Gradient and Divergence Operators

In this chapter we construct an abstract framework for stochastic analysis in continuous time with respect to a normal martingale (M_t)_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 applications, such as logarithmic Sobolev and deviation inequalities, are formulated in this general setting. In the next chapters we will examine concrete examples of operators that can be included in this framework, in particular when (M_t)_t∈R₊ is a Brownian motion or a compensated Poisson process.