formula One of the most useful results from Malliavin calculus is the Clark–Ocone theorem, which allows the process in the martingale representation theorem to be identified explicitly. A simplified version of this theorem is as follows: For satisfying which is Lipschitz and such that F has a strong derivative kernel, in the sense that for in C[0,1] then where H is the previsible projection of F'(x, (t,1]) which may be viewed as the derivative of the function F with respect to a suitable parallel shift of the process X over the portion (t,1] of its domain. This may be more concisely expressed by Much of the work in the formal development of the Malliavin calculus involves extending this result to the largest possible class of functionals F by replacing the derivative kernel used above by the "Malliavin derivative" denoted in the above statement of the result. Skorokhod integral The Skorokhod integral operator which is conventionally denoted δ is defined as the adjoint of the Malliavin derivative thus for u in the domain of the operator which is a subset of , for F in the domain of the Malliavin derivative, we require where the inner product is that on viz The existence of this adjoint follows from the Riesz representation theorem for linear operators on Hilbert spaces. It can be shown that if u is adapted then where the integral is to be understood in the Itô sense. Thus this provides a method of extending the Itô integral to non adapted integrands. Applications The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications for example in stochastic filtering. References Kusuoka, S. and Stroock, D. (1981) "Applications of Malliavin Calculus I", Stochastic Analysis, Proceedings Taniguchi International Symposium Katata and Kyoto 1982, pp 271–306 Kusuoka, S. and Stroock, D. (1985) "Applications of Malliavin Calculus II", J. Faculty Sci. Uni. Tokyo Sect. 1A Math., 32 pp 1–76 Kusuoka, S. and Stroock, D. (1987) "Applications of Malliavin Calculus III", J. Faculty Sci. Univ. Tokyo Sect. 1A Math., 34 pp 391–442 Malliavin, Paul and Thalmaier, Anton. Stochastic Calculus of Variations in Mathematical Finance, Springer 2005, Bell, Denis. (2007) The Malliavin Calculus, Dover. ; ebook Sanz-Solé, Marta (2005) Malliavin Calculus, with applications to stochastic partial differential equations. EPFL Press, distributed by CRC Press, Taylor & Francis Group. Schiller, Alex