Forward Integration
Jorge A. León · 2025
In this chapter, it is shown that the forward integral can be considered as an anticipating integral when the integrator is a semimartingale. That is, it is an extension of Itˆo&s;s stochastic integral that allows us to deal with processes that are not adapted to the underlying filtration as integrands. The forward integral is defined as a limit in probability. So, it is difficult proven that, under suitable conditions of Malliavin calculus, the forward integral is equal to the divergence operator plus an extra term that may depend on the derivative operator. In this way, it is possible to estimate this integral. Some properties of the forward integral are analyzed as the integration by parts formulas and Fubini&s;s type theorems. Special attention is given to the case that the integrator is Brownian motion, or fractional Brownian motion with parameter bigger than one half through the techniques of Malliavin calculus. Introduction The forward integral with respect to Brownian motion is a tool of stochastic analysis that allows us to either deal with problems that apparently should be analyzed trough Itˆo&s;s calculus, or improve the theory of the classical stochastic calculus as it is shown in Chapter 7 . In general, this integral is introduced as a limit in probability. So, in this chapter, it is applied the techniques of Malliavin calculus based on the derivative and divergence operator to establish suitable estimations for the Lp (Ω)-norm of this integral. Some properties of the forward integral are analyzed as the integration by parts formulas and Fubini&s;s type theorems. Special attention is given to the case that the integrator is Brownian motion, or fractional Brownian motion with parameter bigger than one half through the techniques of Malliavin calculus.