A) is the operator of the second quantisation in thespace of Wiener functionals and the differential dγ is the Skorokhoddifferential with respect to the integrator γ =Γ(A)w.As is well-known, Γ(A) can be an operator of conditional expectationin the case when A is a projector [32]. So the relation(8) can be used for deriving the anticipating equationfor optimal filter in the nonsemimartingale case (see [8, 10]).The last equation is a partial stochastic differential equationwith anticipation. The properties of the solutions (existence,smoothness, large deviations) are studied in [13].The ideas mentioned above and facts about the Skorokhodintegral reflect the main steps in the development of this notion.More details can be found in the references. Certainly,the list of references is far from complete but the main ideashave been covered and the cited references can be used to30 EMS Newsletter December 2014
Obituarysupport the study of the rich and beautiful object that is theSkorokhod integral.Bibliography[1] A.A. Dorogovtsev. Boundary problem for the equationswith stochastic differential operators. Theory Probab. Math.Statist., 40(11):23–28, 1989.[2] A.A. Dorogovtsev. Stochastic calculus with anticipating integrands.Ukrainian Math. J., 41(11):1460–1466, 1989.[3] A.A. Dorogovtsev. Stochastic integrals with respect to Gaussianrandom measures. Theory Probab. Math. Statist., 44:53–59, 1992.[4] A.A. Dorogovtsev. One property of the trajectories of the extendedstochastic integrals. Siberian Math. J., 34(5):38–42,1993.[5] A.A. Dorogovtsev. Stochastic Analysis and Random Maps inHilbert Space. VSP, Utrecht, The Netherlands, Tokyo, Japan,1994.[6] A.A. Dorogovtsev. One approach to the non-Gaussianstochastic calculus. J. Appl. Math, and Stoch. Anal., 8(4):361–370, 1995.[7] A.A.Dorogovtsev. Anticipating stochastic equations. Proceedingsof the Institute of Mathematics of the NationalAcademy of Sciences of the Ukraine, 15. Institut Matematiki,Kiev, 1996.[8] A.A. Dorogovtsev. Anticipating equations and filtration problem.Theory Stoch. Proc., 3 (19)(1–2):154–163, 1997.[9] A.A. Dorogovtsev. Stochastic integration and one class Gaussianstochastic processes. Ukrainian Math. J., 50(4):495–505,1998.[10] A.A. Dorogovtsev. Smoothing problem in anticipating scenario.Ukrainian Math. J., 57(9):1218–1234, 2005.[11] A.A. Dorogovtsev. Smoothing problem in anticipating scenario.Ukrainian Math. J., 57(10):1327–1333, 2006.[12] A. Benassi. Calcul stochastique anticipatif: Vartingales hierarchiques.C.R. Acad. Sei., Ser. l, Paris, 311(7):457–460, 1990.[13] A. Millet, D. Nualart, and M. Sanz-Sole. Composition oflarge deviation principles and applications. Ann. Probab.,20(4):1902–1931, 1992.[14] A.S. Ustunel and M. Zakai. Transformation of Measureon Wiener Space. Springer-Verlag, Berlin, Heidelberg, NewYork, 2000.[15] A.V. Skorokhod. Integration in Hilbert Space. Ergebnisseder Mathematik und ihrer Grenzgebiete, Band 79. Springer-Verlag. New York; Heidelberg, 1974.[16] A.V.Skorokhod. One generalization of the stochastic integral.Theory Probab. Appl., 20(2):223–237, 1975.[17] A.Yu. Shevliakov. Stochastic calculus with anticipating integrands.Theory Probab. Math. Statist., 22(11):163–174, 1981.[18] C.A. Tudor. Stochastic calculus with anticipating integrands.Bernoulli, 10(2):313–325, 2004.[19] E.Pardoux and D.Nualart. Stochastic calculus with anticipatingintegrands. Probab. Theory Related Fields, 78:535–581,1988.[20] E. Pardoux and P. Protter. A two-sided stochastic integral andits calculus. Probab. Theory Related Fields, 78:15–19, 1987.[21] A.M. Gomilko and A.A. Dorogovtsev. Localization ofthe extended stochastic integral. Sbornik: Mathematics,197(9):1273–1295, 2006.[22] M.Hitsuda. Formula for Brownian partial derivatives. InThe Second Japan-USSR Symp. on Probab. Theory, Tbilisi;Springer-Verlag, Berlin, New York, pages 111–114, 1972.[23] M. Jolis and M. Sanz-Sole. Integrator properties of the Skorokhodintegral. Stoch. and Stoch. Reports, 41(3):163–176,1992.[24] N.N. Norin. Extended stochastic integral for non-Gaussianmeasures in the locally-convex space. Russian Math.Surveys,41(3):199–200, 1986.[25] D. Nualart. The Malliavin calculus and related topics.Springer-Verlag, New York, 1995.[26] O. Enchev. Stochastic integration with respect to Gaussianrandom measures. In Ph.D. Thesis, Sofia Univ., Sofia, pages52–60, 1983.[27] Shigeyoshi Ogawa. Quelques propriétés de l’intégralestochastique du type noncausal. Japan J. Appl. Math.,1(2):405–416, 1984.[28] O.G. Smolyanov. Differentiable measures on the group offunctions taking values in a compact Lie group. In Abstract ofthe Sixth Intern. Vilnius Conf. on Probab. and Math. Statist.,Vilnius, pages 139–140, 1993.[29] R. Buckhdan. Quasilinear partial stochastic differential equationswith out nonanticipation requirement. Prepr. HumboltUniv., No. 176, Berlin, 1989.[30] R. Buckhdan, P. Malliavin, and D. Nualart. Multidimensionallinear stochastic differential equations in the Skorokhod sense.Stoch. and Stoch. Reports, 62(1–2):117–145, 1997.[31] S. Tindel, C.A. Tudor, and F. Viens. Stochastic evolution equationswith fractional Brownian motion. Probab. Theory RelatedFields, 127(2):186–204, 2003.[32] B. Symon. The P(ϕ) 2 Euclidean (quantum) field theory.Princeton Univ. Press, 1974.[33] V.I. Klyackin. Dynamics of stochastic systems. Phizmathlit,Moscow, 2002.[34] V.V. Baklan. One generalization of stochastic integral. DopovidiAN Ukraine, Ser. A, 41(4):291–294, 1976.[35] S. Watanabe. Stochastic differential equations and Malliavincalculus. Tata Inst, of Pundam. Research, Bombay, 1984.[36] Yu.L. Dalecky and G.Ya. Sohadze. Absolute continuity ofsmooth measures. Funct. Anal, and Appl., 22(2):77–78, 1988.[37] Yu.L. Dalecky and S.N. Paramonova. One formula from Gaussianmeasures theory and estimation of stochastic integrals.Theory Probab. Appl., 19(4):845–849, 1974.[38] Yu.L. Dalecky and S.V. Fomin. Measures and differentialequations in infinite-dimensional space. Kluwer Acad. Publ.Boston, 1983.[39] Yu.L. Dalecky and V.R. Steblovskaya. Smooth measures: absolutecontinuity, stochastic integrals, variational problems. InProc. of the Sixth USSR–Japan Symp. on Probab. Theory andMath. Statist., Kiev – WSPC, pages 52–60, 1991.Professor V. V. Buldygin (5 Nov 1946 – 17 Apr2012) was Head of the Mathematical Analysis andProbability Theory Department at the NationalTechnical University of Ukraine “Kiev PolytechnicInstitute”.A. A. Dorogovtsev [adoro@imath.kiev.ua] headsthe Department of the Theory of Stochastic Processesat the Institute of Mathematics, NationalAcademy of Sciences of Ukraine, Kiev.M. I. Portenko [portenko@imath.kiev.ua] is aleading researcher of the Department of the Theoryof Stochastic Processes at the Institute ofMathematics, National Academy of Sciences ofUkraine, Kiev, and an associate member of theNational Academy of Sciences of Ukraine.Irina Kadyrova [kadyrova@math.msu.edu] isteaching specialist at the Department of Mathematics,Michigan State University, East Lansing,USA.EMS Newsletter December 2014 31
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