Skorokhod integral

An extension of the Itô stochastic integral (cf. Stochastic integral) introduced by A.V. Skorokhod in [a8] in order to integrate stochastic processes that are not adapted to Brownian motion. Suppose that $ W = \{ {W _ {t} } : {t \in [ 0,1 ] } \} $ is a Brownian motion and consider a stochastic process $ u = \{ {u _ {t} } : {t \in [ 0,1 ] } \} $, measurable with respect to $ W $, which is not necessarily adapted (cf. also Optional random process) and satisfies $ {\mathsf E} \int _ {0} ^ {1} {u _ {t} ^ {2} } {dt } < \infty $. The process $ u $ can be developed into a sum of orthogonal multiple stochastic integrals $ u _ {t} = \sum _ {n = 0 } ^ \infty I _ {n} ( f _ {n} ( \cdot,t ) ) $, where $ f _ {n} \in L _ {2} ( [ 0,1 ] ^ {n + 1 } ) $ is symmetric in the first $ n $ variables (see [a2]). The Skorokhod integral of the process $ u $, denoted by $ \delta ( u ) = \int _ {0} ^ {1} {u _ {t} } {dW _ {t} } $, is defined by

$$ \delta ( u ) = \sum _ {n = 0 } ^ \infty I _ {n + 1 } ( {\widetilde{f} } _ {n} ) , $$

provided the above series converges in $ L _ {2} ( \Omega ) $. Here, $ {\widetilde{f} } _ {n} $ denotes the symmetrization of $ f _ {n} $ in all its variables.

In [a1] it is proved that the Skorokhod integral coincides with the adjoint of the derivative operator on the Wiener space (cf. also Wiener space, abstract). Starting from this result, the techniques of stochastic calculus of variations on the Wiener space (see [a4]) have made it possible to develop a stochastic calculus for the Skorokhod integral (see [a6]) which extends the classical Itô calculus introduced in the 1940s. The Skorokhod integral possesses most of the main properties of the Itô stochastic integral. For instance, under suitable hypotheses on the integrand $ u $, the Skorokhod integral is local, the indefinite Skorokhod integral $ \int _ {0} ^ {t} {u _ {s} } {dW _ {s} } $ is continuous and possesses a quadratic variation equal to $ \int _ {0} ^ {t} {u _ {s} ^ {2} } {ds } $, and a change-of-variables formula holds for $ F ( \int _ {0} ^ {t} {u _ {s} } {dW _ {s} } ) $( see [a6]). Multiple Skorokhod integrals are defined in [a7], and the Skorokhod integral is studied in [a3] from the point of view of the white noise analysis.

References

[a1]	B. Gaveau, P. Trauber, "L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel" J. Funct. Anal. , 46 (1982) pp. 230–238
[a2]	K. Itô, "Multiple Wiener integral" J. Math. Soc. Japan , 3 (1951) pp. 157–169
[a3]	H.H. Kuo, A. Russek, "White noise approach to stochastic integration" J. Multivariate Analysis , 24 (1988) pp. 218–236
[a4]	P. Malliavin, "Stochastic calculus of variations and hypoelliptic operators" , Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto 1976 , Wiley (1978) pp. 195–263
[a5]	D. Nualart, "The Malliavin calculus and related topics" , Springer (1995)
[a6]	D. Nualart, E. Pardoux, "Stochastic calculus with anticipating integrands" Probab. Th. Rel. Fields , 78 (1988) pp. 535–581
[a7]	D. Nualart, M. Zakai, "Generalized multiple stochastic integrals and the representation of Wiener functionals" Stochastics , 23 (1988) pp. 311–330
[a8]	A.V. Skorokhod, "On a generalization of a stochastic integral" Th. Probab. Appl. , 20 (1975) pp. 219–233