Anticipating stochastic calculus
As we have seen in Chapter 2, the Skorohod integral is an extension of the Itô integral that allows us to integrate stochastic processes that are not necessarily adapted to the Brownian motion. The adaptability assumption is replaced by some regularity condition. It is possible to develop a stochastic calculus for the Skorohod integral which is similar in some aspects to the classical Itô calculus. In this chapter we present the fundamental facts about this stochastic calculus, and we also discuss other approaches to the problem of constructing stochastic integrals for nonadapted processes (approximation by Riemann sums, development in a basis of L2 ([0,1]), substitution methods). The last section discusses noncausal stochastic differential equations formulated using anticipating stochastic integrals.
KeywordsBrownian Motion Stochastic Differential Equation Continuous Version Stochastic Integral Stochastic Calculus
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