Gradient and Divergence Operators

Abstract

In this chapter we construct an abstract framework for stochastic analysis in continuous time with respect to a normal martingale (Mt)t?R+, using the construction of stochastic calculus presented in Section 2. In particular we identify some minimal properties that should be satisfied in order to connect a gradient and a divergence operator to stochastic integration, and to apply them to the predictable representation of random variables. Some applica- tions, such as logarithmic Sobolev and deviation inequalities, are formulated in this general setting. In the next chapters we will examine concrete exam- ples of operators that can be included in this framework, in particular when (Mt)t?R+ is a Brownian motion or a compensated Poisson process